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Springer Monographs in Mathematics

A. Bonfiglioli • E. Lanconelli • F. Uguzzoni

Stratified Lie Groupsand Potential Theoryfor their Sub-Laplacians

A. Bonfiglioli F. UguzzoniUniversità Bologna, Dip.to Matematica Università Bologna, Dip.to MatematicaPiazza di Porta San Donato 5 Piazza di Porta San Donato 540126 Bologna, Italy 40126 Bologna, Italye-mail: [email protected] e-mail: [email protected]

E. LanconelliUniversità Bologna, Dip.to MatematicaPiazza di Porta San Donato 540126 Bologna, Italye-mail: [email protected]

Library of Congress Control Number: 2007929114

Mathematics Subject Classification (2000): 43A80, 35J70, 35H20, 35A08, 31C05, 31C15,35B50, 22E60

ISSN 1439-7382

ISBN-10 3-540-71896-6 Springer Berlin Heidelberg New YorkISBN-13 978-3-540-71896-3 Springer Berlin Heidelberg New York

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To Professor Bruno Pini

and to our Families

Preface

With this book we aim to present an introduction to the stratified Lie groups and totheir Lie algebras of the left-invariant vector fields, starting from basic and elemen-tary facts from linear algebra and differential calculus for functions of several realvariables. The second aim of this book is to perform a potential theory analysis ofthe sub-Laplacian operators

L =m∑

j=1

X2j ,

where the Xj ’s are vector fields, i.e. linear first order partial differential operators,generating the Lie algebra of a stratified Lie group.

In recent years, these operators have received considerable attention in litera-ture, mainly due to their basic rôle in the theory of subelliptic second order partialdifferential equations with semidefinite characteristic form.

1. Some Historical Overviews

General second order partial differential equations with non-negative and degeneratecharacteristic form have appeared in literature since the early 1900s. They were firststudied by M. Picone, who called them elliptic-parabolic equations and proved thecelebrated weak maximum principle for their solutions [Pic13,Pic27].

The interest in this type of equations in application fields was originally foundby A.D. Fokker, M. Planck and A.N. Kolmogorov. They discovered that partial dif-ferential equations with non-negative characteristic form arise in the mathematicalmodeling of theoretical physics and of diffusion processes [Fok14,Pla17,Kol34].

Since then, over the past half-century, this type of equations appeared in manyother different research fields, both theoretical and applied, including geometric the-ory of several complex variables, Cauchy–Riemann geometry, partial differentialequations, calculus of variations, quasiconformal mappings, minimal surfaces andconvexity in sub-Riemannian settings, Brownian motion, kinetic theory of gases,

VIII Preface

mathematical models in finance and in human vision. We report a short list of refer-ences for these topics at the end of this preface.

A first systematic study of boundary value problems for wide classes of elliptic-parabolic operators was performed by G. Fichera. In 1956 [Fic56a,Fic56b], he provedexistence theorems of weak solutions of the “Dirichlet problem” and found the rightsubset of the boundary on which the data have to be prescribed.

Some years later, several existence and regularity results for elliptic-parabolicoperators were proved by O.A. Oleınik and E.V. Radkevic and by J.J. Kohn andL. Nirenberg (see the monograph [OR73] for a presentation and a wide survey onthis subject). The methods used by these authors required particular assumptionson the Fichera boundary set and led to regularity results strongly depending on theregularity of the boundary data.

1.1. L. Hörmander’s Theorem

The investigations of the local regularity properties of the solutions to elliptic-parabolic equations, that is, regularity properties only depending on the given op-erator, have produced more interesting results. The most beautiful ones have beenobtained for elliptic-parabolic equations with underlying algebraic-geometric struc-tures of sub-Riemannian type. The milestone of these research field is a celebratedtheorem of L. Hörmander proved in 1967.

Theorem 1 (L. Hörmander, [Hor67]). Let X1, . . . , Xm and Y be smooth vectorsfields, i.e. linear first order partial differential operators with smooth coefficients inthe open set Ω ⊆ R

N . Suppose

rank(Lie{X1, . . . , Xm, Y }(x)

) = N ∀ x ∈ Ω. (P.1)

Then the operator

L =m∑

j=1

X2j + Y (P.2)

is hypoelliptic in Ω , i.e. every distributional solution to Lu = f is of class C∞whenever f is of class C∞.

Condition (P.1) simply means that at any point of Ω one can find N linearlyindependent differential operators among X1, . . . , Xm, Y and all their commutators(the Lie algebra generated by {X1, . . . , Xm, Y }).

Hörmander’s work opened up a research field, the most remarkable contribu-tions to which have been given by G.B. Folland, L.P. Rothschild and E.M. Stein.They developed and applied to (P.2) the singular integral theory in nilpotent Liegroups.1

1 The application of this theory also occurs in the developments started from the works byJ.J. Kohn on the ∂-Neumann problem and the ∂b complex.

1. Some Historical Overviews IX

By using these techniques, in 1975, G.B. Folland accomplished a functionalanalytic study of sub-Laplacians on stratified Lie groups [Fol75]. One year laterL.P. Rothschild and E.M. Stein proved their celebrated lifting theorem (see [RS76]),enlightening the basic rôle played by the sub-Laplacians in the theory of second or-der partial differential equations which are sum of squares of vector fields. In forceof this theorem, indeed, we can roughly say that:

Every operator L = ∑mj=1 X2

j satisfying the Hörmander rank condition

(P.1) can be lifted to an operator L “as close as we want” to a sub-Laplacian.

1.2. The Rank Condition

The geometrical meaning of the rank condition (P.1) is clarified by the C. Carathéod-ory, W.L. Chow and P.K. Rashevsky theorem:

If (P.1) is satisfied, then given two points x, y ∈ Ω , sufficiently close, there existsa piecewise smooth curve, contained in Ω and connecting x and y, which is the sumof integral trajectories of the vector fields ±X1, . . . ,±Xm,±Y .

The appearance of (P.1) in Hörmander’s theorem seems to be suggested by somedeep properties of the Kolomogorov operators (see also the Introduction in [Hor67]),which we now aim to discuss.

In studying diffusion phenomena from a probabilistic point of view, A.N. Kol-mogorov showed that the probability density of a system with 2n degrees of freedomsatisfies an equation with non-negative characteristic form

Ku = 0 in R2n × R,

where R2n is the phase-space of the system. A prototype for K is the following

operator

K =n∑

j=1

∂2xj

+n∑

j=1

xj ∂yj− ∂t , (P.3)

where x = (x1, . . . , xn) and y = (y1, . . . , yn) denote the velocity and the positionvectors of the system, respectively. The operator K is “very degenerate”: its secondorder part only contains derivatives with respect to the variables x1, . . . , xn. Never-theless, as Kolmogorov showed, it has a fundamental solution Γ which is smooth outof its pole. This implies that K is hypoelliptic, that is, every distributional solutionto Ku = f is of class C∞ whenever f is of class C∞. The explicit expression of Γ

is given by

Γ (z, t; ζ, τ ) = γ(ζ − E(t − τ)z, t − τ

), z = (x, y), ζ = (ξ, η), (P.4)

where γ (z, t) = 0 if t ≤ 0, and

γ (z, t) = (4π)n√det C(t)

exp

(−1

4

⟨C−1z, z

⟩)if t > 0. (P.5)

X Preface

Here, 〈·, ·〉 stands for the usual inner product in R2n; E(t) and C(t), respectively,

denote the 2n × 2n matrices

E(t) = exp

(−t

(0 0In 0

)), C(t) =

∫ t

0E(s)AE(s)T ds.

Moreover, In denotes the identity matrix of order n and A = (In 00 0

). We explicitly

remark thatC(t) > 0 for every t > 0. (P.6)

This condition makes expression (P.5) meaningful and can be restated in geometrical–differential terms. Indeed, denoting

Xj = ∂xjand Y = ∑n

k=1 xk ∂yk− ∂t ,

it can be proved that (P.6) is equivalent to the following rank condition:

rank(Lie{X1, . . . , Xn, Y }(z, t)) = 2n + 1 ∀ (z, t) ∈ R

2n+1. (P.7)

It is also worthwhile to note that the Kolmogorov operator K can be written as

K =n∑

j=1

X2j + Y. (P.8)

1.3. The Left Translation and Dilation Invariance

The structure (P.4) of Kolmogorov’s fundamental solution suggests the relevance thata Lie group theoretical approach has in the analysis of Hörmander operators. Indeed,from the explicit expression of Γ one realizes that

Γ (z, t; ζ, τ ) = γ((ζ, τ )−1 ◦ (z, t)

),

where ◦ is the following composition law making K := (R2n × R, ◦) a non-commutative Lie group

(z, t) ◦ (z′, t ′) := (z′ + E(t ′) z, t + t ′

),

i.e. more explicitly,

(x, y, t) ◦ (x′, y′, t ′) = (x + x′, y + y′ + t ′ x, t + t ′

).

In K one has (ζ, τ )−1 = (−E(−t) ξ,−τ). It is easy to check that K is invariantw.r.t. the left translations on K and commutes with the following dilations:

dλ(z, t) := (λx, λ3y, λ2t), λ > 0.

For every λ > 0, dλ is an automorphism of K, so that (R2n × R, ◦, dλ) is a homoge-neous Lie group. It can be seen that its Lie algebra is the one generated by the vectorfields Xj = ∂xj

and Y = ∑nk=1 xk ∂yk

− ∂t appearing in (P.8).

1. Some Historical Overviews XI

1.4. The Elliptic Counterpart: Stratified Groups and Sub-Laplacians

For a proper comprehension and appreciation of this type of “parabolic”-type opera-tors such as the above Kolmogorov operator K , it is crucial to possess a deep knowl-edge of their “elliptic” counterpart. This seems unavoidable, also bearing in mind thatthe underlying algebraic–geometric structures of these two different classes of oper-ators are almost identical. Let us go back again, for a moment, to the Kolmogorovoperator (P.3). If in that operator we square the term Y = ∑n

j=1 xj ∂xj+n− ∂t , we

obtain the following “sum of square”-operator (which we may refer to as the “ellipticcounterpart” of K):

L :=n∑

j=1

∂2xj

+(

n∑

j=1

xj ∂xj+n− ∂t

)2

. (P.9)

The characteristic form of L is a non-negative quadratic form with non-trivial kernel.Then L has to be considered as a degenerate elliptic operator. However, it is hypoel-liptic: the Hörmander rank condition (P.7) does not distinguish between L and K!Moreover, L is left-invariant on (R2n × R, ◦) (as we already know, so are the ∂xj

’sand Y ) but, this time, it commutes with the dilations

δλ(z, t) = (λx, λ2y, λ2t), λ > 0.

Also these dilations are automorphisms of (R2n × R, ◦), and G := (R2n × R, ◦, δλ)

becomes a stratified Lie group whose generators are the vector fields ∂xj’s and Y .

Then, according to our general agreement, L is a sub-Laplacian2 on G.

1.5. The Heisenberg Group

In the lower-dimensional case n = 1, the operator (P.9) is

∂2x + (x ∂y − ∂t )

2, (x, y, t) ∈ R3. (P.10)

Up to a change and a relabeling of the variables, this can be written as follows:

(∂2x + 2 y ∂t )

2 + (∂y − 2 x ∂t )2, (x, y, t) ∈ R

3,

which, in turn, is the lower-dimensional version of the celebrated sub-Laplacian onthe Heisenberg group.

The Heisenberg group Hn is the stratified Lie group (R2n+1, ◦) whose composi-

tion law is given by

(z, t) ◦ (z′, t ′) = (z + z′, t + t + 2 Im〈z, z′〉). (P.11)

Here we identify R2n with C

n, and we use the notation

2 All these notions will be properly introduced in Chapter 1.

XII Preface

(z, t) = (z1, . . . , zn, t) = (x1, y1, . . . , xn, yn, t)

for the points of Hn. In (P.11), 〈·, ·〉 stands for the usual Hermitian inner product

in Cn. The dilation

δλ(z, t) = (λz, λ2t) (P.12)

is an automorphism of Hn and the vector fields

Xj = ∂xj+ 2 yj ∂t , Yj = ∂yj

− 2 xj ∂t

are left-invariant on (Hn, ◦). One readily recognizes that the following commutationrelations hold:

[Xj , Yj ] = −4 ∂t (P.13a)

and

[Xj ,Xk] = [Yj , Yk] = [Xj , Yk] = 0 ∀ j = k. (P.13b)

Identity (P.13a) is the canonical commutation relation between momentum and po-sition in quantum mechanics. From (P.13a) it follows that

rank(Lie

{X1, . . . , Xn, Y1, . . . , Yn, ∂t

}) = 2n + 1

at any point of R2n+1. Then, by Hörmander’s Theorem 1, the sub-Laplacian on H

n

ΔHn :=n∑

j=1

(X2j + Y 2

j )

is hypoelliptic. The Heisenberg group and its Lie algebra originally arose in themathematical formalizations of quantum mechanics (see H. Weyl [Weyl31]). Today,they appear in many research fields such as several complex variables, CR geometry,Fourier analysis and partial differential equations of subelliptic type. The Heisenbergsub-Laplacian is undoubtedly the most important prototype of the sub-Laplacians onnon-commutative stratified Lie groups.

1.4. The Lifting Theorem

Obviously, a generic Hörmander operator sum of squares of vector fields is not, ingeneral, a sub-Laplacian on some stratified Lie group. Just consider, as an example,

M = ∂2x + (x ∂y)

2 in R2.

This operator satisfies the Hörmander rank condition, hence it is hypoelliptic. How-ever, there is no Lie group structure in R

2 making M left-invariant on it. Neverthe-less, adding the new variable t , M can be lifted to the operator in (P.10) which isthe sub-Laplacian on (a group isomorphic to) the Heisenberg group H

1. This is thesource idea of the lifting theorem by L.P. Rothschild and E.M. Stein, which states,roughly speaking, that any Hörmander operator sum of squares of vector fields canbe approximated by a sub-Laplacian on a stratified group. This result emphasizes themajor rôle played by the sub-Laplacians in the theory of second order PDE’s withnon-negative and degenerate characteristic form.

2. The Contents of the Book XIII

1.5. Stratified Groups in Sub-Riemannian Geometry

Stratified groups also appear naturally in sub-Riemannian geometry (frequently re-ferred to as “Carnot” geometry). Roughly speaking, stratified groups play a rôle, forsub-Riemannian manifolds, analogous to that played by Euclidean vector spaces forRiemannian manifolds.

More precisely, once it has been provided a suitable notion of tangent space at apoint of a sub-Riemannian manifold, it turns out that (at a regular point) this tangentspace is naturally endowed with a structure of nilpotent Lie group with dilations, astratified Lie group (see J. Mitchell [Mit85] and A. Bellaïche in [BR96]).

Furthermore, the analysis of a left invariant sub-Laplacian on a connected nilpo-tent Lie group (or more generally on a Lie group of polynomial growth) and thegeometry at infinity of this group is described by a canonically associated dilation-invariant sub-Laplacian on a stratified Lie group. See G. Alexopoulos [Ale92,Ale02],S. Ishiwata [Ish03] and N.Th. Varopoulos [Varo00].

2. The Contents of the Book: An Overview

A glance at the contents of the book and at our approach to the subjects is in order.The book is divided into three parts, and every part is, in its turn, subdivided inseveral chapters plus some appendices, if necessary.

2.1. Part I

The first four chapters of Part I are devoted to an elementary and self-contained in-troduction to the stratified Lie groups in R

N . Our presentation does not require aspecialized knowledge neither in algebra nor in differential geometry. The approachis completely elementary, “constructive” whenever possible, abundant in examplesand intended to be understandable by readers with basic backgrounds only in lin-ear algebra and differential calculus in R

N . Subsequently, we present the formaland abstract approach to the stratified Lie groups commonly used in literature, andwe prove the equivalence of the abstract notion of stratified group to the “construc-tive” notion of homogeneous Carnot group. This equivalence is also provided inPart I.

A very special emphasis is given to the examples. We introduce and discussa wide range of explicit stratified Lie groups of arbitrarily large dimension andstep. Some of them have been known in specialized literature for several years,such as the Heisenberg–Kaplan groups, the filiform groups and the Métivier groups.Many others have only appeared very recently, in particular what we shall call theKolmogorov-type groups and the Bony-type groups. Other examples are completelynew, some extracted from geometric control theory.

Our long list of examples is also intended to be appreciated by readers workingin geometry and analysis on Carnot groups. It provides a valuable benchmark set to

XIV Preface

test new special properties of the groups, to exhibit explicit examples and counterex-amples of the “pathologies” and the special features of Carnot groups.

It is also payed a special attention to the Lie algebras of the groups by stressingtheir links with second order partial differential operators of Hörmander type (sumof squares of vector fields). In particular, given such an operator, we show necessaryand sufficient conditions for it to be a sub-Laplacian on a suitable homogeneousstratified Lie group, and we explicitly show how to construct the related compositionlaw. As a byproduct, this enables the reader to build up another plenty of examplesof stratified groups and sub-Laplacians.

Chapter 5 of Part I is dedicated to the analysis of the fundamental solution forthe sub-Laplacians, a central topic of Part I. Here, the mainly used analytic toolsare integration by parts and coarea-formulas. We start from the hypoellipticity ofsub-Laplacians, easy consequence3 of the Hörmander Theorem 1.

From this “assumption” on hypoellipticity, and with the aid of the strong maxi-mum principle, whose proof is postponed to the Appendix of Chapter 5, we deducethe existence of a gauge function d for any given sub-Laplacian L, i.e. the existenceof a positive non-constant homogeneous function d such that d2−Q is L-harmonicaway from the origin. Here Q stands for the homogeneous dimension of the groupon which the sub-Laplacian lives (we always assume that Q ≥ 3).

This property is one of the most striking analogies between L and the classi-cal Laplace operator. We show that this leads to suitable mean value formulas onthe d-balls, extending to this new setting the well-known Gauss theorem for classi-cal harmonic functions. We then use these formulas (which will play a crucial rôlethroughout the book) to prove Liouville-type theorems, Harnack-type inequalities,and a Sobolev–Stein embedding theorem. Furthermore, three sections are devoted tothe following topics: some remarks on the analytic-hypoellipticity of sub-Laplacians,L-harmonic approximations and an integral representation formula for the funda-mental solution.

2.2. Part II

Part II of the book contains an exhaustive potential theory for the sub-Laplacians.Basically, our only starting point is the theorem by G.B. Folland asserting the exis-tence for these operators of a homogeneous and smooth fundamental solution withpole at the origin. This key result allows us to perform a complete potential theorythat parallel the one of the classical Laplace operator.

The lack of explicit Poisson integral formulas forces us to follow an abstractapproach to this theory. For this reason, in Chapter 6 we present some topics fromabstract harmonic space theory, mainly inspired by the ones developed by H. Bauer[Bau66] and C. Costantinescu and A. Cornea [CC72]. This chapter mainly involvesPerron–Wiener–Brelot method for the Dirichlet problem, harmonic minorants andmajorants and balayage theory.

3 We do not go into the proof of this theorem, for it would require techniques very far fromthe ones developed in the book.

2. The Contents of the Book XV

Next, in Chapter 7 we show that every sub-Laplacian equips RN with a structure

of harmonic space satisfying the axioms of the theory presented in Chapter 6. This isaccomplished by using the Harnack-type theorem proved in Chapter 5, and then byshowing the existence of a basis of the topology of R

N formed by L-regular sets, i.e.by sets for which the Dirichlet problem for L is solvable in the usual classical sense(here we follow an idea by J.-M. Bony [Bon69]).

In the subsequent chapters of Part II, we use the full strength of the abstracttheory, together with the remarkable properties of the fundamental solution for L todeal with the arguments listed below:

a) sub-mean characterizations of the L-subharmonic functions, and applications tothe notion of convexity in Carnot groups;

b) Green functions and Riesz representation theorems for L-subharmonic functions,with applications (among which Bôcher-type theorems);

c) maximum principles on unbounded domains;d) L-capacity and L-polar sets, with applications: the Poisson–Jensen formula and

the so-called fundamental convergence theorem;e) L-thinness and L-fine topology, with applications to the Dirichlet problem (and

the derivation of Wiener’s criterion);f) the links between the Hausdorff measure naturally related to the gauge d and the

capacity for L.

In writing this part of the book we were also partially inspired by some mono-graphs on potential theory for the classical Laplace operator—in particular the beau-tiful books by L.L Helms [Helm69], by W.K. Hayman and P.B. Kennedy [HK76] andby D.H. Armitage and S.J. Gardiner [AG01].

2.3. Part III

In Part III, we take up further topics on the algebraic and analytic theory of Carnotgroups. In particular, this part of the book provides:

a) the study of free Lie algebras;b) clear and complete proofs, in several contexts, of the fundamental and remarkable

Campbell–Hausdorff formula4;c) the equivalence of sub-Laplacians under diffeomorphisms;d) the Rothschild–Stein lifting theorem and Folland’s lifting theorem (for stratified

or homogeneous vector fields);e) the study of the algebraic structure of Heisenberg–Kaplan-type groups (also pro-

viding an explicit characterization of them) with a special emphasis to the re-markable form of their fundamental solutions, discovered by G.B. Folland andA. Kaplan (we also present the inversion and the Kelvin transform in the H-typegroups of Iwasawa type);

4 In Chapter 15, we collect four theorems for the Campbell–Hausdorff formula: one for ho-mogeneous vector fields, two for formal power series and one for general smooth vectorfields.

XVI Preface

f) the Carathéodory–Chow–Rashevsky connectivity theorem (for stratified vectorfields) with applications;

g) Taylor’s formula (with Lagrange and with integral remainder) on Carnot groups.

The difficulty of finding “easy” and complete proofs of some of the above men-tioned results in the existing literature is well known. By working with stratifiedvector fields we are able to overcome some of the lengthy steps of the proofs, whilemaintaining a good amount of generality in the final results.

3. How to Read this Book

Besides Ph.D. students, the book is addressed to young and senior researchers. In-deed, one of the main efforts in presenting the material is to use an elementary ap-proach and to reach, step by step, the level of current researches. Many parts of thebook may be used for graduate courses and advanced lectures.

The first four chapters of Part I are addressed to non-specialists in Lie groupand Lie algebra theory. The first two chapters can be skipped by the readers havingfamiliarity with the basics of differential geometry and Lie group theory. The readeralready acquainted with Carnot group theory can pass directly to Chapter 5. In anycase, beginners and specialists in the theory of stratified groups can exploit the firstfour chapters as a source for examples.

Part II is the core of the monograph. The reader with some background in po-tential theory (and interested in the main case of sub-Laplacians) can pass directlyto Chapter 7 and proceed throughout Part II, leaving Chapters 10 and 13 as a furtherreading.

Part III is thought of as a more specialized lecture. Nonetheless, a deep under-standing of, e.g. the Campbell–Hausdorff formula or of Heisenberg-type groups areamongst the main goals of this monograph.

The book provides 21 illustrative figures, 250 exercises (each chapter has its ownsection of exercises) and an index of the basic notation. For the reading convenience,we furnish a synoptic diagram of the structure of the book on page XVII.

3. How to Read this Book XVII

The synoptic diagram of the structure of the book.

XVIII Preface

4. Some References on Theoretical and Applied Related Topics

Here is a short list of references for related topics on analysis on stratified Lie groupsand applications.5

Alexopoulos [Ale02], Altafini [Alt99], Bellaïche and Risler [BR96],Birindelli, Capuzzo Dolcetta and Cutrì [BCC97], Bahri [Bah04,Bah03],Barletta [Bar03], Barletta and Dragomir [BD04],Barletta, Dragomir and Urakawa [BDU01],Birindelli, Capuzzo Dolcetta and Cutrì [BCC98],Brandolini, Rigoli and Setti [BRS98], Capogna [Cap99],Capogna and Cowling [CC69], Capogna and Garofalo [CG98,CG03,CG06],Capogna, Garofalo and Nhieu [CGN00,CGN02], Capuzzo Dolcetta [CD98],Chandresekhar [Cha43], Citti [Cit98], Citti, Lanconelli and Montanari [CLM02],Citti, Manfredini and Sarti [CMS04], Citti and Montanari [CM00],Citti, Pascucci and Polidoro [CPP01], Citti and Sarti [CS06],Citti and Tomassini [CT04] Cowling, De Mari, Korányi and Reimann [CDKR02],Cowling and Reimann [CR03], Danielli, Garofalo, Nhieu and Tournier [DGNT04],Danielli, Garofalo and Salsa [DGS03], Dragomir [Dra01],Franchi, Gutiérrez and van Nguyen [FGvN05],Franchi, Serapioni and Serra Cassano [FSS03a,FSS03b,FSS01],Gamara [Gam01], Gamara and Yacoub [GY01],Garofalo and Lanconelli [GL92], Garofalo and Tournier [GT06],Garofalo and Vassilev [GV00], Golé and Karidi [GK95],Gutiérrez and Lanconelli [GL03], Gutiérrez and Montanari [GM04a,GM04b],Heinonen [Hei95b], Heinonen and Holopainen [HH97],Heinonen and Koskela [HK98], Huisken and Klingenberg [HK99],Jerison and Lee [JL87,JL88,JL89],Juutinen, Lu, Manfredi and Stroffolini [JLMS07],Korányi and Reimann [KR90,KR95],Lanconelli [Lan03], Lanconelli, Pascucci and Polidoro [LPP02],Lanconelli and Uguzzoni [LU00], Lu, Manfredi and Stroffolini [LMS04],Lu and Wei [LW97], Malchiodi and Uguzzoni [MU02],Manfredi and Stroffolini [MS02], Montanari [Mo01],Montanari and Lascialfari [ML01], Montgomery [Mon02],Montgomery, Shapiro and Stolin [MSS97], Monti and Morbidelli [MM05],Monti and Rickly [MR05], Monti and Serra Cassano [MSC01],Petitot and Tondut [PT99], Reimann [Rei01a,Rei01b],Slodkowski and Tomassini [ST91], Stein [Ste81],Uguzzoni [Ugu00], Varopoulos, Saloff-Coste and Coulhon [VSC92].

5 The list is alphabetically ordered and the grouping of the references in different lines isonly meant for typographical readability.

5. Acknowledgments XIX

5. Acknowledgments

The authors would like to thank Chiara Cinti and Andrea Tommasoli for carefulreading some chapters of the manuscript.

We would also like to express our gratitude to Italo Capuzzo Dolcetta, CristianE. Gutiérrez, Guozhen Lu and Juan J. Manfredi for the kind encouraging appreciationof our work.

It is also a pleasure to thank the Springer-Verlag staff for the kind collaboration,in particular Dr. Catriona M. Byrne, Dr. Marina Reizakis and Dr. Susanne Denskus.

Some topics presented in this book have partially appeared in the following pa-pers by joint collaborations of the authors (and of one of us with C. Cinti) [Bon04,BC04,BC05,BL01,BL02,BL03,BL07,BLU02,BLU03,BU04a,BU04b,BU05a].

Bologna, ItalyApril 2007

Andrea BonfiglioliErmanno LanconelliFrancesco Uguzzoni

Contents

Part I Elements of Analysis of Stratified Groups

1 Stratified Groups and Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Vector Fields in R

N : Exponential Maps and Lie Algebras . . . . . . . . . 31.1.1 Vector Fields in R

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Exponentials of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.4 Lie Brackets of Vector Fields in R

N . . . . . . . . . . . . . . . . . . . . . 101.2 Lie Groups on R

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.1 The Lie Algebra of a Lie Group on R

N . . . . . . . . . . . . . . . . . . 131.2.2 The Jacobian Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3 The (Jacobian) Total Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.4 The Exponential Map of a Lie Group on R

N . . . . . . . . . . . . . 231.3 Homogeneous Lie Groups on R

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3.1 δλ-homogeneous Functions and Differential Operators . . . . . 321.3.2 The Composition Law of a Homogeneous Lie Group . . . . . . 381.3.3 The Lie Algebra of a Homogeneous Lie Group on R

N . . . . . 441.3.4 The Exponential Map of a Homogeneous Lie Group . . . . . . . 48

1.4 Homogeneous Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561.5 The Sub-Laplacians on a Homogeneous Carnot Group . . . . . . . . . . . 62

1.5.1 The Horizontal L-gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.6 Exercises of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2 Abstract Lie Groups and Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . 872.1 Abstract Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2.1.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.1.3 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.1.4 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.1.5 Commutators. ϕ-relatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.1.6 Abstract Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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2.1.7 Left Invariant Vector Fields and the Lie Algebra . . . . . . . . . . 1072.1.8 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.1.9 The Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.2 Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212.2.1 Some Properties of the Stratification of a Carnot Group . . . . 1252.2.2 Some General Results on Nilpotent Lie Groups . . . . . . . . . . . 1282.2.3 Abstract and Homogeneous Carnot Groups . . . . . . . . . . . . . . 1302.2.4 More Properties of the Lie Algebra . . . . . . . . . . . . . . . . . . . . . 1382.2.5 Sub-Laplacians of a Stratified Group . . . . . . . . . . . . . . . . . . . . 144

2.3 Exercises of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3 Carnot Groups of Step Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.1 The Heisenberg–Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.2 Homogeneous Carnot Groups of Step Two . . . . . . . . . . . . . . . . . . . . . 1583.3 Free Step-two Homogeneous Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.4 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.5 The Exponential Map of a Step-two Homogeneous Group . . . . . . . . 1663.6 Prototype Groups of Heisenberg Type . . . . . . . . . . . . . . . . . . . . . . . . . 1693.7 H-groups (in the Sense of Métivier) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1733.8 Exercises of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4 Examples of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.1 A Primer of Examples of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . 183

4.1.1 Euclidean Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.1.2 Carnot Groups with Homogeneous Dimension Q ≤ 3 . . . . . 1844.1.3 B-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.1.4 K-type Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.1.5 Sum of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

4.2 From a Set of Vector Fields to a Stratified Group . . . . . . . . . . . . . . . . 1914.3 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

4.3.1 The Vector Fields ∂1, ∂2 + x1∂3 . . . . . . . . . . . . . . . . . . . . . . . . 1984.3.2 Classical and Kohn Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . 2004.3.3 Bony-type Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024.3.4 Kolmogorov-type Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . 2044.3.5 Sub-Laplacians Arising in Control Theory . . . . . . . . . . . . . . . 2054.3.6 Filiform Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2) . . . . . 2104.4.1 Fields not Satisfying Hypothesis (H0) . . . . . . . . . . . . . . . . . . . 2104.4.2 Fields not Satisfying Hypothesis (H1) . . . . . . . . . . . . . . . . . . . 2124.4.3 Fields not Satisfying Hypothesis (H2) . . . . . . . . . . . . . . . . . . . 215

4.5 Exercises of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Contents XXIII

5 The Fundamental Solution for a Sub-Laplacian and Applications . . . . 2275.1 Homogeneous Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2295.2 Control Distances or Carnot–Carathéodory Distances . . . . . . . . . . . . 2325.3 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

5.3.1 The Fundamental Solution in the Abstract Setting . . . . . . . . . 2445.4 L-gauges and L-radial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.5 Gauge Functions and Surface Mean Value Theorem . . . . . . . . . . . . . . 2515.6 Superposition of Average Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 2575.7 Harnack Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2625.8 Liouville-type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

5.8.1 Asymptotic Liouville-type Theorems . . . . . . . . . . . . . . . . . . . . 2745.9 Sobolev–Stein Embedding Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 2765.10 Analytic Hypoellipticity of Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . 2805.11 Harmonic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2875.12 An Integral Representation Formula for Γ . . . . . . . . . . . . . . . . . . . . . . 2915.13 Appendix A. Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

5.13.1 A Decomposition Theorem for L-harmonic Functions . . . . . 3035.14 Appendix B. The Improved Pseudo-triangle Inequality . . . . . . . . . . . 3065.15 Appendix C. Existence of Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 3095.16 Exercises of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Part II Elements of Potential Theory for Sub-Laplacians

6 Abstract Harmonic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3376.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3386.2 Sheafs of Functions. Harmonic Sheafs . . . . . . . . . . . . . . . . . . . . . . . . . 340

6.2.1 Harmonic Measures and Hyperharmonic Functions . . . . . . . . 3416.2.2 Directed Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 342

6.3 Harmonic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3456.3.1 Directed Families of Harmonic and Hyperharmonic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3476.4 B-hyperharmonic Functions. Minimum Principle . . . . . . . . . . . . . . . . 3486.5 Subharmonic and Superharmonic Functions. Perron Families . . . . . . 3536.6 Harmonic Majorants and Minorants . . . . . . . . . . . . . . . . . . . . . . . . . . . 3586.7 The Perron–Wiener–Brelot Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 3596.8 S-harmonic Spaces: Wiener Resolutivity Theorem . . . . . . . . . . . . . . 363

6.8.1 Appendix: The Stone–Weierstrass Theorem . . . . . . . . . . . . . . 3666.9 H-harmonic Measures for Relatively Compact Open Sets . . . . . . . . . 3676.10 S∗-harmonic Spaces: Bouligand’s Theorem . . . . . . . . . . . . . . . . . . . . 3706.11 Reduced Functions and Balayage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3756.12 Exercises of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

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7 The L-harmonic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.1 The L-harmonic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.2 Some Basic Definitions and Selecta of Properties . . . . . . . . . . . . . . . . 3887.3 Exercises of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

8 L-subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3978.1 Sub-mean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3978.2 Some Characterizations of L-subharmonic Functions . . . . . . . . . . . . . 4018.3 Continuous Convex Functions on G . . . . . . . . . . . . . . . . . . . . . . . . . . . 4118.4 Exercises of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

9 Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4259.1 L-Green Function for L-regular Domains . . . . . . . . . . . . . . . . . . . . . . 4259.2 L-Green Function for General Domains . . . . . . . . . . . . . . . . . . . . . . . . 4279.3 Potentials of Radon Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

9.3.1 Potentials Related to the Average Operators . . . . . . . . . . . . . . 4359.4 Riesz Representation Theorems for L-subharmonic Functions . . . . . 4419.5 The Poisson–Jensen Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4459.6 Bounded-above L-subharmonic Functions in G . . . . . . . . . . . . . . . . . 4519.7 Smoothing of L-subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . 4559.8 Isolated Singularities—Bôcher-type Theorems . . . . . . . . . . . . . . . . . . 458

9.8.1 An Application of Bôcher’s Theorem . . . . . . . . . . . . . . . . . . . 4629.9 Exercises of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

10 Maximum Principle on Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . 47310.1 MP Sets and L-thinness at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47310.2 q-sets and the Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47710.3 The Maximum Principle on Unbounded Domains . . . . . . . . . . . . . . . 48210.4 The Proof of Lemma 10.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48310.5 Exercises of Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

11 L-capacity, L-polar Sets and Applications . . . . . . . . . . . . . . . . . . . . . . . . 48911.1 The Continuity Principle for L-potentials . . . . . . . . . . . . . . . . . . . . . . . 48911.2 L-polar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49111.3 The Maria–Frostman Domination Principle . . . . . . . . . . . . . . . . . . . . . 49411.4 L-energy and L-equilibrium Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 49711.5 L-balayage and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50011.6 The Fundamental Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . 51011.7 The Extended Poisson–Jensen Formula . . . . . . . . . . . . . . . . . . . . . . . . 51411.8 Further Results. A Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51911.9 Further Reading and the Quasi-continuity Property . . . . . . . . . . . . . . 52711.10 Exercises of Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

Contents XXV

12 L-thinness and L-fine Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53712.1 The L-fine Topology: A More Intrinsic Tool . . . . . . . . . . . . . . . . . . . . 53712.2 L-thinness at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53812.3 L-thinness and L-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

12.3.1 Functions Peaking at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 54212.3.2 L-thinness and L-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

12.4 Wiener’s Criterion for Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . 54712.4.1 A Technical Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54712.4.2 Wiener’s Criterion for L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

12.5 Exercises of Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

13 d-Hausdorff Measure and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55713.1 d-Hausdorff Measure and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 55713.2 d-Hausdorff Measure and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 56113.3 New Phenomena Concerning the d-Hausdorff Dimension . . . . . . . . . 56913.4 Exercises of Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

Part III Further Topics on Carnot Groups

14 Some Remarks on Free Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57714.1 Free Lie Algebras and Free Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . 57714.2 A Canonical Way to Construct Free Carnot Groups . . . . . . . . . . . . . . 584

14.2.1 The Campbell–Hausdorff Composition � . . . . . . . . . . . . . . . . 58414.2.2 A Canonical Way to Construct Free Carnot Groups . . . . . . . . 586


15 More on the Campbell–Hausdorff Formula . . . . . . . . . . . . . . . . . . . . . . . 59315.1 The Campbell–Hausdorff Formula for Stratified Fields . . . . . . . . . . . 59315.2 The Campbell–Hausdorff Formula for Formal Power Series–1 . . . . . 59915.3 The Campbell–Hausdorff Formula for Formal Power Series–2 . . . . . 60515.4 The Campbell–Hausdorff Formula for Smooth Vector Fields . . . . . . 61015.5 Exercises of Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

16 Families of Diffeomorphic Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . 62116.1 An Isomorphism Turning

∑i,j ai,j XiXj into ΔG . . . . . . . . . . . . . . . 622

16.2 Examples and Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62816.3 Canonical or Non-canonical? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

16.3.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64116.4 Further Reading: An Application to PDE’s . . . . . . . . . . . . . . . . . . . . . 64416.5 Exercises of Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

XXVI Contents

17 Lifting of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64917.1 Lifting to Free Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64917.2 An Example of Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65917.3 An Example of Application to PDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . 66117.4 Folland’s Lifting of Homogeneous Vector Fields . . . . . . . . . . . . . . . . 666

17.4.1 The Hypotheses on the Vector Fields . . . . . . . . . . . . . . . . . . . . 66917.5 Exercises of Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676

18 Groups of Heisenberg Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68118.1 Heisenberg-type Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68118.2 A Direct Characterization of H-type Groups . . . . . . . . . . . . . . . . . . . . 68618.3 The Fundamental Solution on H-type Groups . . . . . . . . . . . . . . . . . . . 69518.4 H-type Groups of Iwasawa-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70218.5 The H-inversion and the H-Kelvin Transform . . . . . . . . . . . . . . . . . . . 70418.6 Exercises of Chapter 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

19 The Carathéodory–Chow–Rashevsky Theorem . . . . . . . . . . . . . . . . . . . . 71519.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified

Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71519.2 An Application of Carathéodory–Chow–Rashevsky Theorem . . . . . . 72719.3 Exercises of Chapter 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730

20 Taylor Formula on Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73320.1 Polynomials and Derivatives on Homogeneous Carnot Groups . . . . . 734

20.1.1 Polynomial Functions on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73420.1.2 Derivatives on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

20.2 Taylor Polynomials on Homogeneous Carnot Groups . . . . . . . . . . . . 74120.3 Taylor Formula on Homogeneous Carnot Groups . . . . . . . . . . . . . . . . 746

20.3.1 Stratified Taylor Formula with Peano Remainder . . . . . . . . . . 75120.3.2 Stratified Taylor Formula with Integral Remainder . . . . . . . . 754


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773

Index of the Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

Part I

Elements of Analysis of Stratified Groups

1

Stratified Groups and Sub-Laplacians

In this first chapter, we introduce the main notation and the basic definitions concern-ing with vector fields in R

N : algebras of vector fields, exponentials of smooth vectorfields, Lie brackets. Then, we introduce the main geometric structure investigatedthroughout the book: the homogeneous Carnot groups.

To this end, we first study Lie groups G on RN and the Lie algebra of their

left-invariant vector fields. Subsequently, we equip G with a homogeneous structureby the datum of a well-behaved group of dilations {δλ}λ>0 on G. The compositionof G thus takes a transparent form, allowing us to study homogeneous Lie groupson R

N by very direct methods. In particular, it will be a simple exercise of calculusto verify that the relevant exponential and logarithmic maps are global polynomialdiffeomorphisms, a result which will throughout account for a very useful tool. Fi-nally, we introduce the notion of homogeneous Carnot group and of sub-Laplacian.A wide number of explicit examples of homogeneous Carnot groups will be given inChapters 3 and 4, after (in Chapter 2) we have analyzed the due relationship betweenabstract Carnot groups and the homogeneous ones.

Indeed, despite our notions of Lie group on RN , homogeneous and homoge-

neous-Carnot group are dependent on a fixed system of coordinates for the group(thus being non-intrinsic notions), every abstract Carnot group is, as we shall see, iso-morphic via the exponential map to its Lie algebra, which is a homogeneous Carnotgroup. This basic fact provides another motivation for this introductory chapter.

1.1 Vector Fields in RNRN

RN . Exponential Maps. Lie Algebras of

Vector Fields

1.1.1 Vector Fields in RNRN

RN

Given N ∈ N, we set, as usual, RN = {(x1, . . . , xN) : x1, . . . , xN ∈ R}. We use any

of the notation

∂j , ∂xj,

∂

∂xj

, ∂/∂ xj

4 1 Stratified Groups and Sub-Laplacians

to indicate the partial derivative operator with respect to the j -th coordinate of RN .

Let Ω ⊆ RN be an open (and non-empty) set. Given an N -tuple of scalar functions

a1, . . . , aN ,aj : Ω → R, j ∈ {1, . . . , N},

the first order linear differential operator

X =N∑

j=1

aj ∂j (1.1)

will be called a vector field on Ω with component functions (or simply, components)a1, . . . , aN . If O ⊆ Ω is an open set and f : O → R is a differentiable function, wedenote by Xf the function on O defined by

Xf (x) =N∑

j=1

aj (x) ∂jf (x), x ∈ O.

Occasionally, we shall also use the notation Xf when

f : O → Rm

is a vector-valued function, to mean the component-wise action of X. More pre-cisely,1

if f (x) =⎛

⎝f1(x)

...

fm(x)

⎞

⎠ , we set Xf (x) =⎛

⎝Xf1(x)

...

Xfm(x)

⎞

⎠ .

1 We warn the reader that points in RN will be usually denoted as N -tuples x =

(x1, . . . , xN ). When this does not lead to confusion, the column-vector notation

x =⎛

⎜⎝x1...

xN

⎞

⎟⎠

will also be allowed. For example, this last notation will be sometimes used (with no riskof misunderstanding) for vector-valued functions f (x) = (f1(x), . . . , fN (x))T , e.g.

I (x) = (x1, . . . , xN )T

will always denote (this time as a rule) the identity map in RN .

However, there are cases in which we shall keep the notation well distinguished: namely,if f : RN → R is a differentiable function, its gradient ∇f = (∂1f, . . . , ∂Nf ) will alwaysbe written as a row-vector.

On the other side, we shall always use the column-vector notation when vectors in RN

appear in matrix calculation. For example, if x, y ∈ RN , xT y will denote the row×column

product

xT y = (x1, . . . , xN )

⎛

⎜⎝y1...

yN

⎞

⎟⎠ =N∑

j=1

xj yj .

1.1 Vector Fields in RN : Exponential Maps and Lie Algebras 5

(Typically, this notation will be used when f = I is the identity map of RN , i.e.

I (x) = (x1, . . . , xN)T ; see (1.2) below.) Furthermore, given a differentiable functionf : O → R

m, we shall denote by

Jf (x), x ∈ O

the Jacobian matrix of f at x.Let C∞(O, R) (for brevity, C∞(O)) be the set of smooth (i.e. infinitely-dif-

ferentiable) real-valued functions. If the components aj ’s of X are smooth, we shallcall X a smooth vector field and we shall often consider X as an operator acting onsmooth functions,

X : C∞(O) → C∞(O), f �→ Xf.

We shall prevalently deal with smooth vector fields. We shall denote by

T (RN)

the set of all smooth vector fields in RN . Equipped with the natural operations,

T (RN) is a vector space over R.We adopt the following (non-conventional) notation: I will denote the identity

map on RN and, if X is the vector field in (1.1), then

XI :=⎛

⎝a1...

aN

⎞

⎠ (1.2)

will be the column vector of the components of X. This notation is obviously con-sistent with our definition of the action of X on a vector-valued function.2 Thus, XI

may also be regarded as a smooth map from RN to itself (that is what some authors

call a “vector field” on RN ).

Often, many authors identify X and XI . Instead, in order to avoid any confusionbetween a smooth vector field as a function belonging to C∞(RN, R

N) and a smoothvector field as a differential operator from C∞(RN) to itself,3 we prefer to use thedifferent notation XI and X as described in (1.2) and (1.1), respectively.

By consistency of notation, we may write

Xf = (∇f ) ·XI,

where ∇ = (∂1, . . . , ∂N )

is the gradient operator in RN , f is any real-valued smooth function on R

N and ·denotes the row×column product. For example, for the following two vector fieldson R

3 (whose points are denoted by x = (x1, x2, x3))

X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 , (1.3a)

we have

X1I (x) =( 1

02 x2

), X2I (x) =

( 01

−2 x1

). (1.3b)

2 Indeed, since I = (I1, . . . , IN ) with Ij (x) = xj , we have X(Ij ) = aj .3 Or even from C∞(RN, R

N) to itself!


1.1.2 Integral Curves

A path γ : D → RN , D being an interval of R, will be said an integral curve of the

smooth vector field X if

γ (t) = XI (γ (t)) for every t ∈ D.

If X is a smooth vector field, then, for every x ∈ RN , the Cauchy problem

{γ = XI (γ ),

γ (0) = x(1.4)

has a unique solutionγX(·, x) : D(X, x) → R

N.

We agree to denote by D(X, x) the greatest open interval of R on which γX(·, x)

exists. For example, if X1 is as in (1.3a), the solution to the relevant problem (1.4) isobtained by solving the following system of ODE’s (we write γ = (γ1, γ2, γ3))

{γ1(t) = 1, γ2(t) = 0, γ3(t) = 2 γ2(t),

γ1(0) = x1, γ2(0) = x2, γ3(0) = x3,

i.e. after simple computations, we obtain D(X1, x) = R and

γX1(t, x) = (x1 + t, x2, x3 + 2 x2 t). (1.5)

Since X is smooth, t �→ γX(t, x) is a C∞ function whose n-th Taylor expansionin a neighborhood of t = 0 is given by

γX(t, x) = x + t X(1)I (x)+ t2

2! X(2)I (x)+ · · · + tn

n! X(n)I (x)

+ 1

n!∫ t

0(t − s)n X(n+1)I (γX(s, x)) ds. (1.6)

Hereafter, for k ∈ N, we denote by X(k) the vector field

X(k) =N∑

j=1

(Xk−1aj )∂xj,

being X0 = I (the identity map) and Xh, h ≥ 1, the h-th order iterated of X, i.e.

Xh := X ◦ · · · ◦X︸︷︷︸h times

.

In other words, we have


X(1)I = XI =⎛

⎝a1...

aN

⎞

⎠ , X(2)I =⎛

⎝X(a1)

...

X(aN)

⎞

⎠ , X(3)I =⎛

⎝X(X(a1))

...

X(X(aN))

⎞

⎠ , . . . .

We remark that Xh is a differential operator of order at most h, whereas X(h) is adifferential operator of order at most 1. To check (1.6) we use (1.4). Writing γ (t)

instead of γX(t, x), (1.4) gives

γ (0) = x, (d/dt)|t=0γ (t) = XI (x) and

d2

dt2

∣∣∣∣t=0

γ (t) = d

dt

∣∣∣∣t=0

(XI)(γ (t)) = JXI (γ (0)) · γ (0) = JXI (x) ·XI (x)

=⎛

⎝∇a1(x) ·XI (x)

...

∇aN(x) ·XI (x)

⎞

⎠ =⎛

⎝Xa1(x)

...

XaN(x)

⎞

⎠ = X(2)I (x).

By iterating this argument, we obtain

γ (k)(0) := dk

dtk

∣∣∣∣t=0

γ (t) = X(k)I (x), k ≥ 2.

Replacing this identity in the Taylor formula

γ (t) = x +n∑

k=1

tk

k!γ(k)(0)+ 1

n!∫ t

0(t − s)nγ (n+1)(s) ds,

we obtain (1.6). Since the identity map I is linear and since the first order part of Xh

coincides with X(h), one hasX(h)I ≡ XhI.

Thus formula (1.6) can be rewritten as

γX(t, x) = x + t XI (x)+ t2

2! X2I (x)+ · · · + tn

n! XnI (x)

+ 1

n!∫ t

0(t − s)n Xn+1I (γX(s, x)) ds. (1.7)

Example 1.1.1. For example, if X1 is as in (1.3a), since

X(1)1 I =

( 10

2 x2

), X

(2)1 I =

( 000

)= X

(k)1 I ∀ k ≥ 3,

we have

γX1(t, x) = x + t X1I (x) =(

x1x2x3

)+ t

( 10

2 x2

)=(

x1 + t

x2x3 + 2 x2 t

),

as we directly found in (1.5). �


1.1.3 Exponentials of Vector Fields

The expansion in (1.7) suggests to use an “exponential-type” notation. For this (morethan notational) reason, we give the following definition.

Definition 1.1.2 (Exponential of a vector field). Let X be a smooth vector fieldon R

N . Following all the above notation, we set

exp(tX)(x) := γX(t, x), (1.8)

where γX(·, x) is the solution of (1.4).This definition makes sense4 for every X ∈ T (RN), for every x ∈ R

N and everyt ∈ D(X, x). (See Fig. 1.1.)

Fig. 1.1. Figure of Definition 1.1.2

Then, being X smooth, for every n ∈ N, we have the expansion

exp(tX)(x) =n∑

k=0

tk

k! XkI (x)

+ 1

n!∫ t

0(t − s)n Xn+1I

(exp(sX)(x)

)ds. (1.9)

In particular, for n = 1,

exp(tX)(x) = x + t XI (x)+∫ t

0(t − s)X2I

(exp(sX)(x)

)ds. (1.10)

If we defineU := {(t, x) ∈ R× R

N | x ∈ RN, t ∈ D(X, x)},

from the basic theory of ordinary differential equations (see, e.g. [Har82]) we knowthat U is open and the map

4 Definition 1.1.2 is well-posed also when X has only Lipschitz-continuous component func-tions, but we shall prevalently deal, as already stated, with smooth vector fields.


U � (t, x) �→ exp(tX)(x) ∈ RN

is smooth. Moreover, from the unique solvability of the Cauchy problem related tosmooth vector fields we get: t ∈ D(−X, x) iff −t ∈ D(X, x) and

exp(−tX)(x) := exp((−t)X)(x) = exp(t (−X))(x), (1.11a)

exp(−tX)(

exp(tX)(x)) = x, (1.11b)

exp((t + τ)X)(x) = exp(tX)(

exp(τX)(x)), (1.11c)

exp((tτ )X)(x) = exp(t (τX))(x), (1.11d)

when all the terms are defined. If D(X, x) = R, identities (1.11a)–(1.11d) hold truefor every t, τ ∈ R. See also the note.5

Remark 1.1.3 (Pyramid-shaped vector fields). For our aims, the vector fields of thefollowing type

X =N∑

j=1

aj (x1, . . . , xj−1) ∂xj(1.12)

will play a crucial rôle. In (1.12), the function aj only depends on the variablesx1, . . . , xj−1, and we agree to let aj (x1, . . . , xj−1) = constant when j = 1. Roughlyspeaking, such a remarkable kind of vector field is “pyramid”-shaped,

X =

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

a1a2(x1)

a3(x1, x2)

a4(x1, x2, x3)...

aN(x1, . . . , xN−1)

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

.

For example, the fields in (1.3b) have this form.For any smooth vector field X of the form (1.12), the map

(x, t) �→ exp(tX)(x)

is well defined for every x ∈ RN and t ∈ R. Indeed, if γ = (γ1, . . . , γN) is the

solution to the Cauchy problem{

γ = XI (γ ),

γ (0) = x, x = (x1, . . . , xN),

then γ1 = a1 and γj = aj (γ1, . . . , γj−1) for j = 2, . . . , N . As a consequence,

γ1(x, t) = x1 + ta1, γj (x, t) = xj +∫ t

0aj (γ1(x, s), . . . , γj−1(x, s)) ds,

5 Strictly speaking, according to the very definition of exp(tX)(x) := γX(t, x), one mayobserve that t and X should be kept separate in notation. Though, we explicitly remark thatidentity (1.11d) justifies the notation “tX” in exp(tX)(x).


and γj (x, t) is defined for every x ∈ RN and t ∈ R. Moreover, γ1(·, t) only depends

on x1, whereas for j = 2, . . . , N , γj (·, t) only depends on x1, . . . , xj . Let us putA1(t) = ta1 and, for j = 2, . . . , N ,

Aj(x, t) = Aj(x1, . . . , xj−1, t) :=∫ t

0aj (γ1(x, s), . . . , γj−1(x, s)) ds.

Then, for every x ∈ RN , t ∈ R,

exp(tX)(x) =

⎛

⎜⎜⎜⎝

x1 + A1(t)

x2 + A2(x1, t)...

xN + AN(x1, . . . , xN−1, t)

⎞

⎟⎟⎟⎠ , (1.13)

and the map x �→ exp(tX)(x) is a global diffeomorphism of RN onto R

N for everyfixed t ∈ R. Its inverse map y �→ L(y, t) is given by

y �→ L(y, t) = exp(−tX)(y). (1.14)

This last statement follows from identity (1.11b). � Remark 1.1.4. Let us consider a smooth function u : R

N → R and the vector fieldin (1.1). Then

Xu(x) = limt→0

u(exp(tX)(x))− u(x)

t∀ x ∈ R

N. (1.15)

Indeed, since exp(tX)(x) = x + tXI (x)+O(t2), the limit on the right-hand side of(1.15) is equal to the following one:

limt→0

u(x + tXI (x))− u(x)

t= ∇u(x) ·XI (x) = Xu(x).

1.1.4 Lie Brackets of Vector Fields in RNRN

RN

Given two smooth vector fields X and Y in RN , we define the Lie-bracket [X, Y ] as

follows[X, Y ] := XY − YX.

This definition is only seemingly deceitful, for it writes [X, Y ] (which is a firstorder differential operator) as a difference of two second order differential operators.Indeed, if X = ∑N

j=1 aj ∂j and Y = ∑Nj=1 bj ∂j , a direct computation shows that

the Lie bracket [X, Y ] is the vector field

[X, Y ] =N∑

j=1

(Xbj − Yaj )∂j .


As a consequence,

[X, Y ]I =⎛

⎝Xb1

...

XbN

⎞

⎠−⎛

⎝Ya1

...

Y aN

⎞

⎠ = JYI ·XI − JXI · YI. (1.16)

For example, if X1, X2 are as in (1.3b) (page 5), we have

[X1, X2] = (X1(−2x1)−X2(2x2)) ∂x3 = −4 ∂x3 .

It is quite trivial to check that (X, Y ) �→ [X, Y ] is a bilinear map on the vector spaceT (RN) satisfying the Jacobi identity

[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0

for every X, Y,Z ∈ T (RN).We shall refer to T (RN) (equipped with the above Lie-bracket) as the Lie algebra

of the vector fields on RN . Any sub-algebra a of T (RN) will be called a Lie algebra

of vector fields. More explicitly, a is a Lie algebra of vector fields if a is a vectorsubspace of T (RN) closed with respect to [ , ], i.e. [X, Y ] ∈ a for every X, Y ∈ a.

We now introduce some other notation on the algebras of vector fields. Given aset of vector fields Z1, . . . , Zm ∈ T (RN) and a multi-index

J = (j1, . . . , jk) ∈ {1, . . . , m}k,we set

ZJ := [Zj1 , . . . [Zjk−1 , Zjk] . . .].

We say that ZJ is a commutator of length (or height) k of Z1, . . . , Zm. If J = j1, wealso say that ZJ := Zj1 is a commutator of length 1 of Z1, . . . , Zm. A commutatorof the form ZJ will also be called nested, in order to emphasize its difference from,e.g. a commutator of the form

[[Z1, Z2], [Z3, Z4]].What is striking is that this last commutator is a linear combination of nested ones,6

as we prove in Proposition 1.1.7.First, we give a definition.

Definition 1.1.5 (The Lie algebra generated by a set). If U is any subset of T (RN),we denote by Lie{U} the least sub-algebra of T (RN) containing U , i.e.

Lie{U} :=⋂

h, where h is a sub-algebra of T (RN) with U ⊆ h.

We definerank

(Lie{U}(x)

) := dimR

{ZI (x) |Z ∈ Lie{U}}.

6 Namely,

[[Z1, Z2], [Z3, Z4]] = −[Z3, [Z4, [Z1, Z2]]] + [Z4, [Z3, [Z1, Z2]]].


Example 1.1.6. Let X1 and X2 be as in (1.3b) (page 5). Since [X1, X2] = −4∂x3 andsince any commutator involving X1, X2 more than twice is identically zero, thenLie{X1, X2} = span{X1, X2, [X1, X2]}, and

rank(Lie{X1, X2}(x)) = 3 for every x ∈ R3

(the well-known Hörmander condition). � The following result holds.

Proposition 1.1.7 (Nested commutators). Let U ⊆ T (RN) be any set of smoothvector fields on R

N . We set

U1 := span{U}, Un := span{[u, v] | u ∈ U, v ∈ Un−1}, n ≥ 2.

Then we haveLie{U} = span{Un | n ∈ N}.

Moreover,[u, v] ∈ Ui+j for every u ∈ Ui , v ∈ Uj .

We explicitly remark that, from the very definition of Un, the vector fields in Un

are linear combination of nested brackets, i.e. brackets of the type

[u1[u2[u3[· · · [un−1, un] · · ·]]]]with u1, . . . , un ∈ U . The above proposition then states that every element of Lie{U}is a linear combination of nested brackets.

To show the idea behind the proof, let us take u1, u2, v1, v2 ∈ U and prove that[[u1, u2], [v1, v2]] is a linear combination of nested brackets. By the Jacobi identity[X, [Y,Z]] = −[Y, [Z,X]] − [Z, [X, Y ]], one has

[[u1, u2]︸︷︷︸X

, [ v1︸︷︷︸Y

, v2︸︷︷︸Z

]] = −[v1, [v2, [u1, u2]]] − [v2, [[u1, u2], v1]]

= −[v1, [v2, [u1, u2]]] + [v2, [v1, [u1, u2]]] ∈ U4.

Proof (of Proposition 1.1.7). We set U∗ := span{Un | n ∈ N}. Obviously, U∗ con-tains U and is contained in any algebra of vector fields which contains U . Hence,we are left to prove that U∗ is closed under the bracket operation. Obviously, it isenough to show that, for any i, j ∈ N and for any u1, . . . , ui , v1, . . . , vj ∈ U , wehave [[u1[u2[· · · [ui−1, ui] · · ·]]]; [v1[v2[· · · [vj−1, vj ] · · ·]]]

] ∈ Ui+j .

We argue by induction on k := i + j ≥ 2. For k = 2 and 3, the assertion is obvious.Let us now suppose the thesis holds for every i+ j ≤ k with k ≥ 4, and prove it alsoholds when i + j = k + 1. We can suppose, by skew-symmetry, j ≥ 3. Exploitingrepeatedly the induction hypothesis and the Jacobi identity, we have

1.2 Lie Groups on RN 13

[u; [v1[v2[· · · [vj−1, vj ] · · ·]]]

]

= −[v1, [[v2, [v3, · · ·]], u]︸︷︷︸length k

] − [[v2, [v3, · · ·]], [u, v1]]

= {element of Uk+1} − [[v1, u], [v2, [v3, · · ·]]]= {element of Uk+1} + [v2, [[v3, · · ·], [v1, u]]︸︷︷︸

length k

] + [[v3, · · ·], [[v1, u]v2]]

= {element of Uk+1} + [[v2, [v1, u]], [v3, · · ·]](after finitely many steps)

= {element of Uk+1} + (−1)j−1[[vj−i , [vj−2, · · · [v1, u]]], vj ]= {element of Uk+1} + (−1)j [vj , [vj−i , [vj−2, · · · [v1, u]]]]∈ Uk+1.

This ends the proof. � Corollary 1.1.8. Let Z1, . . . , Zm ∈ T (RN) be fixed. Then

Lie{Z1, . . . , Zm} = span{ZJ |with J = (j1, . . . , jk) ∈ {1, . . . , m}k , k ∈ N

}.

This (non-trivial) fact (direct consequence of Proposition 1.1.7) will be usedthroughout the next sections, often without mention.

The following notation will be used when dealing with “stratified” (or “graded”)Lie algebras. If V1, V2 are subsets of T (RN), we denote

[V1, V2] := span{[v1, v2] | vi ∈ Vi, i = 1, , 2

}. (1.17)

1.2 Lie Groups on RNRN

RN

1.2.1 The Lie Algebra of a Lie Group on RNRN

RN

We first recall a well-known definition.

Definition 1.2.1 (Lie group on RNRN

RN ). Let ◦ be a given group law on R

N , and supposethat the map

RN × R

N � (x, y) �→ y−1 ◦ x ∈ RN

is smooth. Then G := (RN, ◦) is called a Lie group on RN .

Convention. For the simplicity of notation, we shall assume that the origin 0 of RN

is the identity of G. This assumption is not restrictive. Indeed, if e ∈ G is the identityof G (a Lie group on R

N ), we can consider new coordinates on RN given by the

C∞-diffeomorphism defined by T (x) = x − e. Thus, we obtain a new Lie group onR

N , G = (RN, ∗), where

y ∗ y′ = (y + e) ◦ (y′ + e)− e,


with identity T (e) = 0. Obviously, G and G are isomorphic Lie groups via thediffeomorphism T . Furthermore, we shall see that every homogeneous Lie group onR

N (see Section 1.3) has identity element equal to the origin of RN . Finally, the main

subject of this book, i.e. Carnot groups, are naturally isomorphic to a homogeneousLie group on R

N . This justifies our convention and motivates the study of Lie groupson R

N , i.e. roughly speaking, Lie groups with a global chart.

Fixed α ∈ G, we denote by τα(x) := α ◦ x the left-translation by α on G.A (smooth) vector field X on R

N is called left-invariant on G if

X(ϕ ◦ τα) = (Xϕ) ◦ τα (1.18)

for every α ∈ G and for every smooth function ϕ : RN → R. We denote by g the setof the left-invariant vector fields on G. It is quite obvious to recognize that

for every X, Y ∈ g and for every λ,μ ∈ R,

we have λX + μY ∈ g and [X, Y ] ∈ g. (1.19)

Then, g is a Lie algebra of vector fields, sub-algebra of T (RN). It will be called theLie algebra of G.

Example 1.2.2. The map

(x1, x2, x3) ◦ (y1, y2, y3) =(x1 + y1, x2 + y2, x3 + y3 + 2 (x2 y1 − x1 y2)

)

endows R3 with a structure of Lie group. In several next examples, we shall refer

to H1 = (R3, ◦) as the Heisenberg–Weyl group on R

3. It is a direct computation toshow that the vector fields

X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 (1.20)

are left invariant w.r.t. ◦. Consequently, X1, X2, [X1, X2] ∈ h1, say, the Lie al-gebra of H

1 (this notation is not standard). We shall show that, precisely, h1 =span{X1, X2, [X1, X2]} = Lie{X1, X2}. �

From the theorem of differentiation of composite functions, we easily get thefollowing characterization of left-invariant vector fields on G.

Proposition 1.2.3 (Characterization of g. I). Let G be a Lie group on RN , and let

g be the Lie algebra of G. The (smooth) vector field X belongs to g if and only if

(XI)(α ◦ x) = Jτα (x) · (XI)(x) ∀ α, x ∈ G. (1.21)

As usual, Jτα (x) denotes the Jacobian matrix at the point x of the map τα .

Proof. For every smooth function ϕ on RN , we have

(X(ϕ ◦ τα))(x) = ∇(ϕ ◦ τα)(x) ·XI (x) = ((∇ϕ)(τα(x)) · Jτα (x)

) ·XI (x)

and


(Xϕ)(τα(x)) = (∇ϕ)(τα(x)) ·XI (τα(x)).

Then X ∈ g if and only if

(∇ϕ)(τα(x)) · (Jτα (x) ·XI (x)) = (∇ϕ)(τα(x)) ·XI (τα(x)) (1.22)

for every α, x ∈ RN and for every ϕ ∈ C∞(C∞, R). By choosing ϕ(x) =∑N

j=1 hj xj with hj ∈ R for 1 ≤ j ≤ N , (1.22) gives

hT · Jτα (x) ·XI (x) = hT ·XI (τα(x))

for every h ∈ RN , which obviously implies (1.21). �

Interchanging α with x in (1.21), we obtain (XI)(x ◦ α) = Jτx (α) · (XI)(α) forall α, x ∈ G, so that, when α = 0,

(XI)(x) = Jτx (0) · (XI)(0) ∀ x ∈ G. (1.23)

This identity says that a left-invariant vector field on G is completely determined byits value at the origin (and by the Jacobian matrix at the origin of the left-translation).The following result shows that (1.23) characterizes the vector fields in g.

Proposition 1.2.4. Let G be a Lie group on RN , and let g be the Lie algebra of G.

Let η be a fixed vector of RN , and define the (component functions of the) vector field

X as followsXI (x) := Jτx (0) · η, x ∈ R

N. (1.24)

Then X ∈ g.

Proof. Definition (1.24) gives

XI (α ◦ x) = Jτα◦x (0) · η, α, x ∈ RN. (1.25)

On the other hand, since the composition law on G is associative, we have τα◦x =τα ◦ τx , so that

Jτα◦x (0) = Jτα (x) · Jτx (0).

Replacing this identity in (1.25), we get

XI (α ◦ x) = Jτα (x) · Jτx (0) · η,

which implies, by (1.24), XI (α ◦ x) = Jτα (x) · XI (x). Then, by Proposition 1.2.3,X ∈ g. This ends the proof. � Corollary 1.2.5 (Characterization of g. II). Let G be a Lie group on R

N , and let g

be the Lie algebra of G. The vector field X belongs to g iff

(XI)(x) = Jτx (0) · (XI)(0) ∀ x ∈ G. (1.26)


Proof. If X satisfies (1.26), then, by Proposition 1.2.4 (setting η := XI (0)), we haveX ∈ g. Vice versa, if X ∈ g, then we already showed that (1.26) follows from (1.21)of Proposition 1.2.3. � Example 1.2.6. If G = H

1 (see Example 1.2.2), we have

Jτx (0) =( 1 0 0

0 1 02x2 −2x1 1

).

For example, for X1 = ∂x1 + 2 x2 ∂x3 , we recognize that, for every x ∈ H1,

(X1I )(x) =( 1

02 x2

)=( 1 0 0

0 1 02x2 −2x1 1

)·( 1

00

)= Jτx (0) · (XI)(0).

The same obviously holds, e.g. for the fields X2 = ∂x2 − 2 x1 ∂x3 and [X1, X2] =−4∂x3 .

From Proposition 1.2.3 and identity (1.23) it follows that g is a vector space ofdimension N . Indeed, the following proposition holds.

Proposition 1.2.7 (Characterization of g � RNRN

RN . III). Let G be a Lie group on R

N ,and let g be the Lie algebra of G. The map

J : RN → g, η �→ J (η)

with J (η) defined byJ (η)I (x) = Jτx (0) · η (1.27)

is an isomorphism of vector spaces. In particular,

dim g = N.

Proof. We first observe that J is well defined since, by Proposition 1.2.4, J (η) ∈ g

for every η ∈ RN . Moreover, by identity (1.23), J (RN) = g. The linearity of J is

obvious. Then it remains to prove that J is injective.Suppose J (η) = 0. This means that Jτx (0)·η = 0 for every x ∈ R

N . In particularJτ0(0) · η = 0. On the other hand, since the left-translation τ0 is the identity map,Jτ0(0) · η = η. Then η = 0, and J is one-to-one. � Example 1.2.8. The Lie algebra h1 of G = H

1 (see Example 1.2.2 for the notation) isgiven by span{X1, X2, [X1, X2]}. Indeed, X1, X2, [X1, X2] are three linearly inde-pendent left-invariant vector fields and dim(h1) = 3, as stated in Proposition 1.2.7.Again using the same proposition, we could also argue as follows: X1, X2, [X1, X2]are the vector fields obtained by multiplying Jτx (0) respectively times the basis of R

3

(1, 0, 0)T , (0, 1, 0)T , (0, 0,−4)T . � In what follows, the next remarks will be very useful.


Remark 1.2.9. Let X ∈ g, and denote by η the value of XI at x = 0, i.e. η = XI (0).Then, by identity (1.23), XI (x) = Jτx (0) · η. As a consequence, for every smoothfunction ϕ on R

N ,

d

dt

∣∣∣∣t=0

ϕ(x ◦ (tη)) = d

dt

∣∣∣∣t=0

ϕ(τx(tη))

= ∇ϕ(x) · Jτx (0) · η = ∇ϕ(x) ·XI (x) = (Xϕ)(x).

Then

(Xϕ)(x) = d

dt

∣∣∣∣t=0

ϕ(x ◦ (tη)), η = XI (0). (1.28)

Identity (1.28) characterizes the left-invariant vector fields on G. This followsfrom the next remark.

Remark 1.2.10. Let X be a vector field on RN . Assume that there exists η ∈ R

N suchthat, for every ϕ ∈ C∞(RN, R),

(Xϕ)(x) = d

dt

∣∣∣∣t=0

ϕ(x ◦ (tη)) ∀ x ∈ RN. (1.29)

Then η = XI (0) and X ∈ g.Indeed, by taking ϕ(x) = xj = Ij (x) and x = 0 in (1.29), one gets

(XI)j (0) = d

dt

∣∣∣∣t=0

(tη)j = ηj ,

i.e. XI (0) = η. Then (1.29) and the associativity of ◦ imply

(Xϕ)(α ◦ x) = d

dt

∣∣∣∣t=0

ϕ((α ◦ x) ◦ (tη)) = d

dt

∣∣∣∣t=0

(ϕ ◦ τα)(x ◦ (tη))

= X(ϕ ◦ τα)(x)

for every α, x ∈ G. Then X is left-invariant on G. � Collecting together the above remarks, we have proved the following result.

Proposition 1.2.11 (Characterization of g. IV). Let G be a Lie group on RN , and

let g be the Lie algebra of G. The vector field X belongs to g iff there exists η ∈ RN

such that, for every ϕ ∈ C∞(RN, R),

(Xϕ)(x) = d

dt

∣∣∣∣t=0

ϕ(x ◦ (tη)) ∀ x ∈ RN. (1.30)

In this case η = XI (0).


Remark 1.2.12. For every x ∈ RN and X ∈ g, the following expansion holds

exp(tX)(x) = x ◦ (tη)+ o(t) as t → 0, η = XI (0). (1.31)

Indeed, since XI (x) = Jτx (0) · η,

x ◦ tη = τx(tη) = τx(0)+ tJτx (0) · η + o(t) = x + tXI (x)+ o(t).

Then (1.31) follows from (1.10), see page 8.

We remark that two vector fields can be linearly independent in T (RN) withoutbeing linearly independent at every point. Take, for example, ∂x1 and x1 ∂x2 in R

2.Moreover, two vector fields can be linearly dependent at every point without beinglinearly dependent in T (RN). Take, for example, ∂x1 and x1 ∂x1 in R

2.The following result shows that neither of the previous situations can occur for

left-invariant vector fields on a Lie group G on RN . Indeed, given a family of vector

fields X1, . . . , Xm ∈ g,

the rank of the subset of RN spanned by

{X1I (x), . . . , XmI (x)} is independent of x.

More precisely, we have the following result.

Proposition 1.2.13 (Constant rank). Let G be a Lie group on RN , and let g be the

Lie algebra of G. Let X1, . . . , Xm ∈ g. Then the following statements are equivalent:

(i) X1, . . . , Xm are linearly independent (in g);(ii) X1I (0), . . . , XmI (0) are linearly independent (in R

N );(iii) ∃ x0 ∈ R

N : X1I (x0), . . . , XmI (x0) are linearly independent (in RN );

(iv) X1I (x), . . . , XmI (x) are linearly independent (in RN ) for all x ∈ R

N .

Proof. We first recall that, by identity (1.23),

XjI (x) = Jτx (0) · ηj , with ηj = XjI (0),

for every x ∈ RN . On the other hand, since τx−1 ◦ τx = I ,

Jτx−1 (x) · Jτx (0) = IN.

Hence Jτx (0) is non-singular for every x ∈ RN . Then (ii), (iii) and (iv) are equiv-

alent. The equivalence between (i) and (ii) follows from Proposition 1.2.7. Indeed,with the notation of that proposition, for every j ∈ {1, . . . , m}, Xj = J (ηj ) withηj = XjI (0), and J is an isomorphism of R

N onto g. � Example 1.2.14. The vector fields on R

2 defined by

X1 = ∂x1 , X2 = x1∂x2

do satisfy the so-called Hörmander condition

rank(Lie{X1, X2}(x)) = 2 for every x ∈ R2.

However, since X1 and X2 are independent as vector fields but X1I (0), X2I (0) aredependent as vectors of R

2, X1 and X2 are not left-invariant with respect to anygroup law on R

2.


1.2.2 The Jacobian Basis

From Proposition 1.2.7 it follows that any basis of g is the image via J of a basisof R

N . A natural definition is thus in order.

Definition 1.2.15 (Jacobian basis). Let G be a Lie group on RN , and let g be the Lie

algebra of G. If {e1, . . . , eN } is the canonical basis7 of RN and J is the map defined

in Proposition 1.2.7, we call

{Z1, . . . , ZN }, Zj := J (ej )

the Jacobian basis of g.

(Note. We warn the reader that the notion of Jacobian basis is strictly related tothe fact that, presently, G is a Lie group on R

N , and we are making reference to thefixed Cartesian coordinates on R

N . Hence, the Jacobian basis is not well-posed ongeneral Lie groups. Moreover, if we perform a change of coordinates on R

N , even alinear one, the Jacobian basis changes. Despite this “non-invariant”, non-coordinate-free nature of the Jacobian basis, the reader will soon recognize its usefulness.)

From the very definition of J we obtain

ZjI (x) = Jτx (0) · ej = j -th column of Jτx (0) ∀ x ∈ RN, (1.32)

so that, since Jτx (0) = IN ,ZjI (0) = ej .

From Remark 1.2.10 we also have

(Zjϕ)(x) = d

dt

∣∣∣∣t=0

ϕ(x ◦ tej ) = ∂

∂yj

∣∣∣∣y=0

ϕ(x ◦ y) (1.33)

for every ϕ ∈ C∞(RN) and every x ∈ G.Consequently, the Jacobian basis {Z1, . . . , ZN } of g is given by the N column of

the Jacobian matrixJτx (0)

(whence the name). Moreover, Zj |0 = ∂/∂xj |0 and

(Zjϕ)(x) = (∂/∂yj )|y=0ϕ(x ◦ y) ∀ ϕ ∈ C∞(RN), x ∈ G. (1.34)

Summing up the above results, we have the following equivalent characterizations ofthe Jacobian basis.

7 Id est, for every j ∈ {1, . . . , N},ej = (0, . . . , 1︸︷︷︸

j

, . . . , 0)T .


Proposition 1.2.16 (Jacobian basis). Let G be a Lie group on RN , and let g be the

Lie algebra of G. Let j ∈ {1, . . . , N} be fixed.Then there exists one and only one vector field in g, say Zj , characterized by any

of the following equivalent conditions:

1. Zj |0 = (∂/∂xj )|0, i.e.

(Zjϕ)(0) = ∂ϕ

∂ xj

(0) for every ϕ ∈ C∞(RN, R);

2. for every ϕ ∈ C∞(RN, R), it holds

(Zjϕ)(x) = ∂

∂ yj

∣∣∣∣y=0

(ϕ(x ◦ y)

)for every x ∈ G;

3. if ej denotes the j -th vector of the canonical basis of RN , then

ZjI (0) = ej ;4. the column vector of the component functions of Zj is

ZjI (x) = Jτx (0) · ej = j -th column of Jτx (0);5. for every x ∈ G, we have

(Zjϕ)(x) = d

dt

∣∣∣∣t=0

ϕ(x ◦ (tej )) for every ϕ ∈ C∞(RN, R).

The system of vector fields Z := {Z1, . . . , ZN } is a basis of g, the Jacobian basis.The coordinates of X ∈ g w.r.t.Z are, orderly, the entries of the column vectorXI (0).

(For the proof of the last statement, see Remark 1.2.20.)In the sequel, when we need to endow g with a differentiable structure, we shall

consider the vector space structure of g, making g (in a natural way) a differentiablemanifold. Although the choice of a basis for g is completely immaterial, we shallprevalently fix a system of coordinates on g by choosing the Jacobian basis, thenidentifying g with R

N in a fixed way.

Example 1.2.17 (The Jacobian basis of H1). The Jacobian basis for the Lie algebra

of H1 (see Example 1.2.2) is given by

Z1 = ∂x1 + 2 x2 ∂x3 , Z2 = ∂x2 − 2 x1 ∂x3 , Z3 = ∂x3 ,

since, in this case, the Jacobian matrix at 0 of the left-translation is

Jτx (0) =( 1 0 0

0 1 02x2 −2x1 1

). �


Example 1.2.18 (A non-polynomial nilpotent Lie group on R3). It is a simple exercise

to verify that the following operation

(x1, x2, x3) ◦ (y1, y2, y3)

:= (arcsinh(sinh(x1)+ sinh(y1)), x2 + y2 + sinh(x1)y3, x3 + y3)

endows R3 with a Lie group structure (G, ◦). The Lie algebra g of G is spanned by

the vector fields, coefficient-vectors are given by the columns of the Jacobian matrix

Jτx (0) =( 1

cosh(x1)0 0

0 1 sinh(x1)

0 0 1

)

(the Jacobian basis), i.e. g = span{Z1, Z2, Z3}, where

Z1 = 1

cosh(x1)∂x1

Z2 = ∂x2

Z3 = ∂x3 + sinh(x1) ∂x2 .

Note that G is nilpotent (of step two).

Example 1.2.19 (A non-polynomial non-nilpotent Lie group on R2). It is a simple

exercise to verify that the following operation on R2

(x1, x2) ◦ (y1, y2) = (x1 + y1, y2 + x2 ey1)

defines a Lie group structure, and the Jacobian basis is Z1 = ∂x1 + x2 ∂x2 ,Z2 = ∂x2 . Hence, the relevant Lie algebra is not nilpotent, for [Z2, Z1] = Z2, sothat, inductively,

[· · · [[[Z2, Z1], Z1], Z1] · · ·Z1︸︷︷︸k times

] = Z2 for all k ∈ N.

Remark 1.2.20. Let us consider the map

π : g→ RN, X �→ π(X) := XI (0). (1.35)

From the very definition of π , the following fact follows: if X ∈ g and we writeη := π(X), then we have

Jτx (0) · η = Jτx (0) ·XI (0) = (XI)(x), (1.36)

the last equality following from (1.23). Let now J be the map introduced in Propo-sition 1.2.7. Comparing (1.36) to (1.27), we recognize that

J (π(X)) = J (η) = X, π(J (η)) = η ∀ X ∈ g, η ∈ RN.

Thus π is the inverse map of J .

We explicitly remark that π is the linear map which assigns, to every vector field X

in g, the N -tuple η in RN of the coordinates of X with respect to the Jacobian basis.

The coordinate of X with respect to this basis is simply given by XI (0) (see alsoFig. 1.2).


Fig. 1.2. Figure of Remark 1.2.20

1.2.3 The (Jacobian) Total Gradient

Let G = (RN, ◦) be a Lie group on RN , and let Z1, . . . , ZN be the Jacobian basis8

of the Lie algebra g of G.For any differentiable function u defined on an open set Ω ⊆ R

N , we considera sort of “intrinsic” gradient of u given by (Z1u, . . . , ZNu) (in the sequel, we shallcall it (Jacobian) total gradient). Then it follows from (1.32) that

(Z1u(x), . . . , ZNu(x)) = ∇u(x) · Jτx (0) ∀ x ∈ Ω. (1.37)

On the other hand, since Jτx (0) is non-singular and its inverse is given by Jτx−1 (0),

we can write the Euclidean gradient of u in terms of its total gradient in the followingway

∇u(x) = (Z1u(x), . . . , ZNu(x)) · Jτx−1 (0) ∀ x ∈ Ω. (1.38)

From (1.38) we immediately obtain the following result. We shall follow thenotation of Remark 1.2.3.

Proposition 1.2.21. Let G be a Lie group on RN , and let Z1, . . . , ZN be the rele-

vant Jacobian basis (or any basis for g). Let Ω ⊆ RN be an open and connected

set. A function u ∈ C1(Ω, R) is constant in Ω if and only if its total gradient(Z1u, . . . , ZNu) vanishes identically on Ω .

(Note. A significant improvement of this result will be available in the stratifiedsetting. See, e.g. Proposition 1.5.6).

Proof. From (1.37) and (1.38) it follows that the total gradient of u vanishes at x ∈ Ω

if and only if ∇u(x) = 0. � 8 In this section, we consider the Jacobian basis for the sake of brevity. In fact, Z1, . . . , ZN

can be replaced by any basis X1, . . . , XN of g. Indeed, note that in this case there exists aN ×N non-singular constant matrix M such that

(X1I (x) · · ·XNI (x)) = M · (Z1I (x) · · ·ZNI (x)).


Example 1.2.22. When G = H1, it indeed holds

(Z1u,Z2u,Z3u) = (∂x1u+ 2 x2 ∂x3u, ∂x2u− 2 x1 ∂x3u, ∂x3u)

= (∂x1u, ∂x2u, ∂x3u

) ·( 1 0 0

0 1 02x2 −2x1 1

)= ∇u · Jτx (0),

and, vice versa,

(Z1u,Z2u,Z3u) · Jτx−1 (0) = (Z1u,Z2u,Z3u) ·

( 1 0 00 1 0

−2x2 2x1 1

)= ∇u.

1.2.4 The Exponential Map of a Lie Group on RNRN

RN

The next lemma will be useful to define the notion of Exponential map from g to G,one of the most important tools in the Lie group theory.

Lemma 1.2.23 (Completeness of g). Let (G, ◦) be a Lie group on RN , and let g be

its Lie algebra. Let X ∈ g, and let γ : [t0, t0 + T ] → RN be an integral curve of X.

Then:

(i) α ◦ γ is an integral curve of X for every α ∈ G.(ii) γ can be continued to an integral curve of X on the interval [t0 − T , t0 + 2T ].Proof. (i) For every t ∈ [t0, t0 + T ], we have (by (1.21))

d

dt(α ◦ γ (t)) = d

dt(τα(γ (t))) = Jτα (γ (t)) · γ (t)

= Jτα (γ (t)) ·XI (γ (t)) = X(α ◦ γ (t)).

(ii) Define Γ : [t0 − T , t0 + 2T ] → RN as follows:

Γ (t) :=

⎧⎪⎨

⎪⎩

γ (t0) ◦ (γ (t0 + T ))−1 ◦ γ (t + T ) if t0 − T ≤ t ≤ t0,

γ (t), if t0 ≤ t ≤ t0 + T ,

γ (t0 + T ) ◦ (γ (t0))−1 ◦ γ (t − T ) if t0 + T ≤ t ≤ t0 + 2T .

Then, by (i), Γ is an integral curve of X and, obviously, Γ |[t0,t0+T ] ≡ γ . � From assertion (ii) of this Lemma we immediately obtain the following important

statement: For every X ∈ g, the map

(x, t) �→ exp(tX)(x)

is well-defined for every x ∈ RN and every t ∈ R.

The next corollary easily follows from the assertion (i) of Lemma 1.2.23.


Corollary 1.2.24. Let (G, ◦) be a Lie group on RN , and let g be its Lie algebra. Let

X ∈ g and x, y ∈ G. Then

x ◦ exp(tX)(y) = exp(tX)(x ◦ y) (1.39)

for every t ∈ R. In particular, for y = 0,

exp(tX)(x) = x ◦ exp(tX)(0).

Proof. By Lemma 1.2.23-(i), t �→ x ◦ exp(tX)(y) is an integral curve of X. More-over,

(x ◦ exp(tX)(y))|t=0 = x ◦ y.

Then (1.39) follows. � Definition 1.2.25 (Exponential map). Let G be a Lie group on R

N , and let g be itsLie algebra. The exponential map of the Lie group G is defined by

Exp : g→ G, Exp (X) = exp(1 ·X)(0).

More explicitly, Exp (X) is the value at the time t = 1 of the path γ (t) solution toγ (t) = XI (γ (t)), γ (0) = 0. (See Fig. 1.3.)

From Corollary 1.2.24 and identity (1.11b) (with τ = −t) we get

Exp (−X) ◦ Exp (X) = 0.

Indeed,

Exp (−X) ◦ Exp (X) = Exp (−X) ◦ exp(X)(0) = exp(X)(Exp (−X))

= exp(X)(exp(−X)(0)) = 0.

Then we have(Exp (X))−1 = Exp (−X). (1.40)

Fig. 1.3. Figure of Definition 1.2.25

We give an explicit example of exponential map.


Example 1.2.26 (The Exp map on H1). Let us consider once again the Heisenberg–

Weyl group H1 on R

3. In Example 1.2.8, we showed that a basis for its Lie algebrah1 is given by X1, X2, X3, where X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 andX3 = [X1, X2] = −4∂x3 . Let us construct the exponential map. We set, for ξ ∈ R

3,

ξ ·X := ξ1 X1 + ξ2 X2 + ξ3 X3

= ξ1

( 10

2 x2

)+ ξ2

( 01

−2 x1

)+ ξ3

( 00−4

)=(

ξ1ξ2

−4ξ3 + 2ξ1 x2 − 2ξ2 x1

).

By Definition 1.1.2, for fixed x ∈ H1, we have exp(ξ ·X)(x) = γ (1), where γ (s) =

(γ1(s), γ2(s), γ3(s)) is the solution to{

γ (s) = (ξ ·X)I (γ (s)) = (ξ1, ξ2,−4ξ3 + 2ξ1γ2(s)− 2 ξ2γ1(s)),

γ (0) = x.

Solving the above system of ODE’s, one gets

exp(ξ ·X)(x) =⎛

⎝x1 + ξ1x2 + ξ2

x3 − 4 ξ3 + 2 ξ1 x2 − 2 ξ2 x1

⎞

⎠ .

As a consequence, by Definition 1.2.25, we obtain

Exp (ξ ·X) = exp(ξ ·W)(0) =⎛

⎝ξ1ξ2−4 ξ3

⎞

⎠ ,

so that Exp is globally invertible and its inverse map is given by

Log (y) := (Exp )−1(y) =⎛

⎝y1y2

− 14 y3

⎞

⎠ ·X.

For example, we have

Exp (−ξ ·X) =⎛

⎝−ξ1−ξ2+4 ξ3

⎞

⎠ = −Exp (ξ ·X) = (Exp (−ξ ·X)

)−1,

since the inverse of x in H1 coincides with−x. This fact tests, in this simple example,

the validity of (1.40). �

Remark 1.2.27 (Local invertibility of Exp ). Let {X1, . . . , XN } be a basis of g. Then,for every X ∈ g,

X =N∑

j=1

ξjXj for a suitable ξ = (ξ1, . . . , ξN ) ∈ RN,


Fig. 1.4. The relation between the group G, its algebra g and RN

so that

Exp (X) = exp

(N∑

j=1

ξjXj

)(0).

From the classical theory of ODE’s we know that the map

(ξ1, . . . , ξN ) �→ exp

(N∑

j=1

ξjXj

)(0)

is smooth. Then we can say that the map g � X �→ Exp (X) ∈ G is smooth. Fromthe Taylor expansion (1.10) (page 8) we get

Exp (X) =N∑

j=1

ξj ηj +O(|ξ |2), as |ξ | → 0,

where ηj = XjI (0).Denote by E the matrix whose column vectors are η1, . . . , ηN . Then

JExp (0) = E (we fix on g coordinates related to the Xj ’s).

In particular, if {X1, . . . , XN } = {Z1, . . . , ZN } is the Jacobian basis of g, then

JExp (0) = IN (we fix on g coordinates related to the Zj ’s). (1.41)


As a consequence, Exp is a diffeomorphism from a neighborhood of 0 ∈ g onto aneighborhood of 0 ∈ G. Where defined,9 we denote by Log the inverse map of Exp.

Example 1.2.28. With the notation of Example 1.2.26, we recall that the Jacobianbasis for h1 is given by (see Example 1.2.17)

Z1 = X1, Z2 = X2, Z3 = −1

4X3.

Hence, by the computations in Example 1.2.26, we have

Exp (ξ1Z1+ ξ2Z2+ ξ3Z3) = Exp

(ξ1X1+ ξ2X2− 1

4ξ3X3

)= (ξ1, ξ2, ξ3), (1.42)

and (1.41) is readily verified. � The next proposition is an easy consequence of Corollary 1.2.24 and shows an

important link between the composition law in G and the exponential map.

Proposition 1.2.29 (Exponentiation and composition). Let (G, ◦) be a Lie groupon R

N , and let g be its Lie algebra. Let x, y ∈ G. Assume Log (y) is defined. Then

x ◦ y = exp(Log (y))(x). (1.43)

Proof. Let X = Log (y). This means that

y = Exp (X) = exp(X)(0).

Then, by Corollary 1.2.24, we infer

x ◦ y = x ◦ exp(X)(0) = exp(X)(x).

This is precisely (1.43), and the proof is complete. � By writing y = Exp (X) in (1.43), we obtain

x ◦ Exp (X) = exp(X)(x) for every X ∈ g and every x ∈ G. (1.44)

We give two examples of this proposition, the first is very simple, the second one alittle more elaborated.

Example 1.2.30 (of (1.43) for H1). Let us consider once again the Heisenberg–Weyl

group H1 on R

3. Proceeding with the computations in Example 1.2.26, we have

9 We shall see in Chapter 2 that, in many important situations, such as for connected andsimply connected nilpotent Lie groups, Exp is globally invertible. Any Carnot group is aconnected and simply connected nilpotent Lie group.


Fig. 1.5. Figure of Proposition 1.2.29

exp(Log (y))(x) = exp

((y1, y2,−1

4y3

)·X)

(x)

=⎛

⎝x1 + y1x2 + y2

x3 + y3 + 2 y1 x2 − 2 y2 x1

⎞

⎠ = x ◦ y,

which tests, in this simple example, the validity of (1.43). � Example 1.2.31. Let us consider once again the Lie group G introduced in Exam-ple 1.2.18. The relevant Jacobian basis is {Z1, Z2, Z3}, where

Z1 = 1

cosh(x1)∂x1 , Z2 = ∂x2 , Z3 = ∂x3 + sinh(x1) ∂x2 .

Given ξ = (ξ1, ξ2, ξ3) ∈ R3 and x ∈ G, we set ξ ·Z =∑3

i=1 ξi Zi , so that exp(t (ξ ·Z))(x) = γ (t) is the solution to

γ (t) = ξ1 Z1I (γ (t))+ ξ2 Z2I (γ (t))+ ξ3 Z3I (γ (t)), γ (0) = x,

i.e. more explicitly (writing γ (t) = (γ1(t), γ2(t), γ3(t))),⎧⎪⎨

⎪⎩

γ1(t) = ξ1 (cosh(γ1(t)))−1, γ1(0) = x1,

γ2(t) = ξ2 + ξ3 sinh(γ1(t)), γ2(0) = x2,

γ3(t) = ξ3, γ3(0) = x3.

A direct computation gives⎧⎪⎨

⎪⎩

γ1(t) = arcsinh(sinh(x1)+ ξ1t),

γ2(t) = x2 + (ξ2 + ξ3 sinh(x1))t + ξ1ξ3t2

2 ,

γ3(t) = x3 + ξ3t.


In particular, Exp (ξ · Z) = exp(1 (ξ · Z))(0), i.e.

Exp (ξ · Z) =(

arcsinh(ξ1), ξ2 + 1

2ξ1ξ3, ξ3

),

so that

Log (y1, y2, y3) =(

sinh(y1), y2 − 1

2sinh(y1) y3, y3

)· Z.

Collecting the above facts together, we get

exp(Log (y))(x) = exp

((sinh(y1), y2 − 1

2sinh(y1) y3, y3

)· Z)

(x)

=( arcsinh(sinh(x1)+ sinh(y1))

x2 + (y2 − 12 sinh(y1) y3 + y3 sinh(x1))+ sinh(y1) y3

12

x3 + y3

)

=( arcsinh(sinh(x1)+ sinh(y1))

x2 + y2 + y3 sinh(x1)

x3 + y3

)= x ◦ y,

as stated in (1.43) of Proposition 1.2.29. � Remark 1.2.32 (The composition of G induces a composition on g via Exp: TheCampbell–Hausdorff operation). Suppose that

Exp : g → G and Log : G → g

are globally defined C∞ maps, inverse to each other. We then define on g the opera-tion

X � Y := Log (Exp (X) ◦ Exp (Y )), X, Y ∈ g. (1.45)

It is immediately seen that � defines a Lie group structure on g and

Exp : (g,�)→ (G, ◦)is a Lie-group isomorphism. Indeed, this last fact is obvious from the very definitionof �, whereas the associativity of � on g follows immediately from the associativityof ◦ on G.

One of the most striking facts about Lie algebras and Lie groups is that (undersuitable hypotheses) the operation � on g is well-posed and can be expressed in asomewhat “universal” way as a sum of iterated Lie-brackets of X and Y (see (2.43)and Theorem 2.2.13, page 129). For example, the first few terms are

X � Y = X + Y + 1

2[X, Y ] + 1

12[X, [X, Y ]] − 1

12[Y, [X, Y ]] + · · · . (1.46)

We shall deal extensively on the composition law � throughout the book. We givean example of � when G is the Heisenberg–Weyl group on R

3.


Example 1.2.33 (The operation � for H1). The reader is invited to compare this ex-

ample to the diagram in Fig. 1.6.We use the notation and computations of Example 1.2.26. When G = H

1, wesaw that Exp is globally invertible. Hence (1.45) makes sense. We fix X ∈ h1. IfZ1, Z2, Z3 is the Jacobian basis for h1, and we set, for brevity, ξ := XI (0), we have(see Remark 1.2.20)

X = ξ1 Z1 + ξ2 Z2 + ξ3 Z3 =: ξ · Z.

Analogously, if Y ∈ h1, we set η := YI (0), so that Y = η · Z. Thus, we derive

Log(Exp (X) ◦ Exp (Y )

)

= Log(Exp (ξ · Z) ◦ Exp (η · Z)

)

= (see (1.42)) Log (ξ ◦ η)

= Log (ξ1 + η1, ξ2 + η2, ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1)

(again from (1.42))

= (ξ1 + η1, ξ2 + η2, ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1) · Z= (ξ1 + η1) Z1 + (ξ2 + η2) Z2 + (ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1) Z3. (1.47)

On the other hand, we consider (1.46), truncated to the commutators of length two(sine h1 is nilpotent of step two!), and we explicitly write down X � Y in our case,thus obtaining

(ξ · Z) � (η · Z) = ξ · Z + η · Z + 1

2[ξ · Z, η · Z]

= ξ1 Z1 + ξ2 Z2 + ξ3 Z3 + η1 Z1 + η2 Z2 + η3 Z3

+ 1

2[ξ1 Z1 + ξ2 Z2 + ξ3 Z3, η1 Z1 + η2 Z2 + η3 Z3]

(here we use [Z1, Z2] = −4 Z3, [Z1, Z3] = [Z2, Z3] = 0)

= (ξ1 + η)Z1 + (ξ2 + η2) Z2 + (ξ3 + η3) Z3

+ 1

2

((−4 ξ1 η2 + 4 ξ2 η1) Z3

)

= (ξ1 + η)Z1 + (ξ2 + η2) Z2 + (ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1) Z3,

which equals the last term in (1.47). As a consequence, we have proved that in thiscase it holds


) = X + Y + 1

2[X, Y ].

For the group considered in Examples 1.2.18 and 1.2.31, the reader is invited to testa similar formula. �

1.3 Homogeneous Lie Groups on RN 31

1.3 Homogeneous Lie Groups on RNRN

RN

We begin by giving the definition of homogeneous Lie group (see also E.M. Stein[Ste81]).

Definition 1.3.1 (Homogeneous Lie group (on RNRN

RN )). Let G = (RN, ◦) be a Lie

group on RN (according to Definition 1.2.1). We say that G is a homogeneous (Lie)

group (on RN ) if the following property holds:

(H.1) There exists an N -tuple of real numbers σ = (σ1, . . . , σN), with1 ≤ σ1 ≤ · · · ≤ σN , such that the “dilation”

δλ : RN → RN, δλ(x1, . . . , xN) := (λσ1x1, . . . , λ

σN xN)

is an automorphism of the group G for every λ > 0.

We shall denote by G = (RN, ◦, δλ) the datum of a homogeneous Lie group on RN

with composition law ◦ and dilation group {δλ}λ>0.

(Note. A note similar to that given after Definition 1.2.15 of Jacobian basis ap-plies to the notion of homogeneous Lie group: the notion of homogeneous Lie groupis not coordinate-free and strongly depends on the choice of a fixed system of co-ordinates on R

N . Nonetheless, the reader will soon recognize the suitability of thisnotion.)

The family of dilations {δλ}λ>0 forms a one-parameter group of automorphisms of G

whose identity isδ1 = I,

the identity map of RN . Indeed, we have

δr s(x) = δr

(δs(x)

) ∀ x ∈ G, r, s > 0.

Moreover, (δλ)−1 = δλ−1 . In the sequel, {δλ}λ>0 will be referred to as the dilation

group (or group of dilations) of G.From (H.1) it follows that

δλ(x ◦ y) = (δλ x) ◦ (δλ y) ∀ x, y ∈ G (1.48)

and, if e denotes the identity of G, δλ(e) = e for every λ > 0. This obviously impliesthat e = 0. This is consistent with our previous assumption that the origin is theidentity of G.

For example, the Heisenberg–Weyl group H1 (see Example 1.2.2, page 14) is

a homogeneous Lie group if R3 is equipped with the dilations δλ(x1, x2, x3) =

(λx1, λx2, λ2x3).

Remark 1.3.2. Suppose G = (RN, ◦) is a Lie group on RN such that there exists an

N -tuple of positive real numbers σ = (σ1, . . . , σN) such that

δλ : RN → RN, dλ(x1, . . . , xN) := (λσ1x1, . . . , λ

σN xN)


is an automorphism of the group G for every λ > 0. Then, modulo a permutation ofthe variables of R

N , it is always not restrictive to suppose that σ1 ≤ · · · ≤ σN . Obvi-ously, this permutation of the coordinates does not alter neither (the new permuted)G being a Lie group on R

N nor the (relevant permuted) dilation δλ satisfying (1.48).Moreover, there exists a group of dilations δλ on G such that

δλ(x1, . . . , xN) = (λσ1x1, . . . , λσN xN)

with 1 = σ1 ≤ · · · ≤ σN . Indeed, it suffices to take (once the σj ’s have been orderedincreasingly)

σj := σj/σ1 for every j = 1, . . . , N.

Indeed, with this choice, we have

δλ ≡ dλ1/σ1 ,

and δλ(x ◦ y) = δλ(x) ◦ δλ(y) follows from (1.48), restated for dλ, with λ replacedby λ1/σ1 .

1.3.1 δλ-homogeneous Functions and Differential Operators

Before we continue the analysis of homogeneous Lie groups, we show some basicproperties of homogeneous functions and homogeneous differential operators withrespect to the family {δλ}λ.

In this subsection, no group law is required on RN . Here, we only suppose that it

is given on RN a family of maps δλ of the form

δλ : RN → RN, δλ(x1, . . . , xN) := (λσ1x1, . . . , λ

σN xN), (1.49)

with fixed positive real numbers σ1, . . . , σN . We set σ := (σ1, . . . , σN).A real function a defined on R

N is called δλ-homogeneous of degree m ∈ R if a

does not vanish identically and, for every x ∈ RN and λ > 0, it holds

a(δλ (x)) = λma(x).

A non-identically-vanishing linear differential operator X is called δλ-homo-geneous of degree m ∈ R if, for every ϕ ∈ C∞(RN), x ∈ R

N and λ > 0, itholds

X(ϕ(δλ(x))) = λm(Xϕ)(δλ(x)).

Let a be a smooth δλ-homogeneous function of degree m ∈ R and X be alinear differential operator δλ-homogeneous of degree n ∈ R. Then Xa is a δλ-homogeneous function of degree m− n (unless Xa ≡ 0). Indeed, for every x ∈ R

N

and λ > 0, we have

λn(Xa)(δλ(x)) = X(a(δλ(x))) = X(λma(x)) = λm(Xa)(x).

Given a multi-index α ∈ (N ∪ {0})N , α = (α1, . . . , αN), we define the δλ-length(or δλ-height) of α as


|α|σ = 〈α, σ 〉 =N∑

i=1

αi σi . (1.50)

Definition 1.3.3 (GGG-length of a multi-index. GGG-degree). When G = (RN, ◦, δλ) isa homogeneous Lie group on R

N with its given group of dilations {δλ}λ, we shall usethe notation

|α|Gfor the relevant δλ-length as defined in (1.50). In this case, we shall refer to |α|G asthe G-length (or G-height) of α. Moreover, if p : G → R is a polynomial function(the sum below is intended to be finite)

p(x) =∑

α

cα xα, cα ∈ R,

we say thatdegG (p) := max{|α|G : cα �= 0}

is the G-degree or δλ-(homogeneous) degree of p.

Since x �→ xj and ∂/∂xj , j ∈ {1, . . . , N}, are obviously δλ-homogeneousof degree σj , the function x �→ xα and the differential operator Dα are both δλ-homogeneous of degree |α|σ .

If a is a continuous function, δλ-homogeneous of degree m and a(x0) �= 0 forsome x0 ∈ R

N , then m ≥ 0. Indeed, from a(δλ(x0)) = λma(x0) we get

limλ→0

λm = limλ→0

a(δλ(x0))

a(x0)= a(0)

a(x0).

Moreover, the continuous and δλ-homogeneous of degree 0 functions are preciselythe constant (non-zero) functions. Indeed,

a(x) = a(δλ(x)) = limλ→0+

a(δλ(x)) = a(0).

Let us now consider a smooth and δλ-homogeneous of degree m ∈ R function a

and a multi-index α. Assume that Dαa is not identically zero. Then, since Dαa issmooth and δλ-homogeneous of degree m − |α|σ , it has to be m − |α|σ ≥ 0, i.e.|α|σ ≤ m. This result can be restated as follows:

Dαa ≡ 0 ∀ α such that |α|σ > m.

Thus a is a polynomial function. Let a(x) =∑α∈A aα xα , where A is a finite set of

multi-indices and aα ∈ R for every α ∈ A. Since a is δλ-homogeneous of degree m,we have ∑

α∈Aλm aα xα = λm a(x) = a(δλ(x)) =

∑

α∈Aaα λ|α|σ xα.

Hence λm aα = λ|α|σ aα for every λ > 0, so that |α|σ = m if aα �= 0. Then


a(x) =∑

|α|σ=m

aα xα. (1.51)

It is quite obvious that every polynomial function of the form (1.51) is δλ-homoge-neous of degree m. Thus, we have proved the following proposition.

Proposition 1.3.4 (Smooth δλ-homogeneous functions). Let δλ be as in (1.49).Suppose that a ∈ C∞(RN, R). Then a is δλ-homogeneous of degree m ∈ R ifand only if a is a polynomial function of the form (1.51) with some aα �= 0. As aconsequence, the set of the degrees of the smooth δλ-homogeneous (non-vanishing)functions is precisely the set of the nonnegative real numbers

A = {|α|σ : α ∈ (N ∪ {0})N },with |α|σ = 0 if and only if a is constant.

From the proposition above one easily obtains the following characterization ofthe smooth δλ-homogeneous vector fields.

Proposition 1.3.5 (Smooth δλ-homogeneous vector fields). Let δλ be as in (1.49).Let X be a smooth non-vanishing vector field on R

N ,

X =N∑

j=1

aj (x) ∂xj.

Then X is δλ-homogeneous of degree n ∈ R if and only if aj is a polynomial func-tion δλ-homogeneous of degree σj − n (unless aj ≡ 0). Hence, the degree of δλ-homogeneity of X belongs to the set of real (possibly negative) numbers

Aj = {σj − |α|σ : α ∈ (N ∪ {0})N },whenever j is such that aj is not identically zero.

Proof. A direct computation shows the “if” part of the proposition. Vice versa, ifX(ϕ ◦ δλ) = λn(X ϕ) ◦ δλ, the choice ϕ(x) = xj yields λσj aj (x) = λn aj (δλ(x)),whence aj is a (smooth) δλ-homogeneous function of degree σj − n. By Proposi-tion 1.3.4, aj is a polynomial function. �

For example, the differential operators

X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 (1.52)

on R3 are δλ-homogeneous of degree one with respect to the dilation

δλ(x1, x2, x3) = (λx1, λx2, λ2x3).

Also, the vector fields x31 X1 = x3

1∂x1 + 2 x31x2 ∂x3 and x2 X2 = x2∂x1 − 2 x1x2 ∂x3

are respectively δλ-homogeneous of degrees −2 and 0 w.r.t. the same dilation.


Corollary 1.3.6. Let δλ be as in (1.49). Let X be a smooth non-vanishing vector field.Then X is δλ-homogeneous of degree n ∈ R iff

δλ

(XI (x)

) = λn XI (δλ(x)).

Proof. Let X =∑Nj=1 aj ∂xj

. By Proposition 1.3.5, X is δλ-homogeneous of degreen iff aj (δλ(x)) = λσj−n aj (x) for any j ∈ {1, . . . , N}. This is equivalent to

δλ(XI (x)) = δλ

(a1(x), . . . , aN(x)

)T = (λσ1a1(x), . . . , λσN aN(x)

)T

= λn(a1(δλ(x)), . . . , aN(δλ(x))

)T = λn XI (δλ(x)).

This ends the proof. � As a straightforward consequence, we have the following simple fact.

Remark 1.3.7. Let δλ be as in (1.49). Let X �= 0 be a smooth vector field on RN of

the form

X =N∑

j=1

aj (x) ∂xj.

If X is δλ-homogeneous of degree n ∈ R, then, for every aj non-identically zero,we must have n ≤ σj . As a consequence, it has to be n ≤ σN (i.e. the set of the δλ-homogeneous degrees of the smooth vector fields is bounded above by the maximumexponent of the dilation). Hence, X has the form

X =∑

j≤N, σj≥n

aj (x) ∂/∂xj .

Suppose now n > 0. Since aj is a polynomial function of degree σj − n and n > 0,then aj does not depend on xj , . . . , xN ,

aj (x) = aj (x1, . . . , xj−1)

(we agree to let aj (x1, . . . , xj−1) = constant when j = 1). We already highlightedin the previous section the importance of these “pyramid”-shaped vector fields (seeRemark 1.1.3).

From this remark the next proposition straightforwardly follows.

Proposition 1.3.8. Let δλ be as in (1.49). Let X = ∑Nj=1 aj (x) ∂xj

be a smoothvector field δλ-homogeneous of positive degree. Then its adjoint operator X∗ =−∑N

j=1 ∂j (aj ·) satisfies X∗ = −X and

X2 = div(A · ∇T ), (1.53)

where A is the square matrix (ai aj )i,j≤N . Finally, X has null divergence.


Proof. By the previous remark, the coefficient aj does not depend on xj . Then, forevery smooth function ϕ,

X∗ϕ = −N∑

j=1

∂j (aj ϕ) = −N∑

j=1

aj ∂jϕ = −Xϕ.

Moreover, it holds

X2 =N∑

i,j=1

ai∂i(aj ∂j ) =N∑

i=1

∂i

(N∑

j=1

ai aj ∂j

)= div(A · ∇T ),

where A is as in the statement of the proposition. Finally, div(XI) =∑Nj=1 ∂j (aj ) =

0, since aj is independent of xj . � Vector fields with different degree of δλ-homogeneity are linearly independent if

they do not vanish at the origin. Indeed, the following proposition holds.

Proposition 1.3.9. Let δλ be as in (1.49). Let X1, . . . , Xk ∈ T (RN) be δλ-homo-geneous vector fields of degree n1, . . . , nk , respectively.

If ni �= nj for i �= j and if XjI (0) �= 0 for every j ∈ {1, . . . , k}, then X1, . . . , Xk

are linearly independent.

Proof. Let c1, . . . , ck ∈ R be such that∑k

j=1 cj Xj = 0. Then, for every smoothfunction ϕ,

0 =k∑

j=1

cj Xj (ϕ(δλx)) =k∑

j=1

cj λnj (Xjϕ)(δλx) ∀ x ∈ RN.

If we take ϕ(x) = 〈h, x〉 =∑Nj=1 hj xj , this identity at x = 0 gives

0 =k∑

j=1

cj λnj 〈ηj , h〉 ∀ h ∈ RN, ∀ λ > 0,

where ηj = XjI (0). Equivalently,

0 =⟨

k∑

j=1

cj λnj ηj , h

⟩.

Due to the arbitrariness of h ∈ RN , this gives

∑kj=1 cj λnj ηj = 0 for all λ > 0, so

that (since ni �= nj if i �= j )

cj ηj = 0 for any j ∈ {1, . . . , k}.This implies cj = 0, since, for every j = 1, . . . , k, ηj �= 0 by the hypothesis. �


The following simple proposition will be useful in the sequel.

Proposition 1.3.10. Let δλ be as in (1.49). Let X1, X2 be δλ-homogeneous vectorfields of degree n1, n2, respectively. Then [X1, X2] is δλ-homogeneous of degreen1 + n2 (unless X1 and X2 commute).

As a consequence, if n1, n2 are both positive, then every commutator of X1, X2containing k1 times X1 and k2 times X2 vanish identically whenever k1 n1 +k2 n2 > σN .

Proof. It suffices to note that, for every smooth function ϕ on RN , one has

(X1X2)(ϕ(δλ(x))) = λn2 X1((X2ϕ)(δλ(x))) = λn2+n1 (X1X2)(ϕ(δλ(x))).

This proves the first part of the assertion, since [X1, X2] = X1X2 − X2X1 (and[X1, X2] ≡ 0 iff X1 and X2 commute).

Finally, let X be a commutator of X1, X2 containing k1 times X1 and k2 timesX2. By the first part of this proof, it follows inductively that X is δλ-homogeneousof degree k1 n1 + k2 n2 (unless X ≡ 0). By Remark 1.3.7, we know that if a smoothvector field is δλ-homogeneous of degree n ∈ R, then n ≤ σN . This ends theproof. �

For example, the differential operators X1 = ∂x1 + 2x2∂x3 , X2 = ∂x2 − 2x1∂x3

on the Heisenberg–Weyl group H1 are homogeneous of degree one with respect to

the dilation δλ(x1, x2, x3) = (λx1, λx2, λ2x3), and [X1, X2] = −4∂x3 is indeed δλ-

homogeneous of degree two. Moreover, any commutator of X1, X2 of length ≥ 3vanish identically, as stated in the last part of Proposition 1.3.10.

Corollary 1.3.11. Let G = (RN, ◦, δλ) be a homogeneous Lie group on RN , and

let g be the Lie algebra of G. Let X1, . . . , Xk ∈ g be non-identically vanishing andδλ-homogeneous of degrees n1, . . . , nk , respectively. If ni �= nj for i �= j , thenX1, . . . , Xk are linearly independent.

Proof. Since XjI (x) = Jτx (0)·XjI (0) for every x ∈ RN , and Xj is non-identically

vanishing, then XjI (0) �= 0 for any j ∈ {1, . . . , k}. Hence the assertion follows fromthe previous proposition. � Proposition 1.3.12 (Nilpotence of homogeneous Lie groups on R

NRN

RN ). Let G =

(RN, ◦, δλ) be a homogeneous Lie group on RN , and let g be the Lie algebra of G.

Then G is nilpotent of step ≤ σN , i.e. every commutator of vector fields in g contain-ing more than σN terms vanishes identically.

Moreover, if Zj is the j -th element of the Jacobian basis of g, Zj is δλ-homogeneous of degree σj .

Proof. Let Zj be the j -th element of the Jacobian basis of g. By Proposi-tion 1.2.16-(5), for every ϕ ∈ C∞(RN, R), we have

Zj (ϕ(δλ(x))) = d

dt

∣∣∣∣t=0

(ϕ(δλ(x ◦ (tej )))) = d

dt

∣∣∣∣t=0

(ϕ(δλ(x) ◦ δλ(tej )))

= λσjd

dr

∣∣∣∣r=0

(ϕ(δλ(x) ◦ (r ej ))) = λσj (Zjϕ)(δλ(x)),


i.e. Zj is δλ-homogeneous of degree σj . This proves the last part of the assertion.Since (Z1, . . . , ZN) is a linear basis for g, the above arguments show that

every X ∈ g is a linear combination of δλ-homogeneous smooth vector fields ofdegrees at least σ1 ≥ 1. The first part of the assertion hence follows from Proposi-tion 1.3.10. �

1.3.2 The Composition Law of a Homogeneous Lie Group

By using the elementary properties of the homogeneous functions showed in theprevious section, we shall obtain a structure theorem for the composition law in ahomogeneous Lie group (RN, ◦, δλ). We first prove two lemmas.

Lemma 1.3.13. Let δλ be as in (1.49). Let P : RN × R

N → R be a smooth non-vanishing function such that

P(δλ(x), δλ(y)) = λσj P (x, y) ∀ x, y ∈ RN, ∀ λ > 0,

for some j such that 1 ≤ j ≤ N . Assume also that

P(x, 0) = xj , P (0, y) = yj . (1.54)

Then P(x, y) = x1 + y1 if j = 1 and, if j ≥ 2,

P(x, y) = xj + yj + P (x1, . . . , xj−1, y1, . . . , yj−1),

where P is a polynomial, the sum of mixed monomials in x1, . . . , xj−1, y1, . . . , yj−1.Moreover, P (δλ(x), δλ(y)) = λσj P (x, y). Finally, P(x, y) only depends on the xk’sand yk’s with σk < σj .

Proof. By Proposition 1.3.4, P is a polynomial function of the following type:

P(x, y) =∑

|α|σ+|β|σ=σj

cα,β xα yβ, cα,β ∈ R.

On the other hand, by (1.54),

xj = P(x, 0) =∑

|α|σ=σj

cα,0 xα

andyj = P(0, y) =

∑

|β|σ=σj

c0,β yα.

ThenP(x, y) = xj + yj +

∑

|α|σ+|β|σ=σj , α,β �=0

cα,β xα yβ. (1.55)

We can complete the proof by noticing that the condition |α|σ + |β|σ = σj , α, β �=0 is empty when j = 1, whereas it implies α = (α1, . . . , αj−1, 0, . . . , 0), β =(β1, . . . , βj−1, 0, . . . , 0) when j ≥ 2.

As for the last assertion of the lemma, being α, β �= 0 in the sum in the right-handside of (1.55), the sum itself may depend only on the α’s and β’s with |α|σ , |β|σ <

σj , hence, on the xk’s and yk’s with σk < σj . �


Lemma 1.3.14. Let δλ be as in (1.49). Let Q : RN ×RN → R be a smooth function

such that

Q(δλ(x), δλ(y)) = λm Q(x, y) ∀ x, y ∈ RN, ∀ λ > 0,

where m ≥ 0. Then

x �→ ∂ Q

∂ yj

(x, 0)

is δλ-homogeneous of degree m− σj (unless it vanishes identically).

Proof. By Proposition 1.3.13, Q is a polynomial of the following type

Q(x, y) =∑

|α|σ+|β|σ=m

cα,βxαyβ.

Then, denoting by ej the j -th element of the canonical basis of RN , we have

∂ Q

∂ yj

(x, y) =∑

|α|σ+|β|σ=m

cα,ββjxα yβ−ej ,

so that, since |ej |σ = σj ,

∂ Q

∂ yj

(x, 0) =∑

|α|σ=m−σj ,β=ej

cα,βxα.

This completes the proof. � Now, we are in the position to prove the previously mentioned structure theorem

for the composition law of a homogeneous Lie group on RN .

Theorem 1.3.15 (Composition of a homogeneous Lie group on RNRN

RN ). Let

(RN, ◦, δλ) be a homogeneous Lie group on RN . Then ◦ has polynomial component

functions. Furthermore, we have

(x ◦ y)1 = x1 + y1, (x ◦ y)j = xj + yj +Qj(x, y), 2 ≤ j ≤ N,

and the following facts hold:

1. Qj only depends on x1, . . . , xj−1 and y1, . . . , yj−1;2. Qj is a sum of mixed monomials in x, y;3. Qj(δλx, δλy) = λσj Qj (x, y).

More precisely, Qj(x, y) only depends on the xk’s and yk’s with σk < σj .

Proof. Let j ∈ {1, . . . , N}, and define

Pj : RN × RN → R, Pj (x, y) = (x ◦ y)j .

Since δλ is an automorphism of G, we have


Pj (δλ(x), δλ(y)) = (δλ(x ◦ y))j = λσj (x ◦ y)j = λσj Pj (x, y).

Moreover, since x ◦ 0 = x, 0 ◦ y = y, we have

Pj (x, 0) = xj , Pj (0, y) = yj .

Then the proof follows from Lemma 1.3.13. � For example, the Lie group G = (R3, ◦) considered in Example 1.2.18 is not

homogeneous with respect to any dilation on R3, since its composition

x ◦ y = (arcsinh(sinh(x1)+ sinh(y1)), x2 + y2 + sinh(x1)y3, x3 + y3)

does not fulfill the requirements of Theorem 1.3.15.

Corollary 1.3.16 (Inversion of a homogeneous Lie group on RNRN

RN ). Let G =

(RN, ◦, δλ) be a homogeneous Lie group on RN . Let j ∈ {1, . . . , N}. For every

y ∈ G, we have(y−1)j = −yj + qj (y),

where qj (y) is a polynomial function in y, δλ-homogeneous of degree σj , only de-pending on the yk’s with σk < σj .

Proof. We letp : RN → R, p(y) = y−1.

For every y ∈ G, we have

0 = δλ(y−1 ◦ y) = δλ(y

−1) ◦ δλ(y),

whence δλ(y−1) = (δλ(y))−1, i.e.

δλ(p(y)) = p(δλ(y)) ∀ y ∈ RN. (1.56)

If j ∈ {1, . . . , N} is fixed and pj is the j -th component function of p, (1.56) meansthat pj is δλ-homogeneous of degree σj .

As a consequence, since the inversion is a smooth map (by definition of Liegroup), we can apply Proposition 1.3.4 and infer that pj is a polynomial function,δλ-homogeneous of degree σj .

Now, we exploit the explicit form of the composition in Theorem 1.3.15. If x ◦y = 0, then we have

(�) xj = −yj whenever σj = 1.

Hence, if σj = 2, we have xj = −yj + Qj(x, y), where Qj only depends on thexk’s and yk’s with σk = 1. As a consequence of (�), we infer

xj = −yj + qj (y) whenever σj = 2,

where qj only depends on the yk’s with σk = 1. An inductive argument now provesthat xj = −yj + qj (y), where qj only depends on the yk’s with σk < σj , and theproof is complete. �


Example 1.3.17. Let us consider on R4 the composition law

x ◦ y =

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + y1x2

x4 + y4 + y1x22 + 2 x2y3

⎞

⎟⎟⎠ .

Then ◦ equips R4 with a homogeneous Lie group structure with dilations

δλ(x1, x2, x3, x4) = (λx1, λx2, λ2 x3, λ

3 x4).

Notice that the inversion on this group is given by

y−1 =

⎛

⎜⎜⎝

−y1−y2

−y3 + y1 y2

−y4 − y1 y22 + 2 y2 y3

⎞

⎟⎟⎠ .

Corollary 1.3.18. Let G = (RN, ◦, δλ) be a homogeneous Lie group on RN . Let

j ∈ {1, . . . , N}. For every x, y ∈ G, we have

(y−1 ◦ x)j = xj − yj +∑

k:σk<σj

P(j)k (x, y)(xk − yk),

where P(j)k (x, y) is a polynomial function in x and y only depending on the xk’s and

yk’s with σk < σj .

Proof. We fix j ∈ {1, . . . , N} and y ∈ RN , and we let

fy : RN → R, fy(x) := (y−1 ◦ x)j .

By Theorem 1.3.15, fy is a polynomial function on RN . Hence, Taylor’s formula

centered at ξ ∈ RN gives

fy(x) =∑

α

(Dαfy)(ξ)

α! (x − ξ)α, (1.57)

where the sum is finite. Choose ξ = y and note that the summand with α = 0in (1.57) is fy(ξ) = fy(y) = (y−1 ◦ y)j = 0. Consequently, (1.57) gives

(y−1 ◦ x)j =∑

α �=0

(Dαfy)(y)

α! (x − y)α. (1.58)

Note that any summand in (1.58) contains at least one factor of the type xk − yk

(since α �= 0). On the other hand, by Theorem 1.3.15, we have

(y−1 ◦ x)j = (y−1)j + xj +Qj(y−1, x), (1.59)


where Qj(x, y) only depends on the xk’s and yk’s with σk < σj . Then, by Corol-lary 1.3.16, (1.59) gives

(y−1 ◦ x)j = xj − yj + qj (y)+Qj(y−1, x), (1.60)

where qj (y) is a polynomial function in y, only depending on the yk’s withσk < σj . Note that Qj(y

−1, x) depends on the xk’s with σk < σj and the (y−1)k’swith σk < σj . Exploiting once again Corollary 1.3.16, this proves that Qj(y

−1, x)

depends on the xk’s and the yk’s with σk < σj .As a consequence, (1.60) proves that

(y−1 ◦ x)j = xj − yj + Qj (x, y), (1.61)

where Qj (x, y) is a polynomial function only depending on the xk’s and yk’s withσk < σj . By collecting together (1.58) and (1.61), we immediately get the assertionof the corollary. �

For example, on the Heisenberg–Weyl group H1, we have

y−1 ◦ x = (x1 − y1, x2 − y2, x3 − y3 + 2 (−y2 x1 + y1 x2))

= (x1 − y1, x2 − y2, x3 − y3 − 2 y2(x1 − y1)+ 2 y1(x2 − y2)).

Also, the mapx ◦ y = (x1 + y1, x2 + y2, x3 + y3 + x1 y2)

endows R3 with a homogeneous Lie group structure. Since the inversion is

y−1 = (−y1,−y2,−y3 + y1 y2),

we have

y−1 ◦ x = (x1 − y1, x2 − y2, x3 − y3 + 2 (−y2 x1 + y1 x2))

= (x1 − y1, x2 − y2, x3 − y3 − y1(x2 − y2)).

Note that y−1 ◦ x contains also non-mixed monomials of degree > 1,

y−1 ◦ x = (x1 − y1, x2 − y2, x3 − y3 − y1 x2 + y1y2).

The following result describes in a very explicit way the Jacobian matrix at 0 of theleft-translation τx on a homogeneous Lie group on R

N .

Corollary 1.3.19 (The Jacobian basis of a homogeneous Lie group). Let G =(RN, ◦, δλ) be a homogeneous Lie group on R

N . Then we have

Jτx (0) =

⎛

⎜⎜⎜⎜⎝

1 0 · · · 0

a(1)2 1

. . ....

.... . .

. . . 0a

(1)N · · · a

(N−1)N 1

⎞

⎟⎟⎟⎟⎠, (1.62)


where a(j)i is a polynomial function δλ-homogeneous of degree σi − σj . As a conse-

quence, if we let

Zj = ∂xj+

N∑

i=j+1

a(j)i ∂xi

for 1 ≤ j ≤ N − 1 and ZN = ∂xN,

then Zj is a left-invariant vector field δλ-homogeneous of degree σj . Moreover,

Jτx (0) = (Z1I (x) · · ·ZNI (x)).

In other words, the Jacobian basis Z1, . . . , ZN for the Lie algebra g of G is formedby δλ-homogeneous vector fields of degree σ1, . . . , σN , respectively.

Proof. By Theorem 1.3.15, the Jacobian matrix Jτx (0) takes the form (1.62) with

a(j)i (x) = ∂ Qi

∂ yj

(x, 0).

Then, by Lemma 1.3.14, a(j)i (x) is a polynomial function, δλ-homogeneous of degree

σi − σj . This proves the first part of the corollary. The second one follows fromProposition 1.3.12. � Example 1.3.20. In Example 1.2.6, we showed that the Jacobian matrix of the lefttranslation on H

1 is

Jτx (0) =( 1 0 0

0 1 02x2 −2x1 1

).

We recognize that the three columns of this matrix give raise to the Jacobian basisZ1 = ∂x1 + 2 x2 ∂x3 , Z2 = ∂x2 − 2 x1 ∂x3 and Z3 = ∂x3 and these vector fieldsare homogeneous of degree, respectively, 1, 1, 2 with respect to δλ(x1, x2, x3) =(λx1, λx2, λ

2x3). � The structure Theorem 1.3.15 of the composition law of (RN, ◦, δλ) implies that

the Lebesgue measure on RN is invariant under left and right translations on G.

Indeed, by Theorem 1.3.15, the Jacobian matrices of the functions x �→ α ◦ x andx �→ x ◦ α have the following lower triangular form

⎛

⎜⎜⎜⎜⎝

1 0 · · · 0

� 1. . .

......

. . .. . . 0

� · · · � 1

⎞

⎟⎟⎟⎟⎠.

Then, we have proved the following proposition.

Proposition 1.3.21 (Haar measure on a homogeneous Lie group). Let G be a ho-mogeneous Lie group on R

N . Then the Lebesgue measure on RN is invariant with

respect to the left and the right translations on G.


The above proposition is also restated as: the Lebesgue measure on RN is the

Haar measure for G.If we denote by |E| the Lebesgue measure of a measurable set E ⊆ R

N , we thenhave

|α ◦ E| = |E| = |E ◦ α| ∀ α ∈ G.

We also have that the Lebesgue measure is homogeneous with respect to the dilations{δλ}λ>0. More precisely, as a trivial computation shows,

|δλ(E)| = λQ |E|,where

Q =N∑

j=1

σj . (1.63)

The positive number Q is called the homogeneous dimension of the group G =(RN, ◦, δλ).

For example, in the case of the Heisenberg–Weyl group H1, where τα is given by

τα(x) = (α1 + x1, α2 + x2, α3 + x3 + 2 (α2 x1 − α1 x2)),

and δλ(x1, x2, x3) = (λx1, λx2, λ2x3), we have

Jτα (x) =( 1 0 0

0 1 02α2 −2α1 1

), Jδλ(x) =

(λ 0 00 λ 00 0 λ2

),

so that, for every α, x ∈ H1 and every λ > 0, we have

detJτα (x) = 1, detJδλ(x) = λ4 = λQ,

since the homogeneous dimension of H1 is Q = 1+ 1+ 2 = 4.

1.3.3 The Lie Algebra of a Homogeneous Lie Group on RNRN

RN

The following remark holds.

Remark 1.3.22. Let G be a homogeneous Lie group on RN with Lie algebra g. From

Corollary 1.3.19 we easily obtain the splitting of g as a direct sum of linear spacesspanned by vector fields of constant degree of δλ-homogeneity.

More precisely, let us recall that the exponents σj ’s in the dilation δλ of G (see(H.1) in Definition 1.3.1) satisfy σ1 ≤ · · · ≤ σN and can then be grouped together toproduce real and natural numbers, respectively, say

n1, . . . , nr and N1, . . . , Nr,

such thatn1 < n2 < · · · < nr, N1 +N2 + · · · +Nr = N,


defined by⎧⎪⎪⎪⎨

⎪⎪⎪⎩

n1 = σj for 1 ≤ j ≤ N1,

n2 = σj for N1 < j ≤ N1 +N2,...

nr = σj for N1 + · · · +Nr−1 < j ≤ N1 + · · · +Nr−1 +Nr.

Let Z1, . . . , ZN be the Jacobian basis of g. Define

g1 = span{Zj | 1 ≤ j ≤ N1} and, for i = 2, . . . , r,

gi = span{Zj |N1 + · · · +Ni−1 < j ≤ N1 + · · · +Ni−1 +Ni}.By Corollary 1.3.19, the generators Zj ’s of gi are δλ-homogeneous vector fields ofdegree ni , 1 ≤ i ≤ r . Moreover, we obviously have

g = g1 ⊕ · · · ⊕ gr .

We also explicitly notice that, by Proposition 1.3.9, a vector field X ∈ g is δλ-homogeneous of degree n iff, for a suitable i ∈ {1, . . . , r}, n = ni and X ∈ gi .

In the next section, we shall deal with homogeneous groups in which ni = i for1 ≤ i ≤ r , and the layer (or slice) gi , i ∈ {1, . . . , r}, is precisely generated by thecommutators of length i of the vector fields in g1.

Example 1.3.23. The usual additive group (R3,+) is a homogeneous Lie group ifequipped with the dilation

δλ(x1, x2, x3) = (λ2x1, λπx2, λ

4x3).

The decomposition of the Lie algebra as in Remark 1.3.22 is

span{∂x1} ⊕ span{∂x2} ⊕ span{∂x3}.Moreover, R

4 is a homogeneous Lie group if equipped with the group law

x ◦ y =

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + 2 y1 x2 − 2 y2 x1x4 + y4

⎞

⎟⎟⎠

and the dilation

δλ(x1, x2, x3, x4) = (λx1, λx2, λ2x3, λ

2x4).


g1 ⊕ g2 = span{X1, X2} ⊕ span{∂x3 , ∂x4},where X1 = ∂x1+2 x2 ∂x3 , X2 = ∂x2−2 x1 ∂x3 . Note that (see the notation in (1.17))


[g1, g1] � g2.

Observe that the above (R4, ◦) is isomorphic to the homogeneous Lie group (R4, ∗)with the composition law

ξ ∗ η =

⎛

⎜⎜⎝

ξ1 + η1ξ2 + η2ξ3 + η3

ξ4 + η4 + 2 η1 ξ2 − 2 η2 ξ1

⎞

⎟⎟⎠

and the new group of dilations

δλ(ξ1, ξ2, ξ3, ξ4) = (λξ1, λξ2, λξ3, λ2x4).


g1 ⊕ g2 = span{Z1, Z2, ∂ξ3} ⊕ span{∂ξ4},where Z1 = ∂ξ1 + 2 ξ2 ∂ξ4 , Z2 = ∂ξ2 − 2 ξ1 ∂ξ4 . Note that this time we have

[g1, g1] = g2.

Definition 1.3.24 (Dilations on the Lie algebra of a homogeneous Lie group). LetG = (RN, ◦, δλ) be a homogeneous Lie group on R

N with Lie algebra g and dilation

δλ(x1, . . . , xN) = (λσ1x1, . . . , λσN xN).

We define a group of dilations on g (which we still denote by δλ) as follows: δλ isthe (only) linear (auto)morphism of g mapping the j -th element Zj of the Jacobianbasis for g into λσj Zj .

In other words, if X ∈ g is written w.r.t. the Jacobian basis Z1, . . . , ZN as

X =N∑

j=1

cj Zj , we then have δλ(X) =N∑

j=1

cj λσj Zj .

We immediately recognize that, if π : g→ RN is the map defined by π(X) = XI (0)

(see also Remark 1.2.20), it holds

π(δλ(X)) = δλ(π(X)) ∀ X ∈ g. (1.64)

Indeed, we have

δλ(π(X)) = δλ

(π

(N∑

j=1

cj Zj

))= δλ

(N∑

j=1

cj π(Zj )

)

= δλ

(N∑

j=1

cj (Zj )I (0)

)= δλ(c1, . . . , cN)

= (λσ1 c1, . . . , λσN cN)


and, on the other hand,

π(δλ(X)) = π

(δλ

(N∑

j=1

cj Zj

))= π

(N∑

j=1

cj λσj Zj

)

=N∑

j=1

cjλσj π(Zj ) = (λσ1 c1, . . . , λ

σN cN).

The following simple and very useful fact holds.

Proposition 1.3.25. Let G be a homogeneous Lie group on RN with Lie algebra g.

The dilation on g introduced in Definition 1.3.24 is a Lie algebra automorphism ofg, i.e.

δλ([X, Y ]) = [δλ(X), δλ(Y )] ∀ X, Y ∈ g. (1.65)

Proof. First we remark that, for every i, j ∈ {1, . . . , N},δλ([Zi, Zj ]) = λσi+σj [Zi, Zj ]. (1.66)

Indeed, since Zi and Zj are δλ-homogeneous of degrees σi and σj , respectively, then[Zi, Zj ] is a δλ-homogeneous vector field of degree σi+σj (see Proposition 1.3.10).This implies that, if we express [Zi, Zj ] w.r.t. the Jacobian basis

[Zi, Zj ] =N∑

k=1

ck Zk,

then the sum runs over the k’s such that

σk = σi + σj (1.67)

(here we use, as a crucial tool, Corollary 1.3.11). Consequently,

δλ([Zi, Zj ]) = δλ

(N∑

k=1

ck Zk

)=

N∑

k=1

ckλσkZk

(by (1.67)) = λσi+σj

N∑

k=1

ck Zk = λσi+σj [Zi, Zj ].

This proves (1.66).Let now X =∑N

i=1 xi Zi and Y =∑Nj=1 yj Zj . Then we have

δλ([X, Y ]) = δλ

([N∑

i=1

xi Zi,

N∑

j=1

yj Zj

])

= δλ

(N∑

i,j=1

xi yj [Zi, Zj ])


=N∑

i,j=1

xi yj δλ([Zi, Zj ]) (see (1.66))

=N∑

i,j=1

xi yj λσi+σj ([Zi, Zj ])

=[

N∑

i=1

ci λσi Zi,

N∑

j=1

cj λσj Zj

]= [δλ(X), δλ(Y )].

This completes the proof. �

1.3.4 The Exponential Map of a Homogeneous Lie Group

Let G = (RN, ◦, δλ) be a homogeneous Lie group on RN with Lie algebra g. The ex-

ponential map on g has some remarkable properties, due to the homogeneous struc-ture of G. We prove such properties in what follows.

Let Z1, . . . , ZN be the Jacobian basis of g. By Corollary 1.3.19, Zj is δλ-homogeneous of degree σj and takes the form

Zj =N∑

k=j

a(j)k (x1, . . . , xk−1)∂xk

, (1.68)

where a(j)k is a polynomial function δλ-homogeneous of degree σk−σj and a

(j)j ≡ 1.

We now consider on g the dilation group introduced in Definition 1.3.24, i.e. withabuse of notation (soon justified)

δλ : g −→ g, δλ

(N∑

j=1

ξjZj

):=

N∑

j=1

λσj ξjZj . (1.69)

Remark 1.3.26 (Consistency of the dilations on g and G). The dilation (1.69) is con-sistent with the one in G. More precisely, if Z ∈ g then, for every λ > 0, it holds

δλ(ZI (x)) = (δλZ)I (δλ(x)) ∀ x ∈ G. (1.70)

We first check this identity in the case Z = Zj , j = 1, . . . , N . Since Zj

is δλ-homogeneous of degree σj , by Corollary 1.3.6, we have δλ(Zj I (x)) =λσj (Zj I )(δλ(x)), so that (see (1.69))

δλ(Zj I (x)) = (δλZj )I (δλ(x)).

Then, given Z =∑Nj=1 ξj Zj ∈ g, we have (since δλ is linear on g)

δλ(ZI (x)) =N∑

j=1

ξj δλ(Zj I (x)) =N∑

j=1

ξj

((δλZj )I (δλ(x))

)

=(

N∑

j=1

ξj (δλZj )

)I (δλ(x)) = (δλZ)I (δλ(x)).


From the previous remark, we easily obtain the following lemma.

Lemma 1.3.27. Let G = (RN, ◦, δλ) be a homogeneous Lie group on RN with Lie

algebra g. Denote also by δλ the dilation (1.69) on g.Let γ : [0, T ] → R

N be an integral curve of Z with Z ∈ g. Then Γ := δλ(γ ) isan integral curve of δλ(Z).

Proof. Identity (1.70) gives

Γ = δλ(γ ) = δλ(ZI (γ )) = (δλZ)I (δλ(γ )) = (δλZ)I (Γ ).

This ends the proof. � We are now in the position to prove the following important theorem.

Theorem 1.3.28 (Exponential map of a homogeneous Lie group). LetG = (RN, ◦, δλ) be a homogeneous Lie group with Lie algebra g. Then


are globally defined diffeomorphisms with polynomial component functions (pro-vided g is equipped with its vector space structure and any fixed system of linearcoordinates).

Moreover, denote also by δλ the dilation on g defined in (1.69). Then, for everyZ ∈ g and x ∈ G, it holds

Exp(δλ(Z)

) = δλ(Exp (Z)) and Log (δλ(x)) = δλ(Log (x)). (1.71)

Proof. Let Z ∈ g, Z =∑Nj=1 ξj Zj . From (1.68) we obtain

Z =N∑

k=1

(k∑

j=1

ξj a(j)k (x1, . . . , xk−1)

)∂xk

. (1.72)

Then the system of ODE’s defining Exp (Z) is “pyramid”-shaped, and the first part ofthe theorem follows from Remark 1.1.3. In order to prove the first identity in (1.71),we consider the solution γ to the Cauchy problem

γ = ZI (γ ), γ (0) = 0.

By the very definition of Exp (Z), we have γ (1) = Exp (Z). Let us put Γ = δλ(γ ).By Lemma 1.3.27, Γ is an integral curve of δλ(Z). Moreover, Γ (0) = δλ(γ (0)) =δλ(0) = 0. Then Γ (1) = Exp (δλ(Z)), so that

Exp (δλ(Z)) = Γ (1) = δλ(γ (1)) = δλ(Exp (Z)).

This proves the first identity in (1.71). The second one is trivially equivalent to thefirst one. �


The first part of Theorem 1.3.28 together with (1.40) and Proposition 1.2.29(page 27) give the following corollary.

Corollary 1.3.29. For every x, y ∈ G, we have

x ◦ y = exp(Log (y))(x) and x−1 = Exp (−Log (x)).

Remark 1.3.30 (Exp and Log preserve the mass). If Z is the vector field (1.72), then

ZI (x) =(

ξ1, ξ2 + ξ1 a(1)2 (x1), . . . , ξN +

N−1∑

j=1

a(j)N (x1, . . . , xN−1)

).

This implies (see (1.13), page 10)

Exp (Z) = exp(Z)(0) = (ξ1, ξ2 + B2(ξ1), . . . , ξN + BN(ξ1, . . . , ξN−1)

), (1.73)

where the Bj ’s are suitable polynomial functions. Then the Jacobian matrix of themap

RN � (ξ1, . . . , ξN ) �→ Exp (ξ1 Z1 + · · · + ξN ZN) ∈ R

N

takes the following form ⎛

⎜⎜⎜⎝

1 0 · · · 0

� 1. . .

......

. . .. . . 0

� · · · � 1

⎞

⎟⎟⎟⎠ . (1.74)

Thus, with respect to the Jacobian basis of g and the canonical basis of G ≡ RN ,

Exp preserves the Lebesgue measure. The same property holds for the map Log ,since Log = (Exp )−1.

Equivalently, if Z1, . . . , ZN is the Jacobian basis for g and, as usual, ξ · Z =∑Nj=1 ξj Zj , then Exp and Log have the following remarkable forms

Exp (ξ · Z) =

⎛

⎜⎜⎝

ξ1ξ2 + B2(ξ1)

...

ξN + BN(ξ1, . . . , ξN−1)

⎞

⎟⎟⎠ (1.75a)

and

Log (x) =

⎛

⎜⎜⎝

x1x2 + C2(x1)

...

xN + BN(x1, . . . , xN−1)

⎞

⎟⎟⎠ · Z, (1.75b)

where the Bi’s and Ci’s are polynomial functions (δλ-homogeneous of degree σi)completely determined by the composition law on G.


For example, the above results are readily verified for H1, since in that case (as

we proved in Example 1.2.28, page 27) Exp is represented by the identity matrix(if the algebra of H

1 is equipped with the Jacobian basis). A little more elaboratedexample is given below.

Example 1.3.31. Let us consider on R4 the following composition law (we denote

the points of R4 by x = (x1, x2, x3, x4)):

x ◦ y =

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + y1x2

x4 + y4 + y1x22 + 2 x2y3

⎞

⎟⎟⎠ .

It is readily verified that ◦ equips R4 with a homogeneous Lie group structure, pro-

vided that the dilation group is given by

δλ(x1, x2, x3, x4) := (λx1, λx2, λ2 x3, λ

3 x4).

Let us construct the exponential map. To begin with, the first two vector fieldsX1, X2 of the related Jacobian basis can be found as follows (see (1.33)): for everyϕ ∈ C∞(R4) and every x ∈ R

4, we have

(X1ϕ)(x) = ∂

∂y1

∣∣∣∣y=0

ϕ(x ◦ y), (X2ϕ)(x) = ∂

∂y2

∣∣∣∣y=0

ϕ(x ◦ y),

so that the chain rule straightforwardly gives

X1 = ∂x1 + x2 ∂x3 + x22 ∂x4 , X2 = ∂x2 .

A direct computation shows that

[X1, X2] = −∂3 − 2 x2 ∂4, [X1, [X1, X2]] = 0, [X2, [X1, X2]] = −2 ∂4,

whereas all commutators of length > 3 vanish. We now remark that

X1, X2, [X1, X2] and [X2, [X1, X2]]satisfy the following properties:

– they are left-invariant w.r.t. ◦ (as iterated commutators of left-invariant vectorfields, see also (1.19));

– they are linearly independent vector fields10;– they form a basis of g (since dim(g) = 4, see Proposition 1.2.7).

10 For example, we can notice that their respective evaluations at 0⎛

⎜⎝

1000

⎞

⎟⎠ ,

⎛

⎜⎝

0100

⎞

⎟⎠ ,

⎛

⎜⎝

00−10

⎞

⎟⎠ ,

⎛

⎜⎝

000−2

⎞

⎟⎠

are linearly independent vectors of R4 (and use Proposition 1.2.13).


We now set W1 := X1, W2 := X2, W3 := [X1, X2], W4 := [X2, [X1, X2]], and,for ξ ∈ R

4, we also let ξ ·W := ξ1W1 + ξ2W2 + ξ3W3 + ξ4W4. By the definitionin (1.8), we have exp(ξ ·W)(x) = γ (1), where γ (s) solves

{γ (s) = (ξ ·W)I (γ (s)) = (ξ1, ξ2, ξ1γ2 − ξ3, ξ1γ

22 − 2 ξ3γ2 − 2ξ4),

γ (0) = x.

Solving the above system of ODE’s, one gets

exp(ξ ·W)(x) =

⎛

⎜⎜⎝

x1 + ξ1x2 + ξ2

x3 − ξ3 + 12 ξ1ξ2 + ξ1x2

x4 − 2ξ4 + 13 ξ1ξ

22 − ξ2ξ3 + ξ1x

22 + ξ1ξ2x2 − 2ξ3x2

⎞

⎟⎟⎠ .

As a consequence, by Definition 1.2.25, we obtain

Exp (ξ ·W) = exp(ξ ·W)(0) =

⎛

⎜⎜⎝

ξ1ξ2

−ξ3 + 12 ξ1ξ2

−2ξ4 + 13 ξ1ξ

22 − ξ2ξ3

⎞

⎟⎟⎠ ,

so that the inverse map of Exp is given by

Log (x) =

⎛

⎜⎜⎝

x1x2

−x3 + 12 x1x2

− 12 x4 − 1

12 x1x22 + 1

2 x2x3

⎞

⎟⎟⎠ ·W.

One can now directly check11 the validity of Theorem 1.3.28, Corollary 1.3.29 andRemark 1.3.30. We explicitly remark the vectors Wi’s do not form the Jacobian basisfor g, which, instead, is given by

Z1 = W1, Z2 = W2, Z3 = −W3, Z4 = −1

2W4.

Hence, if we write ξ · Z := ξ1 Z1 + ξ2 Z2 + ξ3 Z3 + ξ4 Z4, we see that

(ξ, ξ2, ξ3, ξ4) · Z =(

ξ, ξ2,−ξ3,−1

2ξ4

)· Z,

11 For example, we see that the inverse of x ∈ G is given by

x−1 = Exp(−Log (x)

)=

⎛

⎜⎜⎝

−x1−x2

−x3 + x1x2−x4 + 2 x2x3 − x1x2

2

⎞

⎟⎟⎠ .


whence, with respect to the Jacobian coordinates, the exponential and the logarithmicmaps are respectively given by

Exp (ξ · Z) =

⎛

⎜⎜⎝

ξ1ξ2

ξ3 + 12 ξ1ξ2

ξ4 + 13 ξ1ξ

22 + ξ2ξ3

⎞

⎟⎟⎠ ,

Log (x) =

⎛

⎜⎜⎝

x1x2

x3 − 12 x1x2

x4 + 16 x1x

22 − x2x3

⎞

⎟⎟⎠ · Z.

Compare this to (1.74). � Theorem 1.3.28 has many important consequences. We collect some of them in

the following remark.

Remark 1.3.32. From Theorem 1.3.28 we infer, in particular, that


are globally defined C∞ maps. Hence, by Remark 1.2.32, the operation on g

X � Y := Log (Exp (X) ◦ Exp (Y )), X, Y ∈ g, (1.76)

defines a Lie group structure isomorphic to (G, ◦). We consider on g the dilation(still denoted by δλ) introduced in Definition 1.3.24. We claim that

δλ is a Lie group automorphism of (g,�),i.e.

δλ(X � Y) = (δλ(X)) � (δλ(Y )) ∀ X, Y ∈ g. (1.77)

Roughly speaking, (g,�, δλ) is a homogeneous Lie group too. To prove the claim,we notice that

δλ(X � Y) = δλ{Log (Exp (X) ◦ Exp (Y ))}(see (1.71)) = Log {δλ(Exp (X) ◦ Exp (Y ))}

= Log {(δλ(Exp (X))) ◦ (δλ(Exp (Y )))}(see (1.71)) = Log {(Exp (δλ(X))) ◦ (Exp (δλ(Y )))}

= (δλ(X)) � (δλ(Y )).

This proves our claim.12

12 We explicitly remark that, if we already knew that � is defined by a “universal” compositionof iterated Lie brackets, we could derive (1.77) from (1.65) in Proposition 1.3.25. Indeed,

δλ(X � Y ) = δλ

(X + Y + 1

2[X,Y ] + 1

12[X, [X,Y ]] + · · ·

)

= δλ(X)+ δλ(Y )+ 1

2[δλ(X), δλ(Y )] + 1

12[δλ(X), [δλ(X), δλ(Y )]] + · · ·

= (δλ(X)) � (δλ(Y )),

the second identity following by a repeated application of (1.65).


We now identify g with RN taking coordinates with respect to the Jacobian basis.

In other words, we consider the map (see Remark 1.2.20)

π : g → RN, X �→ π(X) := XI (0).

Again, we transfer the Lie group structure of (g,�) into a Lie group (RN, ∗) in thenatural way, by setting

ξ ∗ η := π(π−1(ξ) � π−1(η)), ξ, η ∈ RN. (1.78)

As a consequence, (RN, ∗) is isomorphic to (g,�) and hence to (G, ◦). We finallyconsider on R

N the same dilation δλ defined on G (this makes sense, since the un-derlying manifold for G is R

N too).We claim that

(RN, ∗, δλ) is a homogeneous Lie group.

In other words, we have to show that δλ is a Lie group automorphism of (RN, ∗).This follows from the argument below,

δλ(ξ ∗ η) = δλ{π(π−1(ξ) � π−1(η))}(see (1.64)) = π{δλ(π

−1(ξ) � π−1(η))}(see (1.77)) = π{(δλ(π

−1(ξ))) � (δλ(π−1(η)))}

(see (1.64)) = π{(π−1(δλ(ξ))) � (π−1(δλ(η)))} = (δλ(ξ)) ∗ (δλ(η)).

We can summarize the above remarked facts as follows.Given a homogeneous Lie group G = (RN, ◦, δλ), we can consider a somewhat

“more canonical” homogeneous Lie group on RN

C-H(G) := (RN, ∗, δλ)

(which we may call “of Campbell–Hausdorff type”) obtained by the natural identi-fication of the Lie algebra of G (equipped with the Campbell–Hausdorff composi-tion law � in (1.76)) to R

N (via coordinates w.r.t. the Jacobian basis). (See also theFig. 1.6.)

Example 1.3.33. Let us consider the homogeneous Lie group on R3 with the dilation

δλ(x) = δλ(x1, x2, x3) = (λx1, λx2, λ2x3)

and the composition law defined by

x ◦ y =⎛

⎝x1 + y1x2 + y2

x3 + y3 + x1 y2

⎞

⎠ .

The Jacobian basis for g (the Lie algebra of G) is


Fig. 1.6. Figure related to Remark 1.3.32

Z1 = ∂1, Z2 = ∂2 + x1 ∂3, Z3 = [Z1, Z2] = ∂3.

For ξ = (ξ1, ξ2, ξ3) ∈ R3, we fix the notation ξ · Z :=∑3

i=1 ξi Zi ∈ g. It is easy toshow that the exponential and logarithmic maps are given by

Exp (ξ · Z) =⎛

⎝ξ1ξ2

ξ3 + ξ1ξ22

⎞

⎠ , Log (x) =⎛

⎝x1x2

x3 − x1x22

⎞

⎠ · Z.

Hence, the map ∗ considered in (1.78) is given by

ξ ∗ η =⎛

⎝ξ1 + η1ξ2 + η2

ξ3 + η3 + 12 (ξ1η2 − ξ2η1)

⎞

⎠ .

The Jacobian basis related to C-H(G) = (R3, ∗, δλ) is

Z1 = ∂1 − ξ2

2∂3, Z2 = ∂2 + ξ1

2∂3, Z3 = [Z1, Z2] = ∂3.

Now, it is interesting to see what happens if we iterate this “C-G” process. It iseasy to see that, if we consider once again the group obtained from C-H(G) in thesame way (i.e. C-H(C-H(G))), we obtain nothing else than C-H(G) itself (a rigorousformulation of this fact will be given in Proposition 2.2.24).

We remark that G and C-H(G) are isomorphic and the canonical sub-Laplacianof G

ΔG = {(∂/∂x1)}2 + {(∂/∂x2)+ x1(∂/∂x3)}2is “equivalent” to the canonical sub-Laplacian of C-H(G)

ΔC-H(G) ={(∂/∂ξ1)− ξ2

2(∂/∂x3)

}2

+{(∂/∂ξ2)+ ξ1

2(∂/∂x3)

}2

.


1.4 Homogeneous Carnot Groups

We now enter into the core of the chapter by introducing the central definition of thisbook. Our definition here of Carnot group will be properly compared to the classicalone in Section 2.2, page 121.

Definition 1.4.1 (Homogeneous Carnot group). We say that a Lie group on RN ,

G = (RN, ◦), is a (homogeneous) Carnot group or a homogeneous stratified group,if the following properties hold:

(C.1) RN can be split as R

N = RN1 × · · · × R

Nr , and the dilation δλ : RN → RN

δλ(x) = δλ(x(1), . . . , x(r)) = (λx(1), λ2x(2), . . . , λrx(r)), x(i) ∈ R

Ni ,

is an automorphism of the group G for every λ > 0.

Then (RN, ◦, δλ) is a homogeneous Lie group on RN , according to Defini-

tion 1.3.1. Moreover, the following condition holds:

(C.2) If N1 is as above, let Z1, . . . , ZN1 be the left invariant vector fields on G suchthat Zj (0) = ∂/∂xj |0 for j = 1, . . . , N1. Then13

rank(Lie{Z1, . . . , ZN1}(x)) = N for every x ∈ RN .

If (C.1) and (C.2) are satisfied, we shall say that the triple G = (RN, ◦, δλ) is ahomogeneous Carnot group.

We also say that G has step r and N1 generators. The vector fields Z1, . . . , ZN1

will be called the (Jacobian) generators of G, whereas any basis for span{Z1, . . . ,

ZN1} is called a system of generators of G.

(Note. As already remarked for Lie groups on RN and homogeneous ones, the

notion of homogeneous Carnot group is not coordinate-free. This fact will not dis-tract us from recognizing its importance.)

In the sequel, we use the following notation to denote the points of G

x = (x1, . . . , xN) = (x(1), . . . , x(r)) (1.79a)

withx(i) = (x

(i)1 , . . . , x

(i)Ni

) ∈ RNi , i = 1, . . . , r. (1.79b)

Furthermore, we shall denote by g the Lie algebra of G.

Remark 1.4.2 (Equivalent definition of homogeneous Carnot group). An equivalentdefinition of homogeneous Carnot group can be given:

Suppose G = (RN, ◦) is a Lie group on RN , and there exist positive real numbers

τ1 ≤ · · · ≤ τN such that dλ(x) = (λτ1x1, . . . , λτN xN) is a Lie group morphism of

G for every λ > 0. Let g be the Lie algebra of G, and let g1 be the linear subspace

13 See the notation in Definition 1.1.5.

1.4 Homogeneous Carnot Groups 57

of g of the left-invariant vector fields which are dλ-homogeneous of degree τ1. If g1Lie-generates14 the whole g, then G is a homogeneous Carnot group according toDefinition 1.4.1. Precisely, G has step r := τN/τ1, it has m := dim(g1) generators,and it is a homogeneous Lie group with respect to the dilation

δλ = dλ1/τ1 .

Also, set σj := τj /τ1, then {σ1, σ2, . . . , σN } are consecutive integers starting from 1up to r . A sketch of the proof is in order.

As we observed in Remark 1.3.2, δλ is a morphism of (G, ◦), i.e. G = (RN, ◦, δλ)

is a homogeneous Lie group on RN . Obviously, X ∈ g1 if and only if X is δλ-

homogeneous of degree 1. Let ν be the maximum of the integers k’s such that σk = 1.Let us denote by {Z1, . . . , ZN } the Jacobian basis related to G and observe that (byProposition 1.3.12), for every j ≤ N , Zj is δλ-homogeneous of degree σj . We claimthat

(�) ν = dim(g1) =: m, and {Z1, . . . , Zm} is a basis for g1.

Indeed, let X ∈ g1. Then X = ξ1Z1 + · · · + ξNZN for suitable scalars ξj ’s. Since X

is δλ-homogeneous of degree 1, by Corollary 1.3.11 and the definition of ν, it holdsξj = 0 for every j > ν. Hence, g1 is spanned by {Z1, . . . , Zν} whence (this systemof vectors being linearly independent) the claimed (�) holds.

By the assumption Lie(g1) = g and (�), it follows

(��) Lie(Z1, . . . , Zm) = g.

For every j ∈ N, j ≥ 2, let us set (see the notation in (1.17)) gj := [g1, gj−1]. ByProposition 1.3.12, gj = {0} for every j > r := σN . Also, by Proposition 1.3.10,any X ∈ gj is δλ-homogeneous of degree j . Let now j ∈ {m + 1, . . . , N} be fixed.Then, by (��), Zj is a linear combination of nested commutators of Z1, . . . , Zm.But any such commutator is δλ-homogeneous of an integer degree in 1, . . . , r . Thisproves that σj (the δλ-homogeneous degree of Zj ) is integer and (again from Corol-lary 1.3.11) σj ∈ {1, . . . , r}. As a consequence, we have the splitting of R

N , asrequested in (C.1) of Definition 1.4.1, with N1 = m.

Finally, let us prove that (C.2) holds too. This is obvious thanks to (��), since(see the notation in Definition 1.1.5)

rank(g(x)) ≥ rank(Z1I (x), . . . , ZNI (x)

) = rank(Z1I (0), . . . , ZNI (0)

) = N

for every x ∈ G (see Proposition 1.2.13). � Example 1.4.3. The Heisenberg–Weyl group H

1 is a Carnot group of steptwo and two generators. Indeed, it is a homogeneous Lie group (with dilationsδλ(x1, x2, x3) = (λx1, λx2, λ

2x3)). Moreover (since the first two vector fields ofthe Jacobian basis are Z1 = ∂x1 + 2x2∂x3 and Z2 = ∂x2 − 2x1∂x3 ), we have

rank(Lie{Z1, Z2}(x)) = 3 for every x ∈ R3,

14 This means that Lie(g1) = g, see the notation in Proposition 1.1.7.


as we proved in Example 1.1.6. Thus, the above properties (C.1) and (C.2) are ful-filled.

We now give an example of a homogeneous Lie group which is not a Carnotgroup. Let us consider the following composition law on R

2

(x1, x2) ◦ (y1, y2) = (x1 + y1, x2 + y2 + x1 y1).

It can be readily verified that G = (R2, ◦) is a Lie group (here (x1, x2)−1 =

(−x1,−x2 + x21)). Moreover, G is a homogeneous group, if equipped with the di-

lation δλ(x1, x2) := (λx1, λ2x2). Hence (C.1) is satisfied. However, (C.2) is not.

Indeed, if Z1 = ∂x1 + x1 ∂x2 is the first vector field of the Jacobian basis, we have

rank(Lie{Z1}(x)) = 1 �= 2 for every x ∈ R2.

Hence G is not a homogeneous Carnot group.Finally, let us remark that the triple (R2,+, δλ) is a homogeneous Carnot group

if δλ(x1, x2) = (λx1, λx2), whereas if δλ(x1, x2) = (λx1, λ2x2), (R2,+, δλ) is a

homogeneous Lie group but not a Carnot one. � From properties (C.1) and (C.2) of Definition 1.4.1 and the results on the ho-

mogeneous Lie groups showed in Section 1.3 we immediately get the assertionscontained in the following remarks.

Remark 1.4.4. Let (RN, ◦, δλ) be a homogeneous Carnot group. Then ◦ has polyno-mial component functions. Moreover, following the notation in (1.79a) and denotingx ◦ y by ((x ◦ y)(1), . . . , (x ◦ y)(r)), we have

(x ◦ y)(1) = x(1) + y(1), (x ◦ y)(i) = x(i) + y(i) +Q(i)(x, y), 2 ≤ i ≤ r,

where

1. Q(i) only depends on x(1), . . . , x(i−1) and y(1), . . . , y(i−1);2. the component functions of Q(i) are sums of mixed monomials in x, y;3. Q(i)(δλx, δλy) = λiQ(i)(x, y).

Remark 1.4.5. Let (RN, ◦, δλ) be a homogeneous Carnot group. Then we have

Jτx (0) =

⎛

⎜⎜⎜⎜⎝

IN1 0 · · · 0

J(1)2 (x) IN2

. . ....

.... . .

. . . 0J

(1)r (x) · · · J

(r−1)r (x) INr

⎞

⎟⎟⎟⎟⎠, (1.80)

where In is the n × n identity matrix, whereas J(i)j (x) is a Nj × Ni matrix whose

entries are δλ-homogeneous polynomials of degree j − i. In particular, if we let

Jτx (0) = (Z(1)(x) · · ·Z(r)(x)

),

where Z(i)(x) is a N × Ni matrix, then the column vectors of Z(i)(x) define δλ-homogeneous vector fields of degree i: those of the relevant Jacobian basis.


Remark 1.4.6. Let G = (RN, ◦, δλ) be a homogeneous Carnot group with Lie alge-bra g. Let Z1, . . . , ZN be the Jacobian basis of g, i.e.

Zj ∈ g and Zj (0) = ∂xj|0, j = 1, . . . , N.

With a notation consistent with (1.79a) and (1.79b), we shall also denote the Jacobianbasis by

Z(1)1 , . . . , Z

(1)N1; . . . ; Z

(r)1 , . . . , Z

(r)Nr

.

Obviously, Z(1)j = Zj for 1 ≤ j ≤ N1. By Corollary 1.3.19, Z(i)

j is δλ-homogeneousof degree i and takes the form

Z(i)j = ∂/∂x

(i)j +

r∑

h=i+1

Nh∑

k=1

a(i,h)j,k (x(1), . . . , x(h−i)) ∂/∂x

(h)k , (1.81)

where a(i,h)j,k is a δλ-homogeneous polynomial function of degree h − i. In par-

ticular, the Jacobian generators of G, i.e. the vector fields Z(1)1 , . . . , Z

(1)N1

are δλ-homogeneous of degree 1. � Remark 1.4.7. With the notation of the above remark, the Lie algebra g is generatedby Z1, . . . ZN1 ,

g = Lie{Z1, . . . ZN1}. (1.82)

Indeed, the inclusion Lie{Z1, . . . ZN1} ⊆ g is obvious. Since dim(g) = N , in orderto show the opposite inclusion, it is enough to prove that

dim(Lie{Z1, . . . ZN1}) = N.

By condition (C.2), there exists X1, . . . , XN ∈ Lie{Z1, . . . ZN1} such that X1I (0),

. . . , XNI (0) are linearly independent vectors in RN . Then (by Proposition 1.2.13)

X1, . . . , XN are linearly independent in g. Hence

N ≥ dim(Lie{Z1, . . . ZN1}) ≥ N,

and this ends the proof. � Remark 1.4.8 (Stratification of the algebra of a homogeneous Carnot group). Letthe notation of Remark 1.4.6 be employed. Let us denote by W(k) the vector spacespanned by the commutators of length k of Z1, . . . , ZN1 ,

W(k) := span{ZJ | J ∈ {1, . . . , N1}k

}.

Obviously, W(k) ⊆ g, and every Z ∈ W(k) is δλ-homogeneous of degree k. Then, byCorollary 1.3.11 and Proposition 1.3.12, W(k) = {0} if k > r , while

W(k) ⊆ span{Z(k)1 , . . . , Z

(k)Nk} if 2 ≤ k ≤ r . (1.83)


Then, if we agree to let

W(1) = span{Z1, . . . , ZN1} = span{Z(1)1 , . . . , Z

(1)N1},

we havedim(W(k)) ≤ Nk for any k ∈ {1, . . . , r}. (1.84)

On the other hand, by Proposition 1.1.7,

span{W(1), . . . ,W(r)} = Lie{Z(1)1 , . . . , Z

(1)N1}.

Thus, by Remark 1.4.7,

g = span{W(1), . . . , W(r)},so that, since W(h) ∩W(k) = {0} if h �= k (see Corollary 1.3.11), we have

g = W(1) ⊕W(2) ⊕ · · · ⊕W(r).

As a consequence,

dim(g) =r∑

k=1

dim(W(k)).

On the other hand, dim(g) = N =∑rk=1 Nk . Then, by (1.84),

dim(W(k)) = Nk for any k ∈ {1, . . . , r},and, by (1.83),

W(k) = span{Z(k)1 , . . . , Z

(k)Nk} if 1 ≤ k ≤ r .

We also have[W(1),W(i−1)] = W(i) for 2 ≤ k ≤ r (1.85a)

and[W(1),W(r)] = {0}. (1.85b)

Indeed, let us put V1 := W(1) and

Vi := [V1, Vi−1] for i = 2, . . . , r .

By the definition of W(k) and Proposition 1.1.7, Vi ⊆ W(i) for i = 2, . . . , r . Thendim(Vi) ≤ dim(W(i)) = Ni . On the other hand, by Proposition 1.3.12, [V1, Vr ] ={0}, and, by Proposition 1.1.7,

g = Lie{Z(1)1 , . . . , Z

(1)N1} = span{V1, V2, . . . , Vr }.

Then N = ∑ri=1 dim(Vi) ≤ ∑r

i=1 Ni = N . This implies dim(Vi) = Ni for everyi ∈ {1, . . . , r}. As a consequence, Vi = W(i) for every i ∈ {1, . . . , r}, and (1.85a)and (1.85b) hold.


Summing up, we have proved the “stratification” of the Lie algebra g, i.e. thedecomposition

g = W(1) ⊕W(2) ⊕ · · · ⊕W(r)

with

[W(1),W(i−1)] = W(i) for 2 ≤ k ≤ r,

[W(1),W(r)] = {0},where

W(k) = span{Z(k)1 , . . . , Z

(k)Nk} if 1 ≤ k ≤ r .

Remark 1.4.9. Following all the notation and definitions in Remark 1.3.32, if G =(RN, ◦, δλ) is a homogeneous Carnot group, then the Lie group C-H(G) :=(RN, ∗, δλ) obtained by the natural identification of the algebra of G to R

N is ahomogeneous Carnot group too. The proof of this fact is left to the reader as anexercise.

Remark 1.4.10 (Stratified change of basis on a homogeneous Carnot group). Let(RN, ◦, δλ) be a homogeneous Carnot group according to the Definition 1.4.1. Asusual, we denote the points of G by

x = (x(1), . . . , x(r)) with x(i) ∈ RNi

and the dilation group by δλ(x) = (λx(1), . . . , λrx(r)). Let

C(1), . . . , C(r)

be r fixed non-singular matrices with C(i) of dimension Ni × Ni for every i =1, . . . , r . We denote by C the N×N matrix having C(1), . . . , C(r) as diagonal blocksand 0’s elsewhere, i.e.

C =⎛

⎝C(1) · · · 0

.... . .

...

0 · · · C(r)

⎞

⎠ .

Finally, we denote again by C the relevant linear change of basis on RN , i.e. the

linear mapC : RN → R

N, C(x) = C · x.

We define on RN a new composition law ∗ obtained by writing ◦ in the new coordi-

nates defined by ξ = C(x). More precisely, we have

ξ ∗ η := C((C−1(ξ)) ◦ (C−1(η))

) ∀ ξ, η ∈ RN. (1.86)

We claim that H = (RN, ∗, δλ) is a homogeneous Carnot group isomorphic to G =(RN, ◦, δλ). The proof of this (not obvious) assertion is left as an exercise (see alsoSection 16.3 of Chapter 16, page 637).


1.5 The Sub-Laplacians on a Homogeneous Carnot Group

Definition 1.5.1 (Sub-Laplacian on a homogeneous Carnot group). If Z1, . . . ,

ZN1 are the Jacobian generators of the homogeneous Carnot group G = (RN, ◦, δλ),the second order differential operator

ΔG =N1∑

j=1

Z2j (1.87)

is called the canonical sub-Laplacian on G. Any operator

L =N1∑

j=1

Y 2j (1.88)

where Y1, . . . , YN1 is a basis of span{Z1, . . . , ZN1}, is simply called a sub-Laplacianon G. The vector valued operator

∇G = (Z1, . . . , ZN1) (1.89)

will be called the canonical (or horizontal) G-gradient.Finally, if L is as in (1.88), the notation ∇L = (Y1, . . . , YN1) will be used to

denote the L-gradient (or horizontal L-gradient).

Example 1.5.2. The canonical sub-Laplacian of the Heisenberg–Weyl group H1 is

ΔH1 = {∂x1 + 2 x2 ∂x3}2 + {∂x2 − 2 x1 ∂x3}2= (∂x1)

2 + (∂x2)2 + 4(x2

1 + x22) (∂x3)

2 + 4 x2 ∂x1,x3 − 4 x1 ∂x2,x3 .

A (non-canonical) sub-Laplacian on H1 is, for example,

L = {(∂x1 + 2 x2 ∂x3)− (∂x2 − 2 x1 ∂x3)}2 + {∂x2 − 2 x1 ∂x3}2= (∂x1)

2 + 2(∂x2)2 + 4(x2

1 + (x1 + x2)2) (∂x3)

2

− 2∂x1,x2 + 4(x1 + x2) ∂x1,x3 − 4(x1 + (x1 + x2)) ∂x2,x3 .

The following one is a (non-canonical!) sub-Laplacian on the classical additive group(R2,+)

L = {2 ∂x1 − 5 ∂x2}2 + {−∂x1 + 3 ∂x2}2 = 5(∂x1)2 + 34(∂x2)

2 − 26 ∂x1,x2 .

It is not difficult to verify that R4 equipped with the operation

x ◦ y =

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + 12 (y2x1 − y1x2)

x4 + y4 + 12 (y3x1 − y1x3)+ 1

12 (x1 − y1)(y2x1 − y1x2)

⎞

⎟⎟⎠

1.5 The Sub-Laplacians on a Homogeneous Carnot Group 63

and the dilation δλ(x1, x2, x3, x4) := (λx1, λx2, λ2 x3, λ

3 x4) is a homogeneousCarnot group, say G, whose canonical sub-Laplacian is

ΔG ={∂1 − 1

2x2∂3 − 1

2x3∂4 − 1

12x1x2∂4

}2

+{∂2 + 1

2x1∂3 + 1

12x2

1∂4

}2

= ∂11 + ∂22 + 1

4(x2

1 + x22)∂33 + x1∂23 − x2∂13

+ 1

4

((x2

1

6

)2

+(

x3 + x1x2

6

)2)∂44 + x2

1

6∂24 −

(x3 + x1x2

6

)∂14

+ 1

2

(x1

(x2

1

6

)+ x2

(x3 + x1x2

6

))∂34 + 1

6x2 ∂4.

We notice that ΔG also contains a first order (underlined) partial differential opera-tor.15 �

We would like to list some basic properties of the sub-Laplacians, straightforwardconsequences of the properties of the vector fields Z

(1)1 , . . . , Z

(1)N1

. In what follows

L =∑N1j=1 Y 2

j will denote any sub-Laplacian on G.

(A0) L is hypoelliptic, i.e. every distributional solution to Lu = f is of class C∞whenever f is of class C∞ (see Section 5.10, page 280, for further comments).This follows from the celebrated Hörmander hypoellipticity theorem [Hor67,Theorem 1.1], recalled in the Preface (Theorem 1, page VIII) and the fact that,if L =∑N1

j=1 Y 2j , then the following rank-condition holds

rank Lie({Y1, . . . , YN1}(x)

) = N ∀ x ∈ RN.

This is an obvious consequence of hypothesis (C.2) in Definition 1.4.1, page 56.(A1) L is invariant with respect to the left translations on G, i.e. for every fixed

α ∈ G,

L(u(α ◦ x)) = (Lu)(α ◦ x) for every x ∈ G and every u ∈ C∞(RN).

This holds since the Yj ’s are left-translation invariant on G.(A2) L is δλ-homogeneous of degree two, i.e. for every fixed λ > 0,

L(u(δλ(x))) = λ2(Lu)(δλ(x)) for every x ∈ G and every u ∈ C∞(RN).

This holds since the Yj ’s are δλ-homogeneous of degree one, see Remark 1.4.6.

15 This cannot happen for the sub-Laplacians on groups of step two, provided the inversemap on the group is −x, i.e. any sub-Laplacian on a step-two homogeneous Carnot group(whose inverse map is −x) contains only second order coordinate partial derivatives (seeSection 3.2).


(A3) L can be written asL = div(A(x)∇T ), (1.90a)

where div denotes the divergence operator in RN , ∇ = (∂1, . . . , ∂N ), A is the

N ×N symmetric matrix

A(x) = σ(x) σ (x)T (1.90b)

and σ(x) is the N × N1 matrix whose columns are Y1I (x), . . . , YN1I (x). Inother words, A(x) is the Gram matrix of the system of vectors {Y1I (x), . . . ,

YN1I (x)}; hence, since these vectors are linearly independent for every x ∈ G,the rank of A(x) is N1 for every x ∈ G. Now, (1.90a) is a consequence of thefollowing computation

L =N1∑

k=1

Y 2k =

N1∑

k=1

N∑

i=1

(YkI )i(x) ∂i

(N∑

j=1

(YkI )j (x)∂j

)

=N∑

i=1

∂i

{N∑

j=1

(N1∑

k=1

(YkI )i(x) (YkI )j (x)

)∂j

},

since (YkI )i(x) does not depend on xi . Hence, this proves thatL = div(A(x)∇T )

with

A(x) =(

N1∑

k=1

(YkI )i(x) (YkI )j (x)

)

i,j=1,...,N

= σ(x) σ (x)T .

The matrix A takes the following block form

A =(

A1,1 A1,2A2,1 A2,2

), (1.91)

where Ai,j stands for a mi×mj matrix with polynomial entries, with m1 = N1and m2 = N−N1. Furthermore, A1,1 is constant and non-singular. Indeed, fora suitable non-singular matrix B = (bj,k)

N1j,k=1, we have

Yj =N1∑

k=1

bj,k Zk, j = 1, . . . , N1. (1.92)

On the other hand, see (1.81),

Zk = ∂k +N∑

i=N1+1

a(k)i ∂i = ∂k +

N∑

i=N1+1

∂i(a(k)i ·), (1.93)

where the a(k)i ’s are suitable polynomial functions independent of xi . Replac-

ing (1.93) in (1.92) and squaring, we obtain


L =N1∑

j=1

Y 2j =

N1∑

h,k=1

ah,k ∂h,k +∑

h,k≤N, h∨k>N1

∂h(ah,k∂k),

whereA1,1 = (ah,k)h,k≤N1 = BT B

is a constant N1 × N1 matrix. Moreover, when h ∨ k := max{h, k} > N1,then ah,k is a suitable polynomial function. We notice that if L = ΔG, thenB = IN1 , so that A1,1 = IN1 . The above computations also give the expressionof L with respect to the usual coordinate partial derivatives,

L =N1∑

k=1

Y 2k =

N∑

i,j=1

ai,j (x)∂i,j +N∑

j=1

bj (x) ∂j ,

where

ai,j (x) =N1∑

k=1

(YkI )i(x) (YkI )j (x), bj (x) =N1∑

k=1

Yk((YkI )j (x)).

Analogous formulas hold for general sum of squares of vector fields (see Ex. 4at the end of the chapter).

(A4) If x ∈ G is fixed and A(x) is the matrix in (1.90a), then the quadratic form inξ ∈ R

N

qL(x, ξ) := 〈A(x)ξ, ξ 〉is called the characteristic form of L. We have

qL(x, ξ) =N1∑

j=1

〈Yj I (x), ξ 〉2,

so that qL(x, ·) is obtained by formally replacing in L the coordinate deriva-tives ∂1, . . . , ∂N by ξ1, . . . , ξN . This can be easily seen from (1.90b), for wehave

qL(x, ξ) = 〈A(x) ξ, ξ 〉 = 〈σ(x) σ (x)T ξ, ξ 〉

= 〈σ(x)T ξ, σ (x)T ξ 〉 = |σ(x)T ξ |2 =N1∑

j=1

〈Yj I (x), ξ 〉2.

Then qL(x, ξ) ≥ 0 for every x ∈ G and every ξ ∈ RN , i.e. A(x) is positive

semi-definite for every x ∈ G. Moreover, qL(x, ξ) = 0 iff 〈Yj I (x), ξ 〉 = 0 forevery j ∈ {1, . . . , N1}.Hence, for a fixed x ∈ G, the set N(x) of vectors ξ ’s which annihilate thequadratic form related to A(x) is a linear space given by

N(x) := {ξ ∈ RN | qL(x, ξ) = 0} = (

Y1I (x))⊥ ∩ · · · ∩ (YN1I (x)

)⊥. (1.94)


We recall that, by Proposition 1.2.13, since Y1, . . . , YN1 are linearly indepen-dent in g, then Y1I (x), . . . , YN1I (x) are linearly independent in R

N for everyfixed x. Thus, if N1 < N , that is if r ≥ 2, for every x ∈ G there existsξ ∈ R

N \ {0} such that qL(x, ξ) = 0. More precisely, the set of isotropicvectors for the quadratic form related to A(x) is a linear subspace of R

N ofdimension N −N1, equal to the kernel16 of the matrix A(x). This means that

if r ≥ 2, then L is not elliptic at any point of G.

On the other hand, if N1 = N (that is the step r of G is 1) the block A1,1in (1.91) has dimension N × N , while the other blocks disappear. Then Lis a constant coefficient operator of the form L = ∑N

i,j=1 ai,j ∂i,j with A =(ai,j )i,j symmetric and strictly positive definite. Thus, we can summarize theseresults as follows:

The sub-Laplacian L is a second order differential operatorin divergence form with polynomial coefficients.

The characteristic form of L is positive semi-definite.If the step of G is ≥ 2, then L is not elliptic at any point of G.

If the step of G is 1, then L is an elliptic operatorwith constant coefficients.

Example 1.5.3. The canonical sub-Laplacian of the Heisenberg–Weyl groupH

1 has been written in Example 1.5.2: in that case the characteristic form is

q(x, ξ) = (ξ1)2 + (ξ2)

2 + 4(x21 + x2

2)(ξ3)2 + 4x2ξ1ξ3 − 4x1ξ2ξ3

= (ξ1 + 2x2ξ3)2 + (ξ2 − 2x1ξ3)

2.

Hence q(x, ξ) = 0 if and only if

〈(1, 0, 2 x2), ξ 〉 = 〈(0, 1,−2 x1), ξ 〉 = 0,

16 Indeed, given a N × N real symmetric matrix A, we denote by Isotr(A) the set of theisotropic vectors w.r.t. the quadratic form related to A, i.e.

Isotr(A) := {ξ ∈ RN | 〈Aξ, ξ〉 = 0}.

Obviously, it holds Ker(A) ⊆ Isotr(A). In general, the reverse inclusion does not neces-sarily hold (as Isotr(A) is not necessarily a vector space!) as the following example shows:when

A :=(

1 00 −1

),

we have Ker(A) = {(0, 0)}, whereas Isotr(A) = span{(1, 1)} ∪ span{(1,−1)}. However,if A is positive semi-definite, then Ker(A) = Isotr(A). Indeed, let R be a real, symmetricmatrix such that A = R2. If ξ ∈ Isotr(A), it holds

0 = 〈Aξ, ξ〉 = 〈R2ξ, ξ〉 = 〈RT Rξ, ξ〉 = 〈Rξ, Rξ〉 = ‖R ξ‖2.

Hence R ξ = 0, so that Aξ = R2ξ = RRξ = 0. This gives ξ ∈ Ker(A).


so that, for any fixed x ∈ H1, the set N(x) of vectors ξ ’s which annihilate the

relevant quadratic form is (see (1.94))

N(x) = (1, 0, 2 x2)⊥ ∩ (0, 1,−2 x1)

⊥ = span{(−2x2, 2x1, 1)}.Therefore, it is always one-dimensional. As we showed in (A4), N(x) can alsobe found as N(x) = Ker(A(x)), where

A(x) = σ(x) σ (x)T =( 1 0

0 12x2 −2x1

)·(

1 0 2x20 1 −2x1

)

=( 1 0 2x2

0 1 −2x12x2 −2x1 4(x2

1 + x22)

).

This kernel is one-dimensional, as A(x) has rank 2 for every x ∈ H1.

(A5) The sub-Laplacian L is the second order partial differential operator related tothe Dirichlet form

u �→∫|∇Lu|2 dx.

More precisely, let Ω ⊆ RN be an open set, and consider the functional

C∞(Ω, R) � u �→ J (u) = 1

2

∫

Ω

|∇Lu|2 dx, |∇Lu|2 =N1∑

j=1

(Yju)2.

Denoting by 〈 , 〉 the inner product in RN1 , we have

J (u+ h)− J (u) =∫

Ω

〈∇Lu,∇Lh〉 dx + J (h)

for every h ∈ C∞0 (Ω, R). We call critical point of J any function u ∈C∞(Ω, R) such that

∫

Ω

〈∇Lu,∇Lh〉 dx = 0 ∀h ∈ C∞0 (Ω, R).

Then, given u ∈ C∞(Ω, R), we have

u is a critical point of J if and only if Lu = 0 in Ω .

Indeed, since Y ∗j = −Yj , an integration by parts gives

∫

Ω

〈∇Lu,∇Lh〉 dx =N1∑

j=1

∫

Ω

YjuYjh dx = −N1∑

j=1

∫

Ω

(Y 2j u)h dx

= −∫

Ω

(Lu)h dx

for every u ∈ C∞(Ω, R) and h ∈ C∞0 (Ω, R).


1.5.1 The Horizontal L-gradient

We end this section with some useful results on the horizontal L-gradient.

Proposition 1.5.4. Let L = ∑N1j=1 X2

j be a sub-Laplacian on the homogeneousCarnot group G. Let u ∈ C∞(G, R) be such that Xju is a polynomial function ofG-degree not exceeding m for every j = 1, . . . , N1. Then u is a polynomial functionof G-degree not exceeding m+ 1.

Proof. Let Z1, . . . , ZN be the Jacobian basis of g, the Lie algebra of G. Since theXj ’s have polynomial coefficients and g = Lie{X1, . . . , XN1}, pk := Zku is a poly-nomial function, k = 1, . . . , N . Moreover, if we denote

δλ(x) = δλ(x1, . . . , xN) := (λσ1x1, . . . , λσN xN)

with 1 = σ1 ≤ · · · ≤ σN = r , keeping in mind the stratification of g, one easilyrecognizes that

degG pk ≤ m+ 1− σk, k = 1, . . . , N.

Now, by using (1.38) and (1.62) of Section 1.1, we see that

∂xju(x) =

N∑

k=j

pk(x) a(j)k (x−1), j = 1, . . . , N,

where y �→ a(j)k (y) is δλ-homogeneous of degree σk − σj . Then, since δλ(x

−1) =(δλ(x))−1 and the map x �→ x−1 = Exp (−Log x) has polynomial components (seeTheorem 1.3.28 and Corollary 1.3.29), the function x �→ a

(j)k (x−1) is a polynomial

δλ-homogeneous of degree σk − σj . It follows that

x �→ 〈∇u(x), x〉 =N∑

j=1

xj ∂xju(x)

is a polynomial function of G-degree not exceeding m+ 1. Therefore, also

u(x)− u(0) =∫ 1

0

d

dt

(u(tx)

)dt =

∫ 1

0〈∇u(tx), tx〉dt

t

is a polynomial function of G-degree not exceeding m+ 1. � In order to state the next corollary, we introduce a new notation. Let β =

(i1, . . . , ik) be a multi-index with components in the set {1, . . . , N1}. We set

Xβ := Xi1 ◦ · · · ◦Xik and |β| = k.

Corollary 1.5.5. Let the hypotheses of Proposition 1.5.4 hold. Let u ∈ C∞(G, R)

be such thatXβu = 0 ∀ β : |β| = m

for a suitable integer m ≥ 1. Then u is a polynomial function on G of G-degree notexceeding m− 1.


Proof. We argue by induction on m. By Proposition 1.5.6, the assertion holds ifm = 1.

Suppose it holds for m = p, and let us prove that it holds for m = p + 1. Now,if Xβu = 0 for any multi-index β with |β| = p + 1, then

Xγ (Xju) = 0 ∀ γ : |γ | = p

and for every j ∈ {1, . . . , N1}. Hence, by the induction assumption, Xju is a poly-nomial function of G-degree not exceeding p − 1 for every j ∈ {1, . . . , N1}. FromProposition 1.5.4 it follows that u is a polynomial function of G-degree not exceed-ing p. This completes the proof. � Proposition 1.5.6. Let Ω be an open and connected subset of the homogeneousCarnot group G. Let L be any sub-Laplacian on G.

Then a function u ∈ C1(Ω, R) is constant in Ω if and only if the relevant hori-zontal L-gradient ∇Lu vanishes identically on Ω .

Proof. It is obviously non-restrictive to suppose that L = ΔG.Suppose Z1u, . . . , ZN1u vanish identically on Ω . Since the Lie algebra of G is

given byLie{Z1, . . . ZN1}

(see (1.82)), then for every vector field Zj of the Jacobian basis, we have Zju ≡ 0.We end by applying Proposition 1.2.21. � Example 1.5.7. Our proof of Proposition 1.5.6, despite its simplicity, conceals a deepgeometric argument, which can be applied in more general situations (namely, forvector fields satisfying Hörmander’s condition). We describe the underlying geomet-ric idea by an explicit example, leaving to the reader the task to generalize it in moregeneral cases. (See also Chapter 19, where we study the so-called Carathéodory–Chow–Rashevsky connectivity theorem.)

Consider the Heisenberg–Weyl group H1 on R

3. The Jacobian generators of itsLie algebra are the vector fields

X1 = ∂1 + 2x2∂3, X2 = ∂2 − 2x1∂3.

Proposition 1.5.6 then states that if a C1-function u satisfies

X1u = 0 and X2u = 0 in a domain Ω ⊆ R3, (1.95)

then u is constant in Ω . For the sake of brevity, rather than applying some (simple)connectedness argument, we may suppose Ω = R

3.The basic idea is the following one. Suppose that two points x, y ∈ R

3 can bejoined by an integral curve γ of one of the fields ±X1 or ±X2, then u(x) = u(y).Indeed, suppose that γ : [0, T ] → R

3 is such that γ (0) = x, γ (T ) = y and

γ (s) = XI (γ (s)) for all s ∈ [0, T ],


where X is one of the fields ±X1 or ±X2. Then, taking into account (1.95), we have

u(y)− u(x) =∫ T

0

d

d s(u(γ (s))) ds =

∫ T

0〈(∇u)(γ (s)), γ (s)〉 ds =

=∫ T

0

⟨(∇u)(γ (s)),XI (γ (s))

⟩ds =

∫ T

0(Xu)(γ (s)) ds = 0,

whence u(x) = u(y). As a consequence, we can prove that u is constant, if we showthat

any couple of points in H1 can be joined by a finite sequence

of paths which are integral curves of ±X1 and ±X2. (1.96)

We shall refer to (1.96) by saying that H1 is (X1, X2)-connected.

The (X1, X2)-connectedness has a deep motivation, namely the fact that

X1, X2, [X1, X2] are linearly independent.17 (1.97)

In its turn, the fact that (1.97) implies the (X1, X2)-connectedness has a profoundmotivation too, mainly based on the so-called Campbell–Hausdorff formula, as weshall explain at the end of this section.

Before entering into the details of the Campbell–Hausdorff formula, we showhow simple the argument is in the case of H

1. Indeed, let us fix a point P0 :=(x1, x2, x3) ∈ H

1, and let us consider the path γ , integral curve of X1 starting fromthis point. As we showed in (1.5), we have

γ (s) = (x1 + s, x2, x3 + 2 x2 s). (1.98)

We denote this point by P1. We then proceed along the integral curve of X2 startingfrom P1: at the time s, we arrive to the following point (as a simple calculationshows)

P2 = (x1 + s, x2 + s, x3 + 2x2s − 2(x1 + s)s).

Moreover, we proceed along the integral curve of −X1 starting from P2: at the times, we arrive to the following point (it is enough to have in mind (1.98) and replace s

with −s)P3 = (x1, x2 + s, x3 − 2(x1 + s)s − 2s2).

Finally, we proceed along the integral curve of −X2 starting from P3: at the time s,we arrive to the following point (again, it is enough to notice that the integral curveof −X2 at time s coincides with the integral curve of X2 at time −s)

17 When X1, X2 are arbitrary smooth vector fields in R3, the result

(X1, X2, [X1, X2] linearly independent) �⇒ (R3 is [X1, X2]-connected),

is known as Carathéodory’s theorem. For a more general version of this result in RN (but

under some further assumptions on the fields Xi ’s) we refer the reader to Chapter 19.


P4 = (x1, x2, x3 − 4s2).

An analogous calculation shows that, if we start from (x1, x2, x3) and proceed alongthe integral curves of, respectively, X2, X1, −X2, −X1, we arrive to

P4 = (x1, x2, x3 + 4s2).

Being s arbitrary in P4 and P4, this shows that we can join any two points having thesame x1, x2-coordinates and the third one arbitrarily given.

We now start from (x1, x2, x3) and, along an integral curve of X1, at the times we arrive to (x1 + s, x2, x3 + 2x2s). By the preceding argument, keeping fixedthe first two coordinates, we can vary the third one, in order to arrive to the point(x1+ s, x2, x3) (after finitely many integral curves of ±X1, ±X2). Being s arbitrary,this shows that we can join any two points having the same x2, x3 coordinates andthe first one arbitrarily given. Finally, an obvious analogous argument shows thatwe can join any two points having the same x1, x3 coordinates and the second onearbitrarily given. All these facts together prove (1.96).

To end the section, we describe the reason why the validity of (1.97) implies the(X1, X2)-connectedness. For instance, let us denote by Γ (s) the point we obtain if(as we did above) we follow paths of X1, X2, −X1, −X2 for a time s, this is thesame as saying that we follow paths of

sX1, sX2, −sX1, −sX2

for unit time. More explicitly,

Γ (s) = exp(−sX2)(exp(−sX1)(exp(sX2)(exp(sX1)(x)))).

Very simple arguments (showed in the proof of Lemma 5.13.18, page 301) show that(see precisely (5.117), page 302)

lims→0

Γ (s)− x

s2= [X1, X2]I (x). (1.99)

Roughly speaking, (1.99) ensures that, by following suitable integral curves of±X1,±X2, we can arrive as close as we want to the endpoints of the integral curves of thecommutator [X1, X2]. So, if X1, X2, [X1, X2] span R

3 at every point, it is intuitivelyevident that we can connect any two points by suitable integral curves of±X1,±X2.

This argument becomes completely apparent if we make use of the so-calledCampbell–Hausdorff formula. For two vector fields X1, X2 generating an algebranilpotent of step two, the Campbell–Hausdorff formula states that18

exp(X2)(exp(X1)(x)) = exp

(X1 +X2 + 1

2[X1, X2]

)(x). (1.100)

18 Further details on the Campbell–Hausdorff formula can be found in Definition 2.2.11,Theorem 2.2.13 (page 129), in Lemma 4.2.4 (page 194) and, mostly, in Theorem 15.1.1(page 595).


We now apply three times formula (1.100), in order to simplify the above Γ (s). Thefollowing calculation then applies:

Γ (s) = exp(−sX2)(exp(−sX1)(exp(sX2)(exp(sX1)(x))))

= exp(−sX2)

(exp(−sX1)

(exp

(sX1 + sX2 + s2

2[X1, X2]

)(x)

))

= exp(−sX2)

(exp

(sX1 + sX2 + s2

2[X1, X2] − sX1 − s2

2[X2, X1]

)(x)

)

= exp(−sX2)(exp(sX2 + s2[X1, X2])(x))

= exp(sX2 + s2[X1, X2] − sX2)(x) = exp(s2[X1, X2])(x).

This says that, following suitable integral curves of ±X1, ±X2, we can arrivewherever the integral curves of the commutator [X1, X2] do arrive! So, again, ifX1, X2, [X1, X2] span R

3 at every point, we can obviously connect any two pointsby suitable integral curves of ±X1, ±X2.

We explicitly remark that the assumption that X1, X2 generate an algebra nilpo-tent of step two has made the above calculation very transparent. For general smoothvector fields, the Campbell–Hausdorff formula leads to a formula with a remainderterm,

exp(−sX2)(exp(−sX1)(exp(sX2)(exp(sX1)(x))))

= exp(s2[X1, X2] +Ox(s3))(x).

Bibliographical Notes. “Carnot” groups seem to owe their name after an paper byC. Carathéodory [Car09] (related to a mathematical model of thermodynamics) dated1909. The same denomination was then used in the school of M. Gromov [Gro96]and it is nowadays commonly used.

The definition of stratified group given in this chapter is seemingly different fromthe classical one by G.B. Folland [Fol75] and by G.B. Folland & E.M. Stein [FS82](see also M. Gromov [Gro96], P. Pansu [Pan89]).

Indeed, here we focused on stratified groups having an underlying homogeneousstructure. Nonetheless, we shall prove in Chapter 2 that any (abstract) stratified groupis canonically isomorphic to a homogeneous one, so that our definition here is non-restrictive and seems to be more operative and easier to deal with.

A direct approach to homogeneous Carnot groups can also be found in E.M. Stein[Ste81] or in N.T. Varopoulos, L. Saloff-Coste, T. Coulhon [VSC92]; see also P. Ha-jlasz and P. Koskela [HK00].

1.6 Exercises of Chapter 1 73

1.6 Exercises of Chapter 1

Ex. 1) Verify that the following operation

(x1, x2, x3) ◦ (y1, y2, y3)

:= (arcsinh(sinh(x1)+ sinh(y1)), x2 + y2 + sinh(x1)y3, x3 + y3)

endows R3 with a Lie group structure.

Ex. 2) a) Verify that, for every fixed α ∈ R, the following operation

⎛

⎜⎜⎜⎜⎝

x1 + y1,

x2 + y2,

x3 + y3 + 12 (x1 y2 − x2 y1),

x4 + y4 + 12 (x1 y3 − x3 y1)+ α

2 (x2 y3 − x3 y2)

+ 112 (x1 − y1) (x1 y2 − x2 y1)+ α

12 (x2 − y2) (x1 y2 − x2 y1)

⎞

⎟⎟⎟⎟⎠

defines on R4 a homogeneous Carnot group G.

b) Verify that the Jacobian basis for the algebra g of the above G is

Z1 = ∂1 − 1

2x2 ∂3 −

(1

2x3 + 1

12x2(x1 + α x2)

)∂4,

Z2 = ∂2 + 1

2x1 ∂3 +

(−1

2α x3 + 1

12x1(x1 + α x2)

)∂4,

Z3 = [Z1, Z2] = ∂3 +(

1

2x1 + 1

2α x2

)∂4,

Z4 = [Z1, [Z1, Z2]] = [Z1, Z3] = ∂4.

Verify that the only other non-trivial commutator identity is

[Z2, [Z1, Z2]] = α [Z1, [Z1, Z2]].c) Verify that the exponential map for G (written w.r.t. the Jacobian basis)

is the “identity map” in the following sense: if ξ = (ξ1, ξ2, ξ3, ξ4) ∈ R4,

and ξ · Z ∈ g denotes the vector field∑4

i=1 ξi Zi , then

Exp (ξ · Z) = ξ ∈ G.

Ex. 3) With reference to what we proved in Example 1.2.33 (page 30), prove that,for the group considered in Examples 1.2.18 and 1.2.31 (pages 21 and 28) itholds


) = X + Y + 1

2[X, Y ].

Ex. 4) In this exercise, we provide some compact formulas for general sum ofsquares of vector fields. Suppose we are assigned m vector fields of classat least C1 on R

N ,


Xk =N∑

j=1

αj,k ∂j , i.e. as usual, XkI =

⎛

⎜⎜⎝

α1,k

α2,k

...

αN,k

⎞

⎟⎟⎠ , k ≤ m.

Let S be the N ×m matrix whose k-th column vector is given by XkI ,

S :=⎛

⎝α1,1 α1,k α1,m

... · · · ... · · · ...

αN,1 αN,k αN,m

⎞

⎠ = (αj,k) j=1,...,N

k=1,...,m

.

With this notation, for every u ∈ C1(RN, R), we have

∇u · S = (X1u, . . . , Xmu) =: ∇Xu,

where ∇ = (∂1, . . . , ∂N ) is the usual gradient operator, and

∇X := (X1, . . . , Xm)

is the “intrinsic gradient” related to the family {X1, . . . , Xm}. We also definethe N ×N symmetric matrix

A := S · ST =(

m∑

k=1

αi,k αj,k

)

i,j=1,...,N

=: (ai,j )i,j≤N.

For every u ∈ C1(RN, R), it holds

|∇Xu|2 :=m∑

k=1

(Xku)2 = 〈∇Xu,∇Xu〉 = 〈∇u · S,∇u · S〉

= (∇u · S) · (∇u · S)T = ∇u · (S · ST ) · (∇u)T = ∇u · A · (∇u)T ,

i.e.

|∇Xu|2 =N∑

i,j=1

ai,j ∂iu ∂ju.

Show that, for every k = 1, . . . , m, it holds

X2k =

N∑

i,j=1

αi,k αj,k ∂2i,j +

N∑

j=1

(N∑

i=1

αi,k (∂iαj,k)

)∂j .

Derive that, in the coordinate form, the sum of squares related to the vectorfields Xk’s, L :=∑m

k=1 X2k , is given by


L =N∑

i,j=1

ai,j ∂2i,j +

N∑

j=1

bj ∂j ,

with ai,j =m∑

k=1

αi,k αj,k,

bj =m∑

k=1

N∑

i=1

αi,k (∂iαj,k) =m∑

k=1

Xkαj,k.

We note that bj is a sort of “X-divergence” of the j -th row of the matrix S.Besides, show that

X2k =

N∑

i=1

∂i

(N∑

j=1

αi,k αj,k ∂j

)−(

N∑

i=1

∂iαi,k

)·(

N∑

j=1

αj,k∂j

)

=N∑

i=1

∂i

(N∑

j=1

αi,k αj,k ∂j

)− div(XkI) ·Xk.

Deduce that L can be written in the following equivalent ways:

L =N∑

i=1

∂i

(N∑

j=1

m∑

k=1

αi,k αj,k ∂j

)−

N∑

j=1

(m∑

k=1

N∑

i=1

αj,k (∂iαi,k)

)∂j

=N∑

i=1

∂i

(N∑

j=1

ai,j ∂j

)−

N∑

j=1

(m∑

k=1

αj,k div(XkI)

)∂j

= div(A · ∇T )−m∑

k=1

div(XkI) ·Xk

= div(A · ∇T )−⟨S ·⎛

⎝div(X1I )

...

div(XmI)

⎞

⎠ ,∇T

⟩

= div(A · ∇T )− 〈(div(X1I ), . . . , div(XmI)),∇X〉.In particular, the sum of squares L is in divergence form if and only if

∀ j ≤ N,

m∑

k=1

N∑

i=1

αj,k (∂iαi,k) = 0, i.e. S ·⎛

⎝div(X1I )

...

div(XmI)

⎞

⎠ ≡ 0.

(1.101)For example, re-derive that any sub-Laplacian L on a homogeneous Carnotgroup is in divergence form, because in this case it holds div(XkI) =∑N

i=1 ∂iαi,k = 0, for αi,k does not depend on xi . We explicitly remark that, inthe case of non-homogeneous Carnot groups (which will be introduced in the


next chapter), this is not necessarily true, as the following example shows:R

3 equipped with the composition

x ◦ y = (arcsinh(sinh(x1)+ sinh(y1)), x2 + y2 + sinh(x1)y3, x3 + y3)

is a non-homogeneous Carnot group; the first two vector fields of the relevantJacobian basis are

X1 = (cosh(x1))−1 ∂1, X2 = ∂3 + sinh(x1) ∂2,

so that the canonical sub-Laplacian is not a divergence form operator, for(1.101) is not satisfied, being, for j = 1,

α1,1 div(X1I ) = − sinh(x1)/(cosh(x1))3.

Ex. 5) Prove the assertions made in Remark 1.4.9 (page 61).Ex. 6) Prove the following simple formulas of calculus for vector fields. Here,

u, f, g ∈ C2(RN, R), α ∈ C2(R, R), ϕ = (ϕ1, . . . , ϕN) ∈ C2(RN, RN),

X = ∇(·) · XI is a vector field on RN , L = ∑m

j=1 X2j is a sum of squares

of vector fields on RN , ∇L = (X1, . . . , Xm) is the intrinsic gradient related

to L.• X(f g) = (Xf ) g + f (Xg),• ∇L(f g) = (∇Lf ) g + f (∇Lg),• X2(f g) = (X2f ) g + 2 (Xf ) (Xg)+ f (X2g),• L(f g) = (Lf ) g + 2 〈∇Lf,∇Lg〉 + f (Lg),• X(α(u)) = α′(u)Xu,• ∇L(α(u)) = α′(u)∇Lu,• X2(α(u)) = α′′(u) (Xu)2 + α′(u)X2u,• L(α(u)) = α′′(u) |∇Lu|2 + α′(u)Lu,• X(u(ϕ(x))) =∑N

j=1 uj (ϕ(x))Xϕj (x) = 〈(∇u)(ϕ(x)),Xϕ(x)〉,• ∇L(u(ϕ(x))) =∑N

j=1(∂ju)(ϕ(x))∇Lϕj (x); if we use the usual column-vector notation

ϕ(x) =⎛

⎝ϕ1(x)

...

ϕN(x)

⎞

⎠

for vectors in RN , the row-vector notation for the gradients

∇u = (∂1u, . . . , ∂Nu), ∇Lu = (X1u, . . . , Xmu),

and we introduce the X-Jacobian matrix

∂ϕ

∂X(x) :=

⎛

⎝X1ϕ1(x) · · · Xmϕ1(x)

... · · · ...

X1ϕN(x) · · · XmϕN(x)

⎞

⎠ =⎛

⎝∇Lϕ1(x)

...

∇LϕN(x)

⎞

⎠

(which is a N ×m matrix) which can also be denoted by


∇Lϕ(x) = (X1ϕ(x) · · ·Xmϕ(x))

=⎛

⎝X1

⎛

⎝ϕ1(x)

...

ϕN(x)

⎞

⎠ · · ·Xm

⎛

⎝ϕ1(x)

...

ϕN(x)

⎞

⎠

⎞

⎠ ,

then last formula can be rewritten as

∇L(u(ϕ(x))) = (∇u)(ϕ(x)) · ∂ϕ

∂X(x),

or equivalently

∇L(u(ϕ(x))) = (∇u)(ϕ(x)) · ∇Lϕ(x),

which is resemblant to the classical chain rule

∇(u(ϕ(x))) = (∇u)(ϕ(x)) · Jϕ(x),

• prove that

X2(u(ϕ(x))) =N∑

i,j=1

(∂i,j u)(ϕ(x))Xϕi(x)Xϕj (x)

+N∑

j=1

(∂ju)(ϕ(x))X2ϕj (x),

or equivalently, following the previous notation (here Hessu(x) is theusual Hessian matrix of u at x),

X2(u(ϕ(x))) = (Xϕ(x))T · Hessu(ϕ(x)) · (Xϕ(x))

+ (∇u)(ϕ(x)) ·X2ϕ(x),

• deduce from the previous formulas that

L(u(ϕ(x))) =N∑

i,j=1

(∂i,j u)(ϕ(x))⟨∇Lϕi(x),∇Lϕj (x)

⟩

+N∑

j=1

(∂ju)(ϕ(x))Lϕj (x),

which can also be rewritten as

L(u(ϕ(x))) =m∑

k=1

(Xkϕ(x))T · Hessu(ϕ(x)) · (Xkϕ(x))

+ (∇u)(ϕ(x)) · Lϕ(x)


or, introducing the Gram matrix of the system of N vectors in Rm

{∇Lϕ1(x), . . . ,∇LϕN(x)},i.e. the N ×N symmetric matrix

Gϕ(x) := ∂ϕ

∂X(x) ·

(∂ϕ

∂X(x)

)T

=⎛

⎝∇Lϕ1(x)

...

∇LϕN(x)

⎞

⎠ · ((∇Lϕ1(x))T · · · (∇LϕN(x)

)T )

=⎛

⎝〈∇Lϕ1(x),∇Lϕ1(x)〉 · · · 〈∇Lϕ1(x),∇LϕN(x)〉

.... . .

...

〈∇LϕN(x),∇Lϕ1(x)〉 · · · 〈∇LϕN(x),∇LϕN(x)〉

⎞

⎠ ,

the above formula can be rewritten as

L(u(ϕ(x))) = trace(Hessu(ϕ(x)) ·Gϕ(x))+ (∇u)(ϕ(x)) · Lϕ(x).

Ex. 7) Let X be a smooth vector field on some open set Ω ⊆ RN , and let x ∈ Ω .

Let γ be the solution to the system of ODE’s

γ (t) = XI (γ (t)), γ (0) = x,

defined on the open (maximal) interval D(X, x) ⊆ R. Denote (momentarily)γ (t) by E(X, x, t). Let λ ∈ R be arbitrary. Prove the homogeneity relation

E(X, x, λ t) = E(λX, x, t) for all t such that λ t ∈ D(X, x). (1.102)

Let now t0 ∈ D(X, x) be fixed. Take λ = t0 and t = 1 in (1.102) and derivethat (for the arbitrariness of t0)

E(X, x, t) = E(t X, x, 1) for all t ∈ D(X, x). (1.103)

Throughout the following exercise, we shall adopt our usual notationexp(tX)(x) to denote E(X, x, t). Equation (1.103) says that the notation isnot improper.

Ex. 8) In this exercise, we give a generalization of formula (1.7) (page 7). Let Ω ⊆R

N be an open set, and let X be a smooth vector field on Ω . Let also f

be a smooth function on Ω . Finally, we fix x ∈ Ω . Then the function t �→f (exp(tX)(x)) is C∞ near t = 0, and its Taylor expansion at 0 is given by

f (exp(tX)(x)) =n∑

k=0

tk

k! (Xkf )(x)

+ 1

n!∫ t

0(t − s)nXn+1f (exp(sX)(x)) ds. (1.104)


Indeed, prove by induction that it holds

dk

dtk(f (exp(tX)(x))) = (Xkf )(exp(tX)(x)).

Derive from it the (very useful) formula

dk

dtk

∣∣∣∣t=0

(f (exp(tX)(x))) = (Xkf )(x).

Derive from (1.44) that if G = (RN, ◦) is a Lie group on RN and X is left-

invariant on G, then

dk

dtk

∣∣∣∣t=0

(f (x ◦ Exp (tX))) = (Xkf )(x).

Ex. 9) Let Ω ⊆ RN be an open set, and let X be a smooth vector field on Ω . Let us

consider the (autonomous) equation γ = XI (γ ). From general results on theexistence and uniqueness of solution of ODE’s (see, e.g. [Har82]) we knowthat, for every fixed compact set K ⊂ Ω , there exists δ = δ(X,K) > 0 suchthat the solution γ (t, x) to γ = XI (γ ), γ (0) = x exists for every t ∈ ]−δ, δ[and every initial value x ∈ K . From the uniqueness of the solution (and theautonomous nature of the equation) derive that

γ (t, γ (s, x)) = γ (t + s, x), |t | + |s| < δ, x ∈ K. (1.105)

Deduce from this fact that, for every fixed t0 ∈ ]−δ, δ[, the mapping

K � x �→ γ (t0, x)

is injective. (Hint: If γ (t0, x) = γ (t0, y), then γ (t+t0, x) = γ (t, γ (t0, x)) =γ (t, γ (t0, y)) = γ (t + t0, y), and evaluate at t = −t0.)If O is an open set such that O ⊂ Ω is compact, prove that (whenever|t0| < δ(X,O)) the map O � x �→ γ (t0, x) is a C∞-diffeomorphism ontoits image. (Hint: Again thanks to (1.105), the inverse map is given by y �→γ (−t0, x).)

Ex. 10) Let Y1, . . . , Ym be smooth vector fields on an open set Ω ⊆ RN . Let ξ =

(ξ1, . . . , ξm) ∈ Rm. Since the dependence on ξ of ξ1 Y1I + · · · + ξm YmI

is smooth, by general results on ODE’s (see, e.g. [Har82]), given a compactset K ⊂ Ω , we infer the existence of ε > 0 such that the solution γ to

γ =m∑

j=1

ξj Yj I (γ ), γ (0) = x

exists for every x ∈ K , every ξ satisfying |ξ | < ε and every t ∈ ]−ε, ε[(here |ξ | denotes any fixed norm on R

m). Moreover, the dependence of γ

on (x, ξ, t) is smooth. Using (1.103) (and the notation therein), observe that


Fig. 1.7. The flow of the vector field X

E

(∑

j

ξj Yj , x, t

)= E

(∑

j

tξj Yj , x, 1

)for all t ∈ ]−ε, ε[

and derive that the map

(u1, . . . , um) �→ Θx(u) := exp

(∑

j

uj Yj

)(x)

is well-defined (and smooth) on the open ball B = {u ∈ Rm : |u| < ε2}

(for any x ∈ K). Prove that the Jacobian matrix of Θx at u = 0 is the matrixwhose j -th column vector is Yj I (x) or, equivalently, that the differentiald0Θx sends the tangent vector (∂/∂uj )|0 ∈ T0(B) to Yj |x ∈ Tx(Ω). Hint:Use the following fact

∂

∂uj

∣∣∣∣0

exp

(∑

j

uj Yj

)(x) = ∂

∂uj

∣∣∣∣0

(I (x)+

∑

j

uj Yj I (x)+Ou→0(|u|2))

= Yj I (x).

As a corollary, deduce that if m = N and Y1I (x), . . . , YNI (x) are linearlyindependent, then Θx is a C∞-diffeomorphism of a neighborhood of 0 ontoa neighborhood of x. Finally, generalize (1.104) of Exercise 8 (we followthe notation therein) and obtain the following Taylor expansion at u = 0

f

(exp

(m∑

j=1

uj Yj

)(x)

)∼

∞∑

k=0

1

k!

(m∑

j=1

uj Yj

)k

f (x). (1.106)


Hint: Set

F(u) := f

(exp

(m∑

j=1

uj Yj

)(x)

)and G(t, u) := F(tu).

Hence G(t, u) = f (exp(tX)(x)), where X = ∑mj=1 uj Yj . Use Ex. 8 to

derive that∂k

∂tk

∣∣∣∣t=0

G(t, u) =(

m∑

j=1

uj Yj

)k

f (x),

and the Taylor expansion of F at u = 0 follows, as usual, from that of G att = 0.

Ex. 11) (Rellich–Pohozaev identities). We introduce the following notation. IfΩ ⊂ R

N is a domain with boundary regular enough, we denote by ν theouter unit normal to ∂Ω and by dσ the Hausdorff (N − 1)-dimensionalmeasure on ∂Ω . Provide the details for the following proposition.

Proposition 1.6.1. Let G be an arbitrary homogeneous Carnot group, andlet L = ∑m

i=1 X2i be a sub-Laplacian on G. Let Ω be a bounded domain

in G, regular for the divergence theorem. Finally, let Z be a vector field ofclass C1(G). Then, for every ϕ ∈ C2(Ω), we have

2∫

∂Ω

m∑

i=1

Xiϕ 〈XiI, ν〉Zϕ dσ −∫

∂Ω

〈ZI, ν〉 |∇Lϕ|2 dσ

= 2∫

Ω

m∑

i=1

Xiϕ [Xi,Z]ϕ + 2∫

Ω

Lϕ Zϕ −∫

Ω

|∇Lϕ|2 div(ZI). (1.107)

Proof. By applying two times the divergence theorem (and recalling thatX∗i = −Xi), we obtain∫

∂Ω

|∇Lϕ|2 〈ZI, ν〉 dσ =∫

Ω

div(|∇Lϕ|2 ZI)

=∫

Ω

|∇Lϕ|2 div(ZI)+∫

Ω

Z(|∇Lϕ|2)=∫

Ω

|∇Lϕ|2 div(ZI)+ 2∫

Ω

〈Z(∇Lϕ),∇Lϕ〉

=∫

Ω

|∇Lϕ|2 div(ZI)+ 2∫

Ω

m∑

i=1

[ZI,Xi]ϕ Xiϕ

+ 2∫

Ω

m∑

i=1

Xi(Zϕ)Xiϕ

=∫

Ω

|∇Lϕ|2 div(ZI)+ 2∫

Ω

m∑

i=1

[ZI,Xi]ϕ Xiϕ

+ 2∫

∂Ω

m∑

i=1

Zϕ Xiϕ 〈XiI, ν〉 dσ − 2∫

Ω

Zϕ Lϕ.

This ends the proof. �


We now specify the integral identity (1.107) when Z is given by the so-called generator of the translations. Let ◦ be the group law on the Carnotgroup G, and fix z0 ∈ G. We denote by Zz0 the following vector field on G

Zz0I (z) = d

dh

∣∣∣∣h=0

((hz0) ◦ z). (1.108)

Recalling that, for every i = 1, . . . , m,

Xi(z) = (d/dh)h=0(z ◦ (hei))

(here ei is the i-th versor of the canonical basis of RN ), derive that the

bracket [Xi,Zz0 ] vanishes identically. Hint: ◦ is associative and

[Xi,Zz0 ]f (z) = d

dh

∣∣∣∣h=0

d

ds

∣∣∣∣s=0

f ((sz0) ◦ (z ◦ (hei)))

− d

ds

∣∣∣∣s=0

d

dh

∣∣∣∣h=0

f (((sz0) ◦ z) ◦ (hei)).

Furthermore, the divergence of the vector field Zz0 vanishes identically. In-deed, we recall that ◦ has the form z ◦ ζ = ((z ◦ ζ )(1), . . . , (z ◦ ζ )(r)), where

(z◦ζ )(1) = z(1)+ζ (1), (z◦ζ )(j) = z(j)+ζ (j)+Q(j)(z, ζ ), 2 ≤ j ≤ r,

Q(j) being a function with values in RNj and whose components are mixed

polynomials in z and ζ such that Q(j)(δλz, δλζ ) = λjQ(j)(z, ζ ). We thenrecognize that the components of Zz0(z) in the j -th layer have the followingform

(Zz0(z))(j) = z(j)

0 + 〈z(1)0 , q

(j)

1 (z)〉 + · · · + 〈z(j−1)

0 , q(j)

j−1(z)〉,

where q(j)i is a R

Ni -valued function whose components are polynomialsδλ-homogeneous of degree j − i. In particular, (Zz0(z))(j) does not dependon zj , whence div(Zz0) = 0. From Proposition 1.6.1 and the above remarks,the next result immediately follows.

Proposition 1.6.2. Let G be a homogeneous Carnot group, and let L =∑mi=1 X2

i be a sub-Laplacian on G. Let Ω be a bounded domain in G, reg-ular for the divergence theorem. Finally, for a fixed z0 ∈ G, let Zz0 be thevector field defined in (1.108). Then, for every ϕ ∈ C2(Ω), we have

2∫

∂Ω

m∑

i=1

Xiϕ 〈XiI, ν〉Zz0ϕ dσ −∫

∂Ω

〈Zz0I, ν〉 |∇Lϕ|2 dσ

= 2∫

Ω

Lϕ Zz0ϕ.


Ex. 12) (Maps commuting with a sum of squares).a) Consider a second order differential operator

L :=N∑

i,j=1

ai,j (x) ∂2i,j +

N∑

i=1

bi(x) ∂i

with coefficients ai,j and bi in C2(RN) such that ai,j = aj,i for all i

and j . Consider ψ ∈ C2(RN, RN). Prove that the necessary and suffi-

cient conditions in order to have

L(u ◦ ψ) = (L u) ◦ ψ ∀ u ∈ C2(RN) (1.109)

are the following ones:⎧⎪⎪⎨

⎪⎪⎩

bk(ψ) = Lψk ∀ k = 1, . . . , N,

ai,j (ψ) =N∑

r,s=1

ar,s∂rψi∂sψj ∀ i, j = 1, . . . , N.(1.110)

(Hint: Take u(x) = xk and then u(x) := xi xj .) If Jψ denotes the usualJacobian matrix of ψ and

A(x) := (ai,j (x)

)1≤i,j≤N

, B(x) :=⎛

⎜⎝b1(x)

...

bN(x)

⎞

⎟⎠ ,

so thatL = trace

(A(x) · Hess

)+ 〈∇, B(x)〉,then system (1.110) can be rewritten as

{B(ψ(x)) = Lψ(x)

A(ψ(x)) = Jψ(x) · A(x) · (Jψ(x))T .(1.111)

b) Suppose furthermore that L is a sum of squares L = ∑p

k=1 X2k with

Xk =∑Ni=1 σ

(k)i ∂i , so that

L =N∑

i,j=1

(p∑

k=1

σ(k)i σ

(k)j

)∂2i,j +

N∑

i=1

(p∑

k=1

Xk σ(k)i

)∂i .

Prove that the necessary and sufficient conditions for (1.109) to hold are⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

p∑

k=1

(Xk σ(k)i )(ψ) = Lψi ∀ i = 1, . . . , N,

p∑

k=1

σ(k)i (ψ) σ

(k)j (ψ) = 〈∇Lψi,∇Lψj 〉 ∀ i, j = 1, . . . , N,

(1.112)


where ∇Lu := (X1u, . . . , Xpu). Set, as usual,

XkI :=⎛

⎜⎝σ

(k)1...

σ(k)N

⎞

⎟⎠ .

Considering the N × p matrix

S(x) := (σ(j)i (x))i≤N, j≤p = (X1I (x) · · ·XpI (x)),

we have

A(x) =(

p∑

k=1

σ(k)i σ

(k)j

)

i,j≤N

= S(x) · (S(x))T ,

so that

Jψ(x) · A(x) · (Jψ(x))T = (Jψ(x) · S(x)) · (Jψ(x) · S(x))T .

Moreover,

Jψ(x) · S(x) = (Xjψi(x))i≤N, j≤p =⎛

⎜⎝∇Lψ1

...

∇LψN

⎞

⎟⎠ .

As a consequence,

(Jψ(x) · S(x)) · (Jψ(x) · S(x))T = (〈∇Lψi(x),∇Lψj(x)〉)i,j≤N .

Moreover,

B(x) =(

p∑

k=1

Xk σ(k)i

)

i≤N

=⎛

⎜⎝

∑p

k=1 Xk σ(k)1

...∑p

k=1 Xk σ(k)N

⎞

⎟⎠ .

Finally, (1.112) becomes{

B(ψ(x)) = Lψ(x),

A(ψ(x)) = (〈∇Lψi(x),∇Lψj(x)〉)i,j≤N .

Ex. 13) Consider the Lie group (and the notation) in Example 1.2.19 (page 21).Show that, for every ξ1, ξ2 ∈ R, the integral curve of ξ1Z1 + ξ2Z2 startingat (0, 0) is

γ (t) ={

(ξ1t, (ξ2 eξ1t − ξ2)/ξ1) if ξ1 �= 0,

(0, ξ2t) if ξ1 = 0.


Equivalently, considering the function

f : R → R, f (z) := ez − 1

zif z �= 0, f (0) := 1,

we haveγ (t) = (ξ1 t, ξ2 t f (ξ1t)).

Derive that

Exp : g → G, exp(ξ1Z1 + ξ2Z2) = (ξ1, ξ2 f (ξ1)).

Prove that Exp is smooth (indeed, real analytic!) and globally invertible.Find the Log function. (Hint: The function f is analytic and invertible.)

2

Abstract Lie Groups and Carnot Groups

The aim of this chapter is to prove that, up to a canonical isomorphism, the classi-cal definition of stratified group (or Carnot group) (see Definition 2.2.3) coincideswith our definition of homogeneous Carnot group, as given in Chapter 1. To thisaim, we begin by recalling some basic facts about abstract Lie groups, providing allthe terminology and the main results about manifolds, tangent vectors, left-invariantvector fields, Lie algebras, homomorphisms, the exponential map. Our exposition inSection 2.1 is self-contained and is intended to provide the topics from differentialgeometry and Lie group theory, which are strictly necessary to read this book.

In Section 2.2, we provide the cited equivalence between the two notions ofCarnot group. This is accomplished by showing that the Lie algebra g of the ab-stract Carnot group G possesses a natural structure of homogeneous Carnot group.Indeed, the group operation on g is the one induced by that of G via the exponentialmap, and the dilations on g are modeled on its stratification. A central rôle here willbe played by the Campbell–Hausdorff formula, that we assume in this chapter byrecalling (without proofs) some of its abstract and very general properties. Nonethe-less, we shall devote Chapter 15 to provide a self-contained investigation of such aremarkable formula in the significant case of homogeneous stratified groups.

2.1 Abstract Lie Groups

2.1.1 Differentiable Manifolds

Let N ∈ N, and let us define, for i = 1, . . . , N , the coordinate projections on RN

(whose points will be denoted by ξ = (ξ1, . . . , ξN ) ∈ RN with ξ1, . . . , ξN ∈ R)

πi : RN −→ R, πi(ξ) := ξi .

Definition 2.1.1 (N -dimensional locally Euclidean space). An N -dimensional lo-cally Euclidean space M is a Hausdorff topological space such that every point of M

88 2 Abstract Lie Groups and Carnot Groups

has a neighborhood in M homeomorphic to an open subset of RN . If ϕ is a homeo-

morphism between a connected open set U ⊆ M and an open subset of RN , we say

that ϕ : U → RN is a coordinate map,

xi := πi ◦ ϕ : U → R

is a coordinate function, and the pair (U, ϕ) (sometimes also denoted by (U, x1,

. . . , xN)) is a coordinate system or a chart. If m ∈ U and ϕ(m) = 0, we say that thecoordinate system is centered at m.

Definition 2.1.2 (Differentiable manifold). A C∞ differentiable structure F on alocally Euclidean space M is a collection of coordinate systems

{(Uα, ϕα) : α ∈ A }with the following properties:

• ⋃α∈A Uα = M;

• ϕα ◦ ϕ−1β is C∞ for every α, β ∈ A (whenever it is defined);

• F is maximal w.r.t. the second property in the sense that if (U, ϕ) is a coordinatesystem such that ϕ ◦ϕ−1

α and ϕα ◦ϕ−1 are C∞ for every α ∈ A, then (U, ϕ) ∈ F .

An N -dimensional C∞ differentiable manifold is a couple (M,F), where M is a sec-ond countable N -dimensional locally Euclidean space and F is a C∞ differentiablestructure.

As usual, when we say “M is an N -dimensional C∞ differentiable manifold”,we leave unsaid that M is equipped with the fixed datum of a C∞ differentiablestructure F on M .

Let M and M ′ be differentiable manifolds of dimension N and N ′, respectively,and let f : M −→ M ′. Then we say that f is C∞ in m ∈ M if, for every (or, equiva-lently, for at least one) coordinate system (U, ϕ) of M and for every (or, equivalently,for at least one) coordinate system (U ′, ϕ′) of M ′ such that m ∈ U and f (m) ∈ U ′,the function

ϕ′ ◦ f ◦ ϕ−1 : ϕ(U) ⊆ RN −→ ϕ′(U ′) ⊆ R

N ′

is C∞ in a neighborhood of ϕ(m). Let now μ(t) be a C∞ function defined on areal interval and with values in M . We say that μ(t) is a curve passing through m

at the time t0 if μ(t0) = m. We say that a C∞ real valued function f defined in aneighborhood of m ∈ M is horizontal in m if

d f (μ(t))

d t

∣∣∣∣t=0

= 0

for every curve passing through m at the time t = 0.

Remark 2.1.3. The following characterizations hold:(I). It is immediate to see that a real-valued function f is C∞ in a neighborhood

of m ∈ M if and only if there exist a coordinate system (U, ϕ) with m ∈ U and a

2.1 Abstract Lie Groups 89

(ordinary) C∞ real-valued function f on (the open subset of RN ) ϕ(U) such that

f = f ◦ ϕ on U . (For example, any coordinate map xi = πi ◦ ϕ is C∞ on U for πi

is smooth.)(II). Analogously, μ is a curve passing through m ∈ M at t0 if and only if there

exist a coordinate system (U, ϕ) with m ∈ U and a (ordinary) C∞ curve

ν : ]a, b[ ⊆ R −→ ϕ(U) ⊆ RN

with t0 ∈]a, b[, ν(t0) = ϕ(m) and μ = ϕ−1 ◦ ν on ]a, b[.(III). Furthermore, we show that f is horizontal in m ∈ M if and only if there

exist a coordinate system (U, ϕ) with m ∈ U and a C∞ real-valued function f onϕ(U) such that f = f ◦ ϕ on U and the ordinary gradient

(∇f )(ϕ(m))

is null.Indeed, suppose this fact holds. Let μ be any curve passing through m at

t = 0. By (II), there exist a coordinate system (V ,ψ) with m ∈ V and a C∞ curveν : ]a, b[ −→ ψ(U) with 0 ∈ ]a, b[, ν(0) = ψ(m) and μ = ψ−1 ◦ ν on ]a, b[. Then,for ε > 0 sufficiently small (so that ν(t) ∈ ψ(U ∩ V ) for every t ∈ ]−ε, ε[), thecomposition

f ◦ μ = f ◦ ϕ ◦ ψ−1 ◦ ν

is well defined and smooth on ]−ε, ε[ and, by the ordinary chain rule, it holds

d f (μ(t))

d t

∣∣∣∣t=0

= (∇f )(ϕ(ψ−1(ν(0)))

) · Jϕ◦ψ−1(ν(0)) · ν(0) = 0,

for (∇f )(ϕ(ψ−1(ν(0)))) = (∇f )(ϕ(m)) = 0. This proves that f is horizontal inm. Vice versa, suppose f is horizontal in m. By (I), there exist a coordinate system(U, ϕ) with m ∈ U and a C∞ real-valued function f on ϕ(U) such that f = f ◦ ϕ

on U . We aim to prove that (∇f )(ϕ(m)) = 0. Let h ∈ RN be any fixed vector. Let

ε > 0 be small enough, so that, for every |t | < ε, we have

ν(t) := ϕ(m) + t h ∈ ϕ(U).

Then, by (II), μ := ϕ−1 ◦ ν is a C∞ curve passing through m at t = 0. Hence, beingf horizontal in m, we have

0 = d f (μ(t))

d t

∣∣∣∣t=0

= d

d t

∣∣∣∣t=0

(f ◦ ϕ ◦ ϕ−1 ◦ ν

)(t)

= (∇f )(ν(0)) · ν(0) = ⟨(∇f )(ϕ(m)), h

⟩.

Consequently, due to the arbitrariness of h, this proves that (∇f )(ϕ(m)) = 0.(IV). Finally, we prove that f is horizontal in m ∈ M if and only if for every

coordinate system (V ,ψ) with m ∈ V , the C∞ real-valued function f = f ◦ ψ−1

on ψ(V ) satisfies


(∇f )(ψ(m)) = 0.

The “if” part follows by (III). Vice versa, suppose f is horizontal in m ∈ M . By (III),there exist a coordinate system (U, ϕ) with m ∈ U and a C∞ real-valued function g

on ϕ(U) such that f = g ◦ ϕ on U and (∇g)(ϕ(m)) = 0. As a consequence,

(∇f )(ψ(m)) = (∇(f ◦ ψ−1))(ψ(m)) = (∇(g ◦ ϕ ◦ ψ−1))(ψ(m))

= (∇g)(ϕ ◦ ψ−1 ◦ ψ(m)) · Jϕ◦ψ−1(ψ(m))

= (∇g)(ϕ(m)) · Jϕ◦ψ−1(ψ(m)) = 0,

for (∇g)(ϕ(m)) = 0. �Example 2.1.4. (i). Any constant function on M is trivially horizontal at every pointof M . The same is true for a function vanishing near m.

(ii). Let m ∈ M , and fix any coordinate system (U, ϕ) with m ∈ U . Let i, j ∈{1, . . . , N} be chosen. Consider the relevant coordinate functions xi , xj , i.e. xi =πi ◦ ϕ and xj = πj ◦ ϕ on U . Denote x0 := ϕ(m). We show that

f := (xi − x0i ) (xj − x0

j )

is horizontal at m. By (III) of Remark 2.1.3, in order to prove that f is horizontal atm, it suffices to show that

∇(f ◦ ϕ−1)(ϕ(m)) = 0.

But this is obvious since, for every u ∈ ϕ(U),

(f ◦ ϕ−1)(u) = ((πi ◦ ϕ − x0

i ) (πj ◦ ϕ − x0j ))(ϕ−1(u))

= (ui − x0i ) (uj − x0

j ),

and trivially∂

∂ uk

∣∣∣∣x0

((ui − x0

i )(uj − x0j )) = 0

for every k = 1, . . . , N .(iii). If f is horizontal at m ∈ M , f (m) = 0 and g is any smooth function

in a neighborhood of m, then f g is horizontal at m ∈ M . Indeed, let (U, ϕ) be acoordinate system with m ∈ M . By (IV) of Remark 2.1.3, we need to show that∇((f g) ◦ (ϕ−1))(ϕ(m)) = 0. This is obvious since

∇((f g) ◦ (ϕ−1)) = (g ◦ ϕ−1)∇(f ◦ (ϕ−1)

)+ (f ◦ ϕ−1)∇(g ◦ (ϕ−1))

and recalling that ∇(f ◦ ϕ−1)(ϕ(m)) = 0 (for f is horizontal in m) and f (m) = 0by hypothesis. �

As we shall see in the proof of Proposition 2.1.7, the functions in (i), (ii), (iii) ofExample 2.1.4 are “infinitesimally” the only functions worth considering.


2.1.2 Tangent Vectors

Definition 2.1.5 (Tangent vector, space and bundle). Let M be an N -dimensionalC∞ differentiable manifold. A tangent vector v at m ∈ M is a linear functional,defined on the collection of the real-valued functions C∞ in some neighborhood ofm, such that

v(f ) = 0

whenever f is horizontal in m.We denote by Mm the set of the tangent vectors at m ∈ M , and we say that Mm

is the tangent space to M at m. We finally set

T (M) :=⋃

m∈M

{m} × Mm = {(m, v) : m ∈ M, v ∈ Mm

}. (2.1)

T (M) is called the tangent bundle to M .

Remark 2.1.6. Suppose f is a real-valued C∞ function defined in a neighborhood ofm ∈ M . Hence, there exists a coordinate system (U, ϕ) such that m ∈ U , and f isdefined on U . Consider the open set ϕ(U) ⊆ R

N . For n ∈ {1, 2, 3, 4}, denote by Bn

the Euclidean ball centered at ϕ(m) with radius n ε, where ε > 0 is small enough,so that B4 is contained in ϕ(U). Let χ be a smooth cut-off function on R

N such thatχ ≡ 1 on B1 and χ ≡ 0 on R

N \ B2. Then, consider the function on M

f (n) :={

χ(ϕ(n)) f (n) if n ∈ ϕ−1(B3),

0 otherwise,n ∈ M.

It is easy to see that f ≡ f on the set ϕ−1(B1), open neighborhood of n in M , andthat ϕ is defined and smooth on M (note that χ ◦ϕ is C∞ on U by Remark 2.1.3-(I)).

Since f − f vanishes in a neighborhood of m, by Example 2.1.4-(i), we havev(f − f ) = 0 for every tangent vector v at m. Hence, by linearity, v(f ) = v(f ).

This shows that in the definition of vector field it is not restrictive to supposethat a tangent vector at a point is a linear functional defined on the collection of thereal-valued functions C∞ in M .

We remark that Mm is a vector space with the usual operations of sum of func-tionals and multiplication of a functional times a scalar factor. Denoting by dim M

the dimension N of the differentiable manifold M , we have the following result.(Note. The reader will recognize in the proof below that a tangent vector at a

point is indeed a linear “differential” operator of the first order defined on the C∞functions on M .)

Proposition 2.1.7. Let M be an N -dimensional C∞ differentiable manifold. Thendim(Mm) = N = dim M .

Proof. Let v ∈ Mm. Let (U, ϕ) be a coordinate system such that m ∈ U . Let f bean arbitrary C∞ function defined in a neighborhood of ϕ(m). We set


v(f ) := v(f ), where f := f ◦ ϕ.

By Remark 2.1.3-(I), v is meaningful, and it is easily seen that v is a linear functionaldefined on the functions C∞ in a neighborhood of x0 := ϕ(m). In particular, v(precisely, a suitable restriction of v) is linear on the vector space V of the real-valuedlinear functions a(x) defined on R

N (where N := dim M; note that V = (RN)∗, theusual dual space of R

N ). Now, any linear functional on V is uniquely represented bya vector λ ∈ R

N , i.e.v(a(u)) = a(λ).

Let now f = f (u) be any function C∞ in a neighborhood of x0. By Taylor expan-sion with the integral remainder, we have

f (u) = f (x0) + ⟨∇f (x0), u − x0⟩

+N∑

i,j=1

(ui − x0i )(uj − x0

j )

∫ 1

0(1 − t)

∂2 f

∂ui∂uj

(x0 + t (u − x0)

)dt

=: f (x0) − a(x0) + a(u) + R(u)

for every u near x0. By Example 2.1.4, we see that f (x0) − a(x0) and R ◦ ϕ arehorizontal in m. Indeed, the summand in the (i, j)-sum defining R ◦ϕ is the productof

(ui − x0i )(uj − x0

j ) ◦ ϕ = (xi − x0i )(xj − x0

j )

(where xi, xj are the relevant coordinate functions), which is horizontal and vanish-ing in m, times the smooth function near m

U � n �→ g(n) :=∫ 1

0(1 − t)

∂2 f

∂ui∂uj

(x0 + t (ϕ(n) − x0)

)dt.

We now set f = f ◦ ϕ, so that

f = f (x0) − a(x0) + a ◦ ϕ + R ◦ ϕ

= a ◦ ϕ + {a horizontal function in m}.As a consequence of the definition of tangent vector, we infer

v(f ) = v(a ◦ ϕ) = v((a ◦ ϕ) ◦ ϕ−1)

= v(a(u)) = ⟨∇(f ◦ ϕ−1)(x0), λ⟩. (2.2)

Now, it is easy (and is left as an exercise) to prove that the map

Mm � v �→ λ ∈ RN

is linear, injective and surjective. This completes the proof. �


Remark 2.1.8. It is easy to see that v ∈ Mm if and only if for every coordinate map(U, ϕ) with m ∈ U (or, equivalently, for at least one such coordinate map) thereexists λ = λ(ϕ,m) ∈ R

N such that

v(f ) = ⟨∇(f ◦ ϕ−1)(ϕ(m)), λ⟩

for every f , real valued and C∞ near m.

The “only if” part follows from the proof of Proposition 2.1.7 (see in particular(2.2)). Suppose now that there exists (U, ϕ) and λ as above. Let f be horizontal inm ∈ M . We need to show that

⟨∇(f ◦ ϕ−1)(ϕ(m)), λ⟩ = 0.

By Remark 2.1.3, the function f = f ◦ ϕ−1 satisfies (∇f )(ϕ(m)) = 0. This defi-nitely suffices for what we needed to prove. �Remark 2.1.9. Remark 2.1.8 furnishes another very useful characterization of tan-gent vectors (actually, an alternative definition frequently used in literature). Ifm ∈ M , then v ∈ Mm if and only if there exists a C∞ curve on M passing throughm at t0 such that

v(f ) = d

d t

∣∣∣∣t0

(f (μ(t))) for every f , real valued and C∞ near m.

Indeed, if v is defined in this way, we have, for any given coordinate map (U, ϕ),m ∈ U ,

v(f ) = d

d t

∣∣∣∣t0

(f (μ(t))) = d

d t

∣∣∣∣t0

((f ◦ ϕ−1) ◦ (ϕ ◦ μ)

)(t)

=⟨∇(f ◦ ϕ−1)(ϕ(m)),

d

d t(ϕ ◦ μ)(t0)

⟩,

whence v satisfies the condition in Remark 2.1.8 with

λ = d

d t(ϕ ◦ μ)(t0).

Vice versa, suppose v ∈ Mm. Fix any coordinate map (U, ϕ) with m ∈ U . Then,again by Remark 2.1.8, there exists λ ∈ R

N such that

( ) v(f ) = ⟨∇(f ◦ ϕ−1)(ϕ(m)), λ⟩

for every f , real valued and C∞ near m. Consider the curve on M given by (see (II)in Remark 2.1.3)

μ(t) := ϕ−1(ϕ(m) + t λ), |t | < ε,

with ε > 0 small enough. Obviously, μ passes through m at t = 0. Moreover,


d

d t

∣∣∣∣0(f (μ(t))) = d

d t

∣∣∣∣0

((f ◦ ϕ−1) ◦ (ϕ ◦ μ)

)(t)

=⟨∇(f ◦ ϕ−1)(ϕ(m)),

d

d t

∣∣∣∣0

(ϕ(m) + t λ

)(0)

⟩

= ⟨∇(f ◦ ϕ−1)(ϕ(m)), λ⟩.

Comparing to ( ), we have proved our assertion. �Definition 2.1.10 (Partial derivatives on M). Let M be an N -dimensional C∞ dif-ferentiable manifold. Let (U, ϕ) be a coordinate system with coordinate functionsx1, . . . , xN (xi := πi ◦ ϕ), and let m ∈ U . For every i ∈ {1, . . . , N} we define atangent vector, denoted

∂

∂ xi

∣∣∣∣m

∈ Mm,

by setting∂

∂ xi

∣∣∣∣m

(f ) := ∂

∂ ξi

∣∣∣∣ϕ(m)

(f ◦ ϕ−1)(ξ) (2.3)

for every C∞ function f defined in a neighborhood of m.

We remark that f �→ ((∂/∂ xi)|m)(f ) actually defines an element of Mm (seeRemark 2.1.8 with λ = the i-th element of the canonical basis of R

N ). Note that thedefinition of (∂/∂ xi)|m is not coordinate-free: despite the notation, forgetful of ϕ, itdepends on the coordinate map ϕ, as xi itself.

Example 2.1.11. With the notation of the above definition,

∂

∂ xi

∣∣∣∣m

(xj ) = δi,j (of Kronecker).

Indeed, xj ◦ ϕ−1 = πj , so that (2.3) gives

∂

∂ xi

∣∣∣∣m

(xj ) = ∂

∂ ξi

∣∣∣∣ϕ(m)

(ξj ) = δi,j .

Remark 2.1.12. (i). If v ∈ Mm, then (by collecting together Remark 2.1.8 and Exam-ple 2.1.11) we have the following suggesting formula:

v =N∑

i=1

v(xi) · ∂

∂ xi

∣∣∣∣m

, xi := πi ◦ ϕ, (2.4)

and then(∂/∂ x1)|m, . . . , (∂/∂ xN)|m

is a basis for Mm.(ii). In particular, two tangent vectors v, w ∈ Mm coincide if and only if


v(πi ◦ ϕ) = w(πi ◦ ϕ) ∀ i = 1, . . . , N,

for every (or, equivalently, for at least one) coordinate system (U, ϕ) such thatm ∈ U .

(iii). The above formula (2.4) allows to represent a tangent vector in a very ex-plicit way, once a coordinate system has been fixed. Indeed, if ϕ : U −→ R

N is acoordinate map, ϕi is the i-th component of ϕ, (ξ1, . . . , ξN ) are the coordinates onR

N , m ∈ M , f is a C∞ function defined in U and v ∈ Mm, then we have

v(f ) =N∑

i=1

v(ϕi) · ∂(f ◦ ϕ−1)

∂ ξi

(ϕ(m)). (2.5)

2.1.3 Differentials

Definition 2.1.13 (Differential at a point). Let ψ : M −→ M ′ be a C∞ map be-tween two differentiable manifolds, and let m ∈ M . The differential of ψ at m is thelinear map

dmψ : Mm −→ M ′ψ(m)

defined as follows: if v ∈ Mm, dmψ(v) is the tangent vector in M ′ψ(m) acting in the

following way: if f is a C∞ function in a neighborhood of ψ(m), we set

(dmψ(v)

)(f ) := v(f ◦ ψ). (2.6)

Even if many authors use to write dψ instead of dmψ , we shall keep the notationwell distinguished, preserving “dψ” for a suitable further notion.

Remark 2.1.14. We verify that (2.6) actually defines a tangent vector at ψ(m). Let f

be a C∞ function in a neighborhood of ψ(m), and let (U, ϕ), (U ′, ϕ′) be coordinatesystems in M and M ′, respectively, with m ∈ U , ψ(m) ∈ U ′. By Remark 2.1.8, thereexists λ ∈ R

N such that

v(g) = ⟨∇(g ◦ ϕ−1)(ϕ(m)), λ⟩

for every g, C∞ near m. Hence, we have (we denote λ as a column vector)

(dmψ(v)

)(f ) = v(f ◦ ψ) = v

(f ◦ (ϕ′)−1 ◦ ϕ′ ◦ ψ ◦ ϕ−1 ◦ ϕ

)

= ⟨∇(f ◦ (ϕ′)−1 ◦ ϕ′ ◦ ψ ◦ ϕ−1)(ϕ(m)), λ⟩

= ∇(f ◦ (ϕ′)−1)(ϕ′(ψ(m))) · Jϕ′◦ψ◦ϕ−1(ϕ(m)) · λ

= ⟨∇(f ◦ (ϕ′)−1)(ϕ′(ψ(m))),Jϕ′◦ψ◦ϕ−1(ϕ(m)) · λ

⟩.

This shows that

(dmψ(v)

)(f ) = ⟨∇(f ◦ (ϕ′)−1)(ϕ′(ψ(m))

), λ′⟩, (2.7a)


whereλ′ = Jϕ′◦ψ◦ϕ−1(ϕ(m)) · λ. (2.7b)

Again thanks to Remark 2.1.8, this proves that dmψ(v) is a tangent vectorat ψ(m). �Remark 2.1.15. We remark that, by (2.4), (2.7a) and (2.7b), if (U, x1, . . . , xN) and(U ′, y1, . . . , yN ′) are coordinate systems at m ∈ M and ψ(m) ∈ M ′, respectively,we have

dmψ

(∂

∂ xi

∣∣∣∣m

)=

N ′∑

j=1

∂

∂ xi

∣∣∣∣m

(yj ◦ ψ) · ∂

∂ yj

∣∣∣∣ψ(m)

. (2.8)

Indeed, (2.8) follows from (2.7b) by taking f = πj ◦ ϕ′, λ = the i-th element ofthe standard basis of R

N and by recalling that yj ◦ ψ = πj ◦ ϕ′ ◦ ψ and

∂

∂ xi

∣∣∣∣m

(yj ◦ ψ) =(∂i(πj ◦ ϕ′ ◦ ψ ◦ ϕ−1)

)(ϕ(m)) =

(Jϕ′◦ψ◦ϕ−1(ϕ(m))

)

j,i.

Remark 2.1.16 (Transformation of tangent vectors via a differential). In general,a vector field v at m transforms under the differential of ψ as follows:

dmψ

(N∑

i=1

v(πi ◦ ϕ) · ∂

∂ xi

∣∣∣∣m

)

=N ′∑

j=1

(N∑

i=1

v(πi ◦ ϕ) · ∂i

(πj ◦ ϕ′ ◦ ψ ◦ ϕ−1)

)∂

∂ yj

∣∣∣∣ψ(m)

=N ′∑

j=1

(Jϕ′◦ψ◦ϕ−1(ϕ(m)) · v(ϕ)

)

j

∂

∂ yj

∣∣∣∣ψ(m)

, (2.9)

where

v(ϕ) =⎛

⎜⎝v(π1 ◦ ϕ)

...

v(πN ◦ ϕ)

⎞

⎟⎠

and (U, ϕ), (U ′, ϕ′) are, respectively, coordinate systems at m ∈ M , ψ(m) ∈ M ′(M, M ′ are differentiable manifolds of dimensions N, N ′, respectively).

Remark 2.1.17 (Differential of the composition and of the inverse). Let ψ : M −→M ′, φ : M ′ −→ M ′′ be C∞ maps between differentiable manifolds M, M ′, M ′′.Then it is easily seen that φ ◦ ψ is C∞, and we have

dm(φ ◦ ψ) = dψ(m)φ ◦ dmψ ∀ m ∈ M. (2.10)

Furthermore, let ψ : M −→ M ′ be a C∞ map between two differentiable man-ifolds. Suppose ψ−1 : M ′ −→ M is a C∞ map too. In this case, we say that ψ is adiffeomorphism.


It is immediate to observe that if ψ is a diffeomorphism and m ∈ M , then dmψ

is also invertible (for every m ∈ M) and

dψ(m)(ψ−1) : M ′

ψ(m) −→ Mm

is the inverse function ofdmψ : Mm −→ M ′

ψ(m).

Definition 2.1.18 (dψ as a map on the tangent bundle). Let ψ : M → M ′ be aC∞ map between two differentiable manifolds M, M ′. We set

dψ : T (M) → T (M ′), dψ(m, v) := (ψ(m), dmψ(v)). (2.11)

Note that, whereas dmψ is a map from Mm to M ′ψ(m) (for any fixed m ∈ M), dψ

is a map from T (M) to T (M ′).

2.1.4 Vector Fields

The following definition is one of the most important in differential geometry.

Definition 2.1.19 (Vector field). Let Ω ⊆ M be an open subset of a differentiablemanifold M . A vector field X on Ω is an application

X : Ω −→ T (M)

such that,X(m) = (m, v(m)) ∈ T (M) ∀ m ∈ Ω.

Equivalently, we have

X(m) = (m, v(m)), where v(m) ∈ Mm for every m ∈ Ω.

In order to avoid any confusion, we make explicit the following conventionalidentification of a vector field with its projection onto the second argument. Thisidentification is frequently tacitly employed in literature.

Convention–Notation. If T (M) is the tangent bundle of a differentiable mani-fold M , and, for every m ∈ M , v ∈ Mm, we set π(m, v) := v, then the followingmap is well posed on T (M):

π : T (M) →⋃

m∈M

Mm, (m, v) �→ v.

In the sequel, if X is a vector field on an open set Ω ⊆ M , we shall use the notationX(m) for the map

X : Ω → T (M), m �→ X(m),

whereas Xm will denote the map


Ω →⋃

m∈M

Mm, m �→ Xm := (π ◦ X)(m).

So, the above positions can be summarized as

X(m) = (m,Xm) for every m ∈ M . (2.12)

Finally, if f is a C∞ function on Ω and X is a vector field on Ω , we shall denote(with an abuse of notation) by X(f ) or shortly Xf the function on Ω whose value atm is Xm(f ), i.e.

Xf : Ω → R, (Xf )(m) := Xm(f ). (2.13)

Definition 2.1.20 (Smooth vector field). Let X be a vector field defined on a man-ifold M . We say that X is C∞ (or smooth) if, for every open set Ω ⊆ M and everysmooth real-valued function f on Ω , the function Xf as defined in (2.13) is smoothon Ω .

Remark 2.1.21. It is straightforward to verify that X is a smooth vector field on M ifand only if, for every coordinate system (U, x1, . . . , xN), the functions a1, . . . , aN

defined on U by

Xm =N∑

i=1

ai(m) · ∂

∂ xi

∣∣∣∣m

(see (2.12) and (2.4)), are C∞ functions on U .Following the above notation, we have

ai(m) = Xm(xi), where xi = πi ◦ ϕ.

Recalling (2.5), we see that, if (U, ϕ) is a coordinate system, a smooth vector fieldacts on a function f ∈ C∞(U) in the following way:

Xm(f ) =N∑

i=1

ai(m) · {∂i(f ◦ ϕ−1)}(ϕ(m))

=N∑

i=1

Xm(πi ◦ ϕ) · {∂i(f ◦ ϕ−1)}(ϕ(m)), (2.14)

where a1, . . . , aN ∈ C∞(U) are fixed functions depending on X and on the coordi-nate map. This shows that a vector field X on M is smooth iff, for every coordinatesystem (U, ϕ), the functions

m �→ Xm(π1 ◦ ϕ), . . . , Xm(πN ◦ ϕ)

are C∞ on U . �Remark 2.1.22 (Smooth vector fields as operators on C∞(M, R)). Let X be a smoothvector field on a differentiable manifold M . Besides a map from M to T (M), it ispossible to identify X with the map


X : C∞(M, R) → C∞(M, R), f �→ Xf,

where (see (2.13))

Xf : M → R, m �→ (Xf )(m) = Xmf.

We denote by X (M) the set of the smooth vector fields considered as linear operators(i.e. endomorphisms) on C∞(M, R). Note that X (M) is a vector space over R.

The correspondence between X as a vector field on M and as an endomorphismon C∞(M, R) is faithful in the sense that, if X, Y are vector fields such that Xf ≡Yf for every f ∈ C∞(M, R), then X = Y . (Indeed, take any m ∈ M , any coordinatesystem (U, ϕ) around m, any i ∈ {1, . . . , N} and choose f = πi ◦ ϕ. Then applyRemark 2.1.12-(ii).)

Remark 2.1.23. Definition 2.1.13 defines dmψ as a map from Mm to M ′ψ(m). More-

over, in Definition 2.1.18, we introduced a natural map denoted by dψ between thetangent bundles T (M) and T (M ′). It may be thought that a third natural map can bedefined between X (M) and X (M ′) by mapping X ∈ X (M) into the vector field Y

such that Yψ(m) = dmψ(Xm). Unfortunately, in general, this defines a “vector field”only on the points of ψ(M) and not on the whole M ′. However, if ψ is a diffeomor-phism, this can be done as we describe below.

Definition 2.1.24 (dψ as a map on X (M)). Suppose ψ : M → M ′ is a C∞-diffeomorphism of differentiable manifolds M, M ′. We set

dψ : X (M) −→ X (M ′), X �→ dψ(X),

where, for every f ′ ∈ C∞(M ′, R),

{dψ(X)}m′(f ′) = dψ−1(m′)ψ(Xψ−1(m′))(f′) ∀ m′ ∈ M ′. (2.15a)

Since ψ is onto, (2.15a) is equivalent to set

{dψ(X)}ψ(m) = dmψ(Xm) for every m ∈ M . (2.15b)

We say that X ∈ X (M) and the above defined dψ(X) ∈ X (M ′) are ψ-related (seealso Definition 2.1.34). We leave to the reader the verification that dψ(X) is indeeda smooth vector field on M ′ according to the definition of X (M ′) in Remark 2.1.22.The following example shows that the above mapping X �→ dψ(X) appears natu-rally when a “change of variable” occurs.

Example 2.1.25 (Related vector fields via a diffeomorphism). We consider a sim-ple but significant example. Let T : R

N → RN be a C∞-diffeomorphism (i.e. an

invertible map, smooth together with its inverse function) from RN onto itself. We

consider on the domain of T a fixed system of Cartesian coordinates (x1, . . . , xN),and we equip the image of T (which still coincides with R

N = T (RN)) with the newsystem of coordinates y defined by y = T (x). We use the following notation


T : (RN, x) → (RN, y).

To every smooth vector field X on (RN, x), there corresponds a vector field X on(RN, y) in a natural way, roughly speaking, by expressing X in the new coordinatesy. Namely, given any f = f (y), f ∈ C∞((RN, y), R), we set

Xy(f ) := XT −1(y)(f ◦ T ),

also written asXf (y) := X(f ◦ T )(T −1(y)),

or equivalently,(Xf )(T (x)) = X(f ◦ T )(x). (2.16)

With reference to (2.16), we say that T turns X into X, or that X and X are T -related.Roughly speaking, X is the representation1 of X in the new system of coordinatesdefined by y = T (x).

1 A simple example is in order. Let R2 be equipped with coordinates x = (x1, x2) and

consider the linear change of coordinates given by

y = (y1, y2) = T (x1, x2) := (2x1 − x2, −5x1 + 3x2).

Following the above definition, the ordinary partial derivatives X1 := ∂x1 and X2 := ∂x2

are turned into the operators X1 and X2, respectively, where

X1f (y) = ∂

∂x1

∣∣∣∣∣x=T −1(y)

f (2x1 − x2, −5x1 + 3x2)

= 2∂y1f (y1, y2) − 5 ∂y2f (y1, y2),

X2f (y) = ∂

∂x2

∣∣∣∣∣x=T −1(y)

f (2x1 − x2, −5x1 + 3x2)

= −∂y1f (y1, y2) + 3 ∂y2f (y1, y2).

Since the Jacobian matrix of T equals

JT (x1, x2) =(

2 −1−5 3

),

then X1 and X2 are the vector fields whose component functions at y are respectively givenby

JT (T −1(y)) · X1I (T −1(y)) =(

2 −1−5 3

)·(

10

),

JT (T −1(y)) · X2I (T −1(y)) =(

2 −1−5 3

)·(

01

).

Hence ∂x1 is turned by T into 2 ∂y1 − 5 ∂y2 and ∂x2 is turned by T into − ∂y1 + 3 ∂y2 .Consequently, the ordinary Laplace operator Δ = (∂x1 )

2 + (∂x2 )2 is turned by T into the

following second order constant coefficient differential operator of elliptic type (which is asub-Laplacian on (R2, +)!)


If we write, as usual, X = ∇ · XI , the chain rule then gives

Xf (y) = ∇(f ◦ T )(T −1(y)) · XI (T −1(y))

= (∇yf )(y) · JT (T −1(y)) · XI (T −1(y)).

In other words, for every y ∈ (RN, y), one has

Xy = ∇y · (XI )(y) with (XI )(y) = JT (T −1(y)) · XI (T −1(y)), (2.17a)

or equivalently, for every x ∈ (RN, x),

XT (x) = ∇T (x) · (XI )(T (x)) with (XI )(T (x)) = JT (x) · XI (x). (2.17b)

Keeping in mind (2.9) of Remark 2.1.16, (2.17b) shows that

XT (x) = dxT (Xx), (2.17c)

which gives a significant interpretation of the differential map (at least in the presentcase when T is a diffeomorphism):

Given a vector field X, dxT (Xx) is the tangent vector at T (x) which representsXx with respect to the change of variable y = T (x). �

In what follows, we introduce an important definition. The adjectives “regular”and “smooth” will always mean “of class C∞”.

Definition 2.1.26 (Tangent vector to a curve). Let μ : [a, b] → M be a regularcurve. The tangent vector to the curve μ at time t is defined by

μ(t) := dtμ

(d

d r

∣∣∣∣r=t

)∈ Mμ(t). (2.18)

Hence, fixed t ∈ [a, b], if f is C∞ near μ(t), we have

μ(t)(f ) = d

d r

∣∣∣∣r=t

(f (μ(r))).

Remark 2.1.27. Note that, as Remark 2.1.9 shows, any tangent vector at a point of M

can be represented as the tangent vector to a certain curve at a suitable time.

Definition 2.1.28 (Integral curve). Let X be a smooth vector field on the differen-tiable manifold M . A regular curve μ : [a, b] −→ M is called an integral curve ofX if

μ(t) = Xμ(t) for every t ∈ [a, b]. (2.19)

Δ = 5(∂y1 )2 + 34(∂y2 )

2 − 26 ∂y1 ∂y2 .

In other words, Δ is simply the ordinary Laplace operator expressed in a new system ofcoordinates y in R

2. In Chapter 16, we show that every second order constant coefficientdifferential operator of elliptic type in R

N is simply the ordinary Laplace operator ex-pressed in a suitable new system of coordinates via a linear change of basis.


More explicitly, (2.19) means that

d

d r

∣∣∣∣r=t

(f (μ(r))) = X(f )(μ(t))

for every smooth function f on M and every t ∈ [a, b].We remark that if Xm = ∑N

i=1 ai(m)(∂/∂ xi)|m and (U, ϕ) is a coordinate sys-tem such that μ(t) ∈ U , then (2.19) is also equivalent to

d

d r

∣∣∣∣r=t

{ϕi(μ(r))

} = ai(μ(t)),

which becomes (setting γ (t) := ϕ(μ(t)) ∈ RN )

γi (t) = (ai ◦ ϕ−1)(γ (t)), i = 1, . . . , N, (2.20)

which is an ODE on RN . Equivalently, making more explicit the rôle of X, μ :

[a, b] → M is an integral curve of X if and only if, whenever (U, ϕ) is a coordinatesystem and (t0, t1) ⊂ [a, b] is such that μ(t) ∈ U for every t ∈ (t0, t1), it holdsμ(t) = ϕ−1(γ (t)), where

γ (t) = Xϕ−1(γ (t))(ϕ) ∀ t ∈ (t0, t1)

and Xm(ϕ) = (Xm(π1 ◦ ϕ), . . . , Xm(πN ◦ ϕ)).

Remark 2.1.29. By the above observation on the coordinate form of Definition 2.1.28and by recalling the existence “in small” for smooth systems of ODE’s, we inferthat, given any smooth vector field X on M and fixed any m ∈ M , there exists oneand only one integral curve of X passing through m at time 0. Hence the followingdefinition is well-posed.

Definition 2.1.30 (Complete vector field). Let X be a smooth vector field on thedifferentiable manifold M . We say that X is complete if, for every m ∈ M , theintegral curve μ of X such that μ(0) = m is defined on the whole R (i.e. its maximalinterval of definition is R).

2.1.5 Commutators. ϕ-relatedness

In the sequel, we denote by C∞(M, R) or, shortly, C∞(M) the set of the smoothreal-valued functions defined on a differentiable manifold M . It is immediate toobserve that if X is a smooth vector field on M and f ∈ C∞(M, R), we haveXf ∈ C∞(M, R). We explicitly recall that, here and in the sequel, we use the nota-tion in (2.13):

Xf : M → R, (Xf )(m) = Xm(f ).

As a consequence, the following definition is well posed.


Definition 2.1.31 (Commutators). Let X and Y be smooth vector fields on a differ-entiable manifold M . We define a vector field on M (called the commutator of X

and Y ) in the following way:

[X, Y ] : M → T (M), [X, Y ](m) := (m, [X, Y ]m),

where[X, Y ]m(f ) := Xm(Yf ) − Ym(Xf ) (2.21)

for every m ∈ M and every f ∈ C∞(M, R).

Definition 2.1.31 is well posed as it follows from (i) in the proposition below.

Proposition 2.1.32. If X, Y and Z are smooth vector fields on M , we have:

(i) [X, Y ] is a smooth vector field on M;(ii) [X, Y ]m = −[Y,X]m for every m ∈ M;

(iii) [[X, Y ], Z]m + [[Y,Z], X]m + [[Z,X], Y ]m = 0 for every m ∈ M .

Proof (Sketch). The verification that these facts hold true, can be performed via acomputation in a coordinate map. This reduces the above identities (ii) and (iii) toidentities between usual differential operators in R

N .For instance, let us prove (i). Let m ∈ M be fixed and choose any coordinate map

(U, ϕ) such that m ∈ M . According to Definition 2.1.20, we have

Xm(f ) =N∑

i=1

Xm(πi ◦ ϕ){∂i(f ◦ ϕ−1)

}(ϕ(m)).

As a consequence,

Xm(Yf ) =N∑

i,j=1

Xm(πi ◦ ϕ)

{∂

∂ ξi

(Yϕ−1(ξ)(πj ◦ ϕ) ∂ξj

(f ◦ ϕ−1)(ξ))}

(ϕ(m))

=N∑

i,j=1

Xm(πi ◦ ϕ) ·{∂ξi

(Yϕ−1(ξ)(πj ◦ ϕ)

)∂ξj

(f ◦ ϕ−1)(ξ)

+ Yϕ−1(ξ)(πj ◦ ϕ)∂2

∂ ξiξj

(f ◦ ϕ−1)(ξ)

}

ξ=ϕ(m)

.

By reversing the rôles of X and Y and then subtracting, we obtain (note, before thesecond equality sign, the cancellation of the second order terms)

Xm(Yf ) − Ym(Xf )

=N∑

i,j=1

Xm(πi ◦ ϕ){∂ξi

(Yϕ−1(ξ)(πj ◦ ϕ)

)∂ξj

(f ◦ ϕ−1)(ξ)}ξ=ϕ(m)


−N∑

i,j=1

Ym(πi ◦ ϕ){∂ξi

(Xϕ−1(ξ)(πj ◦ ϕ)

)∂ξj

(f ◦ ϕ−1)(ξ)}ξ=ϕ(m)

+N∑

i,j=1

Xm(πi ◦ ϕ) Ym(πj ◦ ϕ)

(∂2

∂ ξiξj

(f ◦ ϕ−1)

)(ϕ(m))

−N∑

i,j=1

Ym(πi ◦ ϕ)Xm(πj ◦ ϕ)

(∂2

∂ ξiξj

(f ◦ ϕ−1)

)(ϕ(m))

=N∑

j=1

{N∑

i=1

{Xm(πi ◦ ϕ) ∂ξi

(Yϕ−1(ξ)(πj ◦ ϕ)

)(ϕ(m))

− Ym(πi ◦ ϕ) ∂ξi

(Xϕ−1(ξ)(πj ◦ ϕ)

)(ϕ(m))

}}

∂ξj(f ◦ ϕ−1)(ϕ(m)).

Thus, [X, Y ]m is actually a tangent vector at m (see, for instance, Remark 2.1.8),and, following the notation in Definition 2.1.20, we have proved that, in a coordinatesystem (U, ϕ) around m,

[X, Y ]m(f ) =N∑

j=1

cj (m) ∂j (f ◦ ϕ−1)(ϕ(m)) with

cj (m) = [X, Y ]m(πj ◦ ϕ) = Xm

(Y(πj ◦ ϕ)

)− Ym

(X(πj ◦ ϕ)

) ∀ j ≤ N. (2.22)

We see that the maps cj ’s are smooth on U , so that, by Remark 2.1.21, [X, Y ] is asmooth vector field on M .

Now, (iii) in the assertion can be proved with an analogous coordinate-comput-ation, whereas (ii) is trivial. �Remark 2.1.33 (Commutators in X (M)). Consider the alternative definition ofsmooth vector field as an element of X (M), see Remark 2.1.33. The commu-tator operation rewrites as an operation on X (M) in the following way: GivenX, Y ∈ X (M), we consider the operator on C∞(M, R) defined by

[X, Y ] : C∞(M, R) → C∞(M, R), f �→ [X, Y ]f,

where([X, Y ]f )(m) := [X, Y ]mf = Xm(Yf ) − Ym(Xf ).

Then, obviously, [X, Y ] ∈ X (M) is the operator on C∞(M, R) related to the (usual)vector field [X, Y ].

With this meaning of the commutation, Proposition 2.1.32 rewrites as: If X, Y

and Z belong to X (M), we have:

(i) [X, Y ] ∈ X (M);(ii) [X, Y ] = −[Y,X];

(iii) [[X, Y ], Z] + [[Y,Z], X] + [[Z,X], Y ] = 0.


Definition 2.1.34 (ϕ-relatedness). Let ϕ : M → M be a C∞ function between twodifferentiable manifolds. The vector fields X on M and X on M are called ϕ-relatedif we have

dϕ ◦ X = X ◦ ϕ. (2.23a)

Here, dϕ is intended as a map from T (M) to T (M) (see Definition 2.1.18) and,as usual, the vector fields are maps from the underlying differentiable manifolds tothe relevant tangent bundles. Hence, (2.23a) is intended as an equality of functionsfrom M to T (M). Hence, ϕ-relatedness is equivalent to saying that the followingdiagram is commutative

M

X

ϕ

M

X

T (M)dϕ

T (M).

Since, for every m ∈ M , we have

(dϕ ◦ X)(m) = dϕ(m,Xm) = (ϕ(m), dmϕ(Xm)),

whereas(X ◦ ϕ)(m) = X(ϕ(m)) = (ϕ(m), Xϕ(m)),

then (2.23a) can be rewritten as the collection of identities

dmϕ(Xm) = Xϕ(m) ∀ m ∈ M (2.23b)

between tangent vectors at ϕ(m), i.e. between two functionals on C∞(M, R).Finally, condition (2.23a) is equivalent to the following ones:

Xm(f ◦ ϕ) = Xϕ(m)(f ), (2.23c)

X(f ◦ ϕ) = (Xf ) ◦ ϕ (2.23d)

for every m ∈ M and for every f smooth in a neighborhood of ϕ(m) in M .

Remark 2.1.35. When ϕ is a diffeomorphism, for every X ∈ X (M), there alwaysexists a (unique) X ∈ X (M) which is ϕ-related to X (see Remark 2.1.23), namely,with the notation of Definition 2.1.24, X = dϕ(X).

With reference to (2.17c), we have an interpretation of ϕ-relatedness when ϕ :R

N → RN is a diffeomorphism: X and X are ϕ-related if and only if X is the

expression of X in the new coordinates defined by the new Cartesian coordinatesy = ϕ(x). �Remark 2.1.36 (ϕ-relatedness in X (M)). Consider the alternative definition of vectorfield as in Remark 2.1.22. Then, the notion of ϕ-relatedness of vector fields rewritesas follows: X ∈ X (M) and X ∈ X (M) are ϕ-related if, for every f ∈ C∞(M, R),the functions X(f ◦ ϕ), (Xf ) ◦ ϕ on M do coincide, i.e.

Xm(f ◦ ϕ) = Xϕ(m)f for every m ∈ M,

or, equivalently, (2.23b) holds.

The following simple result will be soon of a crucial importance.


Proposition 2.1.37 (ϕ-relatedness and commutators). Let ϕ : M → M be a C∞function between two differentiable manifolds. Let X, Y be smooth vector fields onM and X, Y be smooth vector fields on M . If X is ϕ-related to X and Y is ϕ-relatedto Y , then [X, Y ] is ϕ-related to [X, Y ].Proof. We have to prove

dϕ ◦ [X, Y ] = [X, Y ] ◦ ϕ,

provided dϕ◦X = X◦ϕ and dϕ◦Y = Y ◦ϕ. To this end, let m ∈ M and f ∈ C∞(M).By the equivalent restatement (2.23c) of ϕ-relatedness, we have to show that

( ) dmϕ([X, Y ]m

)(f ) = [X, Y ]ϕ(m)(f ).

By definition of dmϕ, we have

dmϕ([X, Y ]m

)(f ) = [X, Y ]m(f ◦ ϕ)

(see (2.21)) = Xm

(Y(f ◦ ϕ)

)− Ym

(X(f ◦ ϕ)

)

(see (2.23d)) = Xm

((Y f ) ◦ ϕ

)− Ym

((Xf ) ◦ ϕ

)

= dmϕ(Xm)(Y (f )

)− dmϕ(Ym)(X(f )

)

(see (2.23b)) = Xϕ(m)

(Y (f )

)− Yϕ(m)

(X(f )

)

(see (2.21)) = [X, Y ]ϕ(m)(f ).

This gives ( ) thus ending the proof. �

2.1.6 Abstract Lie Groups

Definition 2.1.38 (Lie group). A Lie group G is a differentiable manifold G alongwith a group law ∗ : G × G −→ G such that the applications

G × G � (x, y) �→ x ∗ y ∈ G, G � x �→ x−1 ∈ G

are smooth.2

In the following, we shall always denote by e the identity of (G, ∗). Moreover,fixed σ ∈ G, we denote by τσ the left translation on G by σ , i.e. the map

G � x �→ τσ (x) := σ ∗ x ∈ G.

In case, when more than only one composition law is involved, we may also writeτ ∗σ instead of τσ .

2 The notion of “smoothness” of a map from G × G should be properly defined. The prod-uct of two differentiable manifolds is indeed endowed with a differentiable structure in anatural way. Here, we just remark that a function f : G × G → R is smooth if, for everycouple of coordinate systems (U, ϕ) and (V , φ) on G, the function

RN × R

N ⊇ ϕ(U) × φ(V ) � (u, v) �→ f(ϕ−1(u) ∗ φ−1(v)

)∈ R

is smooth. For more details see, e.g. [War83].


Definition 2.1.39 (Lie algebra). A (real) Lie algebra is a real vector space g with abilinear operation [·, ·] : g × g −→ g (called (Lie) bracket) such that, for every X,Y , Z ∈ g, we have:

1. (anti-commutativity) [X, Y ] = −[Y,X];2. (Jacobi identity) [[X, Y ], Z] + [[Y,Z], X] + [[Z,X], Y ] = 0.

A very remarkable fact is that, given any Lie group, there exists a certain finite-dimensional Lie algebra such that the group properties are reflected into propertiesof the algebra. For instance (as we shall see later on), any connected and simply con-nected Lie group is completely determined (up to isomorphism) by its Lie algebra.Therefore, the study of a Lie group is often reduced to the study of its Lie algebra.

Remark 2.1.40. Straightforwardly adapting the proof of Proposition 1.1.7 in Sec-tion 1.1 (see page 12), we have: if X1, . . . , Xm are elements of an (abstract) Liealgebra, then a system of generators of Lie{X1, . . . , Xm} is given by the commuta-tors

XI := [Xi1, [Xi2 , [Xi3, . . . [Xik−1, Xik ] . . .]]],where {i1, i2, . . . , ik} ⊆ {1, . . . , m} and I = (i1, i2, . . . , ik), k ∈ N. Indeed, theproof of Proposition 1.1.7 is only based on anti-commutativity and the Jacobi iden-tity.

2.1.7 Left Invariant Vector Fields and the Lie Algebra

Definition 2.1.41 (Left invariant vector fields). Let G be a Lie group. A smoothvector field X on G is called left invariant if, for every σ ∈ G, X is τσ -related toitself, i.e.

dτσ ◦ X = X ◦ τσ . (2.24)

Here dτσ is intended as a map from T (G) to itself (as in Definition 2.1.18). Asshown by the remarks after Definition 2.1.34, condition (2.24) is equivalent to thefollowing one:

(dxτσ )(Xx) = Xσ∗x ∀ x, σ ∈ G. (2.25a)

Applying (2.25a) at the identity e, it follows immediately that if X is a left invariantvector field, we have

deτσ (Xe) = Xσ ∀ σ ∈ G, (2.25b)

which proves that a left invariant vector field is determined by its action at the origin.Equality (2.25b) is actually equivalent3 to (2.25a). Moreover, (2.25a) can also bewritten as

3 Indeed, by applying (2.25b) with σ replaced by σ ∗x, subsequently (2.10) and finally usingagain (2.25b) with σ replaced by x, we have

Xσ∗x = deτσ∗x(Xe) = dxτσ

(deτx(Xe)

)= dxτσ (Xx).


Xx(f ◦ τσ ) = Xσ∗x(f )

for every x, σ ∈ G and every f ∈ C∞(G, R), (2.25c)

or again as (the most commonly used)

Xx

(y �→ f (σ ∗ y)

) = (Xf )(σ ∗ x), (2.25d)

i.e. comparing it to (1.18) (page 14), the very analogue of the usual left-invariance.Before giving the following central Definition 2.1.42, we pause a moment in

order to recall the multiple ways a smooth vector field can be thought of. A smoothvector field on G is a map X : G → T (G) such that, for every x ∈ G, it holdsX(x) = (x,Xx), where Xx ∈ Gx for every x ∈ G and such that, for every f ∈C∞(G, R), the function x �→ Xx(f ) is smooth on G. A smooth vector field can beidentified to the operator

X : C∞(G, R) −→ C∞(G, R),

f �→ Xf : G → R,

x �→ Xxf.

The set of the vector fields, as the above described operators, is denoted by X (G).Obviously, the set of the left invariant operators on G gives rise to a relevant subsetin X (G), following the above identification.

We are ready to give the following central definition.

Definition 2.1.42 (Algebra of a Lie group). Let G be a Lie group. Then the subsetof X (G) of the smooth left invariant vector fields on G is called the (Lie) algebraof G. It will be denoted by g.

More precisely, following Remark 2.1.22, we henceforth identify a left invariantvector field X on G with the following operator

X : C∞(G, R) → C∞(G, R)

such that, for every f ∈ C∞(G, R), the function Xf on G is defined by

(Xf )(x) := Xxf ∀ x ∈ G.

Hence, g is a (linear) set of endomorphisms on C∞(G, R),

g ⊆ X (G).

Note that, from the left invariance of X ∈ g, we have

(Xf )(x) = X(f ◦ τx)(e) ∀ x ∈ G ∀ f ∈ C∞(G, R).

Along with the above definition of the algebra of a Lie group, there is a wide com-monly used identification of g with Ge described in the following theorem.

Theorem 2.1.43 (The Lie algebra of a Lie group). Let G be a Lie group and g beits algebra. Then we have:


(i) g is a vector space, and the map

α : g −→ Ge,

X �→ α(X) := Xe

is an isomorphism between g and the tangent space Ge to G at the identity e

of G. As a consequence, dim g = dim Ge = dim G;(ii) The commutator of smooth left invariant vector fields (see also Remark 2.1.33)

is a smooth left invariant vector field;(iii) g with the commutation operation is a Lie algebra.

Proof. (i). It is evident that g (thought of as a subset of X (G)) is a vector space andthat α is linear.

Let us prove that α is injective. If α(X) = α(Y ), we have (see (2.25b))

Xσ = dτσ (Xe) = dτσ (α(X)) = dτσ (α(Y )) = dτσ (Ye) = Yσ ,

i.e. X = Y .Let us prove that α is surjective. If v ∈ Ge, we set

Xσ := (deτσ )(v) for every σ ∈ G.

Here, deτσ is the differential of the map τσ at the identity e of G. By definition, wehave Xσ ∈ Gσ , i.e. σ �→ (σ,Xσ ) is a vector field on G. Moreover, it is not difficult4

to prove that X is smooth. On the other hand, X is left invariant since

Xσ∗x = deτσ∗x(v) = (dxτσ ◦ deτx)(v) = dxτσ (Xx),

which gives (2.25a). Finally, we have

4 Indeed, let us prove that, for every f ∈ C∞(G, R), G � σ �→ Xσ f is smooth. Fixσ0 ∈ G, and let (U, ϕ) be a coordinate system around σ0. We have to prove that the mapϕ(U) � u �→ (Xf )(ϕ−1(u)) is smooth. By definition, it holds

(Xf )(ϕ−1(u)) = Xϕ−1(u)(f ) = (deτϕ−1(u))(v)(f ) = v(f ◦ τϕ−1(u)) = (�).

By Remark 2.1.8, fixed a coordinate system (E, χ) around e, there exists λ = λ(χ, e, v) ∈R

N such that

(�) =⟨λ, ∇

(f ◦ τϕ−1(u) ◦ χ−1

)(χ(e))

⟩=⟨λ, ∇v

(f (ϕ−1(u) ∗ χ−1(v))

)(χ(e))

⟩.

By the very definition of Lie group,

ϕ(U) × χ(E) � (u, v) �→ f (ϕ−1(u) ∗ χ−1(v))

is smooth, so that

ϕ(U) � u �→ ∇v

(f (ϕ−1(u) ∗ χ−1(v))

)(χ(e))

is smooth, and this ends the proof that X is smooth.


α(X) = Xe = deτe(v) = v,

i.e. α is surjective. Since α is an isomorphism of vector spaces, we have dim g =dim Ge = dim G (see Proposition 2.1.7).

(ii). Let σ ∈ G be arbitrary. Let X and Y ∈ g, i.e. X is τσ -related to itself and Y

is τσ -related to itself. From Proposition 2.1.37 it follows that [X, Y ] is τσ -related toitself, i.e. [X, Y ] ∈ g.

(iii). The last statement of the theorem follows from Proposition 2.1.32 (see alsoRemark 2.1.33). �

Incidentally, in the above proof, we have explicitly written the inverse map of thenatural identification

α : g −→ Ge,

X �→ α(X) := Xe.

Indeed, the inverse of α is given by

α−1 : Ge −→ g, v �→ X,

where Xσ = (deτσ )(v) for every σ ∈ G. (2.26)

Example 2.1.44 (The Lie algebra of (R,+)). It is obvious that the Lie algebra r ofthe usual Euclidean Lie group (R,+) is

span

{d

d r

},

whered

d r: C∞(R, R) → C∞(R, R), f �→ f ′.

With the usual formalism Xx for vector fields, this rewrites as

d

d r

∣∣∣∣t

f = f ′(t) for every t ∈ R.

Remark 2.1.45 (Left invariance in coordinates). We now make explicit the left in-variance condition in terms of the coefficients ai of the coordinate form of a smoothvector field

Xσ =N∑

i=1

ai(σ ) · ∂

∂ xi

∣∣∣∣σ

.

We know that a smooth vector field X is left invariant on G if and only if (see (2.25b))

( ) deτσ (Xe) = Xσ ∀ σ ∈ G.

In turn (see Remark 2.1.21), fixed a coordinate system (E, χ) around the identity e,( ) is equivalent to state that, for every σ ∈ G and for every (or, equivalently, for atleast one) coordinate system (U, ϕ) around σ , it holds


N∑

i=1

Xσ (ϕi) · ∂

∂ ui

(f ◦ ϕ−1)(ϕ(σ )) =N∑

i=1

Xe(χi) · ∂

∂ vi

(f ◦ τσ ◦ χ−1)(χ(e))

for every smooth function f around σ (or, equivalently, for the N smooth functionsϕ1, . . . , ϕN ). Writing f = f ◦ ϕ−1 ◦ ϕ, on the right-hand side the above becomes

N∑

i=1

Xσ (ϕi) · ∂

∂ ui

(f ◦ ϕ−1)(ϕ(σ ))

=N∑

j=1

{N∑

i=1

Xe(χi)∂

∂ vi

(ϕj ◦ τσ ◦ χ−1)(χ(e))

}· ∂

∂ uj

(f ◦ ϕ−1)(ϕ(σ ))

=N∑

j=1

{N∑

i=1

(Jϕ◦τσ ◦χ−1(χ(e))

)

j,iXe(χi)

}· ∂

∂ uj

(f ◦ ϕ−1)(ϕ(σ ))

=N∑

j=1

(Jϕ◦τσ ◦χ−1(χ(e)) · Xe(χ)

)

j· ∂

∂ uj

(f ◦ ϕ−1)(ϕ(σ )).

Here, Xe(χ) = (Xe(χ1), . . . , Xe(χN))T . Comparing the far left-hand and right-handsides, we rewrite all as the matrix identity

Jϕ◦τσ ◦χ−1(χ(e)) · Xe(χ) = Xσ (ϕ). (2.27)

Here, Xe(ϕ) = (Xe(ϕ1), . . . , Xe(ϕN))T .As a consequence, we have proved that a smooth vector field X on the Lie group

G is left invariant if and only if, fixed a coordinate system (E, χ) around the identitye, for every σ ∈ G, and for every (or, equivalently, for at least one) coordinate system(U, ϕ) around σ , it holds

Xσ =N∑

i=1

ai(σ )∂

∂ xi

∣∣∣∣σ

,

where

⎛

⎜⎝a1(σ )

...

aN(σ )

⎞

⎟⎠ = Jϕ◦τσ ◦χ−1(χ(e)) ·⎛

⎜⎝Xe(χ1)

...

Xe(χN)

⎞

⎟⎠ . (2.28)

Formula (2.28) is particularly useful for those Lie groups admitting a single globalcoordinate system defined on the whole group (such as Carnot groups or GLN(R),to give some example).

Remark 2.1.46 (Bracket in the identity). Since the Lie algebra can be identified toGe, it is interesting to analyze what form takes the bracket as seen as an operationon Ge.

Let (E, χ) be a fixed coordinate system around the identity e ∈ G. We know that(see (2.14) and (2.22))


Xef =N∑

j=1

λj ∂j |χ(e)(f ◦ χ−1),

Yef =N∑

j=1

μj ∂j |χ(e)(f ◦ χ−1),

[X, Y ]ef =N∑

j=1

νj ∂j |χ(e)(f ◦ χ−1),

where λj = Xe(χj ), μj = Ye(χj ), and

νj = Xe(Yχj ) − Ye(Xχj )

=∑

i

λi∂ui|χ(e)

(Yχ−1(u)χj

)−∑

i

μi∂ui|χ(e)

(Xχ−1(u)χj

)

(after a lengthy computation using also (2.28))

=N∑

i,k=1

(λiμk − μiλk)∂

∂ ui

∣∣∣∣χ(e)

∂

∂ vk

∣∣∣∣χ(e)

{χj

(χ−1(u) ∗ χ−1(v)

)}.

2.1.8 Homomorphisms

Definition 2.1.47 (Homomorphisms). Let (G, •) and (H, ∗) be Lie groups. A mapϕ : G −→ H is a homomorphism of Lie groups if it is C∞ and if

ϕ(x • y) = ϕ(x) ∗ ϕ(y) ∀ x, y ∈ G.

A map ϕ is an isomorphism of Lie groups if it is a homomorphism of Lie groupsand a diffeomorphism of differentiable manifolds. An isomorphism of G onto itself iscalled an automorphism of G.

Let (g, [·, ·]1) and (h, [·, ·]2) be Lie algebras. A map ϕ : g −→ h is a homomor-phism of Lie algebras if it is linear and if

ϕ([X, Y ]1

) = [ϕ(X), ϕ(Y )]2 ∀ X, Y ∈ g.

A map ϕ is an isomorphism of Lie algebras if it is a bijective homomorphism of Liealgebras. An isomorphism of g onto itself is called an automorphism of g.

Example 2.1.48. Suppose (G, •) is a Lie group and M is a differentiable manifold.Suppose T : G → M is a C∞-diffeomorphism. We can consider on M a Lie groupstructure naturally induced by • via T . Precisely, we equip M with the followingcomposition law

M × M � (x, y) �→ x ∗ y := T(T −1(x) • T −1(y)

) ∈ M. (2.29)

It is an easy exercise to verify that ∗ defines on M a Lie group structure, T : (G, •) →(M, ∗) is a Lie-group isomorphism, the identity of (M, ∗) is T (eG) (eG being theidentity of (G, •)) and, finally, the inverse of x in M with respect to ∗ is given by


x−1 = T((

T −1(x))−1

).

Now, let X be a •-left invariant vector field on G. We showed in Example 2.1.25 howto relate to X a vector field X on M simply setting (see, for instance (2.17c))

Xy = dT −1(y)T (XT −1(y)) for every y ∈ M . (2.30)

This obviously ensures that X and X are T -related according to Definition 2.1.34.Moreover, we claim that

X is ∗-left invariant. (2.31)

This is equivalent to

Xy(f (σ ∗ ·)) = Xσ∗y(f ) ∀ y, σ ∈ M ∀ f ∈ C∞(M, R).

Now, this last equality follows from the following calculation

Xy(f (σ ∗ ·)) =(see (2.30)) = {

dT −1(y)T (XT −1(y))}(f (σ ∗ ·))

= (XT −1(y))(f (σ ∗ T (·)))

(see (2.29)) = (XT −1(y)){f(T (T −1(σ ) • (·)))}

(X is •-left inv.) = (XT −1(σ )•T −1(y)){f(T (·))}

(see (2.29)) = (XT −1(σ∗y)){f(T (·))}

= (dT −1(σ∗y)T (XT −1(σ∗y))

)(f )

(see (2.30)) = Xσ∗y(f ).

Roughly speaking, this gives the following fact.Let T : R

N → RN be a C∞-diffeomorphism defining on R

N a change of vari-able y = T (x). Suppose the domain (RN, x) is equipped with a group law •, anddefine on the image (RN, y) the induced group law ∗ as in (2.29). Then, if X is aleft-invariant vector field w.r.t. • on (RN, x), the vector field X which expresses X

in the y-coordinates is a left-invariant vector field w.r.t. ∗ on (RN, y). �Let ϕ : G −→ H be a homomorphism of Lie groups. Since ϕ sends the identity

of G to the identity of H, the differential deϕ of ϕ is a linear map from Ge to He (forbrevity we denote by e the identity element both in G and in H).

Now, by means of the natural identification between Ge and the algebra g of G

and between He and the algebra h of H (see Theorem 2.1.43), deϕ thus induces anatural linear map between g and h, which we denote by dϕ.

In other words, the following definition naturally arises.

Definition 2.1.49 (Differential of a homomorphism). If ϕ : G −→ H is a homo-morphism of Lie groups, we define

dϕ : g −→ h


as the linear map such that, for every X ∈ g, dϕ(X) is the only element of h satisfying

{dϕ(X)}eH= deG

ϕ(XeG) (∈ HeH

). (2.32)

At this point, some confusion in the notation may occur. Indeed, Definition 2.1.49defines the differential of a Lie-group homomorphism ϕ : G → H as a map fromthe Lie-algebra g to the Lie-algebra h; but we already defined (see Definition 2.1.18)dϕ as a map between the relevant tangent bundles, when ϕ is smooth, and dϕ (seeDefinition 2.1.24) as a map between the smooth vector fields as operators, when ϕ isa diffeomorphism.

The following theorem shows that the three definitions are consistent (see alsoRemark 2.1.51).

Theorem 2.1.50. Let G and H be Lie groups with associated algebras g and h, re-spectively. Let ϕ : G −→ H be a homomorphism of Lie groups. Then:

(i) for every X ∈ g, we have that X and dϕ(X) are ϕ-related, i.e.{dϕ(X)

}ϕ(x)

= dxϕ(Xx) ∀ x ∈ G; (2.33)

Condition (2.33) characterizes dϕ(X), i.e. dϕ(X) is the only left invariant vectorfield Y ∈ h such that Yϕ(x) = dxϕ(Xx) for every x ∈ G;

(ii) dϕ : g −→ h is a homomorphism of Lie algebras.

Proof. Let X ∈ g be arbitrary. Let X := dϕ(X). We have to prove that (see Re-mark 2.1.36 and (2.23b))

( ) Xϕ(σ ) = dσ ϕ(Xσ ) ∀ σ ∈ G.

Let σ ∈ G. We shall use the notation τGσ and τH

ϕ(σ ) for the left translations on G

and on H and the notation eG, eH for the identity elements in G and H, respectively.Moreover, the group laws on G and H will be respectively denoted by ◦G and ◦H.Since ϕ is a Lie group homomorphism, we have

(τH

ϕ(σ ) ◦ ϕ)(·) = ϕ(σ) ◦H ϕ(·) = ϕ(σ ◦G ·) = (ϕ ◦ τG

σ )(·),i.e. the following identity of maps on G holds

τH

ϕ(σ ) ◦ ϕ ≡ ϕ ◦ τG

σ . (2.34)

As a consequence, we have (also recall that X and X are left invariant)

Xϕ(σ ) =(see (2.25b)) = deH

τH

ϕ(σ )(XeH)

(see (2.32)) = deHτH

ϕ(σ )(deGϕ(XeG

)) = deG(τH

ϕ(σ ) ◦ ϕ)(XeG)

(see (2.34)) = deG(ϕ ◦ τG

σ )(XeG) = dσ ϕ(deG

τG

σ (XeG))

(see (2.25b)) = dσ ϕ(Xσ ).

This is precisely ( ), and (i) is proved.


Finally, we have to prove

dϕ([X, Y ]) = [dϕ(X), dϕ(Y )] ∀ X, Y ∈ g.

By means of the first part (i) of the assertion, we have that X (resp. Y ) is ϕ-related to dϕ(X) (resp. dϕ(Y )). Thus, by Proposition 2.1.37, [X, Y ] is ϕ-related to[dϕ(X), dϕ(Y )], i.e.

dxϕ([X, Y ]x) = [dϕ(X), dϕ(Y )]ϕ(x) ∀ x ∈ G.

In particular, for x = eG, we have

deGϕ([X, Y ]eG

) = [dϕ(X), dϕ(Y )]eH.

Now, by definition, dϕ([X, Y ]) is the unique vector field in h whose value in eH isdeG

ϕ([X, Y ]eG), then dϕ([X, Y ]) = [dϕ(X), dϕ(Y )], and the theorem is completely

proved. �Remark 2.1.51 (Consistency of the notation “dϕ”). Let G and H be Lie groups withassociated algebras g and h, respectively. Let ϕ : G → H be a homomorphism ofLie groups. In Definition 2.1.18, dϕ was defined as the map

dϕ : T (G) → T (H), dϕ(x,Xx) = (ϕ(x), Yϕ(x)),

where Yϕ(x) = dxϕ(Xx),

Hence, by Theorem 2.1.50, dϕ, as defined in Definition 2.1.49, is just the “projec-tion” of the previously defined dϕ on the second component.

In case ϕ is also an isomorphism, we have defined dϕ in Definition 2.1.24.A comparison of (2.15b) and (2.33) shows that dϕ and dϕ of Definition 2.1.49 actu-ally coincide. �Remark 2.1.52. (i) From Theorem 2.1.50 it immediately follows that if ϕ : G −→ H

is a Lie group isomorphism then dϕ : g −→ h is a Lie algebra isomorphism.(ii) Let us now suppose we are given two Lie groups on R

N as defined in Chap-ter 1: (G, ◦) and (H, ∗). Let the associated Lie-algebras be g and h, respectively.Suppose it is given a Lie group isomorphism ϕ from G to H. As usual, the identityelements in G and H are supposed to be the origin 0 of R

N . Then dϕ sends the ◦-Jacobian basis of g in a basis of h that in 0 coincides with the column vectors of thematrix Jϕ(0).

Indeed, if Z1, . . . , ZN is the Jacobian basis related to (G, ◦) and if f ∈C∞(H, R), we have

{dϕ(Zk)}0(f ) = {d0ϕ((Zk)0)}(f ) = {Zk}0(f ◦ ϕ)

= ∂xk|0(f ◦ ϕ) =

N∑

j=1

(∂ξjf )(0) · (∂xk

ϕj )(0).

In particular, dϕ sends the ◦-Jacobian basis of g in the ∗-Jacobian basis of h if andonly if Jϕ(0) is the identity matrix.


(iii) Let F, G, H be three Lie groups with Lie algebras f, g, h. Let

ϕ : F → G, ψ : G → H

be Lie-group homomorphisms. Then d(ψ ◦ ϕ) = dψ ◦ dϕ. Indeed, for every X ∈ f,d(ψ ◦ ϕ)(X) is the only element of h such that

( )(d(ψ ◦ ϕ)(X)

)eH

= deF(ψ ◦ ϕ)(XeF

).

But also (dψ ◦ dϕ)(X) when applied on eH coincides with the right-hand side of ( ).Indeed,

((dψ ◦ dϕ)(X)

)eH

= (dψ(dϕ(X)

))eH

= deGψ((

dϕ(X))eG

)

= deGψ(deF

ϕ(XeF)) = deF

(ψ ◦ ϕ)(XeF). �

2.1.9 The Exponential Map

We begin with a simple but crucial result on Lie groups, following from an ODE’sresult.

We recall that, according to Definition 2.1.30, a smooth vector field X on a Liegroup G is complete if, for every x ∈ G, the integral curve μ of X such that μ(0) = x

is defined on the whole R.

Proposition 2.1.53 (Completeness of the left invariant vector fields). The left in-variant vector fields on a Lie group G are complete.

Proof. Let X ∈ g be fixed, g being the Lie algebra of G. By simple prolongationresults for ODE’s, it is enough to prove that there exists ε = ε(X) > 0 such that, forevery x ∈ G, the integral curve μx of X such that μx(0) = x is defined on [−ε, ε].

Let e be the identity of G. Let ε > 0 be such that the integral curve μe of X withμe(0) = e is defined on [−ε, ε]. Then, it is an immediate consequence of the leftinvariance of X to derive that the map

[−ε, ε] � t �→ (τx ◦ μe)(t) =: ν(t) ∈ G

coincides with the above μx . Indeed, ν(0) = τx(μe(0)) = τx(e) = x and (see theDefinition 2.1.26 for ν(t))

ν(t) = dt ν

(d

d r

∣∣∣∣r=t

)= dt (τx ◦ μe)

(d

d r

∣∣∣∣r=t

)

= (dμe(t)τx ◦ dtμe

)( d

d r

∣∣∣∣r=t

)

= dμe(t)τx(μe(t)) = dμe(t)τx(Xμe(t))

(see (2.24)) = X(τx◦μe)(t) = Xν(t).



Definition 2.1.54 (The exponential curve expX(t)). Let G be a Lie group with Liealgebra g. Let X ∈ g be fixed. By Proposition 2.1.53, the integral curve μ(t) of X

passing through the identity of G when t = 0 is defined on the whole R. We set

expX(t) := μ(t).

By the very Definition 2.1.28 of integral curve, we have

expX : R → G with

⎧⎨

⎩

expX(0) = eG,

dt expX

(d

d r

∣∣∣∣r=t

)= XexpX(t) ∀ t ∈ R.

(2.35)

In terms of functionals on C∞(G, R), (2.35) can be written more explicitly as(

d

dr

)

r=t

{f (expX(r))

} = XexpX(t)(f ) ∀ f ∈ C∞(G, R). (2.36a)

In particular, when t = 0,(

d

dr

)

r=0

{f (expX(r))

} = Xe(f ) ∀ f ∈ C∞(G, R). (2.36b)

Again from (2.35) with t = 0 we infer

d0 expX

{(d

dt

)

0

}= Xe. (2.36c)

Remark 2.1.55 (The exponential curve as a homomorphism). It is a simple exerciseon ODE’s (see also Ex. 9 at the end of Chapter 1) to verify that

expX : (R,+) −→ (G, ∗)

is a Lie group homomorphism. In other words, it holds

expX(r + s) = expX(r) ∗ expX(s) for every r, s ∈ R.

Hence, Definition 2.1.49 can be applied. The differential of expX is the followingmap (here r is the Lie algebra of (R,+))

d expX : r −→ g is such that λd

dt�→ λ X ∀ λ ∈ R,

i.e. it holds

d expX

(d

dt

)= X. (2.37)

Indeed, by definition of the differential of a homomorphism, d expX(d/dt) is theunique vector field of g such that

{d expX

(d

dt

)}

e

= d0 expX

{(d

dt

)

0

}.

The above right-hand side equals Xe thanks to (2.36c), whence (2.37). �


For future reference, we collect some other useful formulas for expX(t), imme-diate consequence of the facts proved above.

Theorem 2.1.56. Let (G, ∗) be a Lie group with algebra g. Let X ∈ g. Then:

(i) expX(r + s) = expX(r) ∗ expX(s) for every r, s ∈ R;(ii) expX(−t) = (expX(t))−1 for every t ∈ R;

(iii) expX(0) = e;(iv) R � t �→ expX(t) ∈ G is a smooth curve;(v) expX(t) is the unique integral curve of X passing through the identity at time

zero, so that, for every x ∈ G,

t �→ x ∗ (expX(t))

is the unique integral curve of X passing through x at time zero.

Note that (i) in the above theorem jointly with the left invariance of X and (2.36c)gives back5 (2.36b).

We are ready to give the fundamental definition.

Definition 2.1.57 (Exponential map). Let (G, ∗) be a Lie group with Lie algebra g.Following the notation in Definition 2.1.54, we set

Exp : g −→ G,

X �→ Exp(X) := expX(1).

Exp is called the exponential map (related to the Lie group G).

The following results hold.

Proposition 2.1.58. Let (G, ∗) be a Lie group with Lie algebra g. For every X ∈ g,we have

(i) Exp(t X) = expX(t) for every t ∈ R;(ii) Exp((r + s)X) = Exp(r X) ∗ Exp(sX) for every r, s ∈ R;

(iii) Exp(−tX) = (Exp(tX))−1, for every t ∈ R.

5 Indeed, a direct computation gives(

d

dr

)

r=t

{f (expX(r))

}=(

d

dr

)

r=0

{f (expX(r + t))

}

=(

d

dr

)

r=0

{f (expX(t) ∗ expX(r))

}=(

d

dr

)

r=0

{(f ◦ τexpX(t))(expX(r))

}

={

d0 expX

{(d

dr

)

r=0

}}(f ◦ τexpX(t)) = Xe(f ◦ τexpX(t)) = XexpX(t)(f ).

Here we used (2.36c) and (2.25c).


Proof. Fix t ∈ R and consider the curve s �→ μ(s) := expX(s t). We have μ(0) =expX(0) = e. By Theorem 2.1.56, we have

μ(s) = t XexpX(s t) = t Xμ(s).

Thus μ is the integral curve of tX, and, again by Theorem 2.1.56, we have μ(s) =exptX(s), i.e.

expX(s t) = exptX(s).

For s = 1, we have expX(t) = exptX(1) = Exp (t X).

Therefore, this yields

Exp ((r + s)X) = expX(r + s) = expX(r) ∗ expX(s) = Exp (r X) ∗ Exp (s X),

and moreover,

Exp (−t X) = expX(−t) = (expX(t)

)−1 = (Exp (t X)

)−1.

The proposition is thus completely proved. �Theorem 2.1.59. Let G and H be Lie groups with associated algebras g and h. Wedenote by ExpG and ExpH the exponential maps related to G and to H, respectively.Finally, let ϕ : G −→ H be a Lie group homomorphism. Then the following diagramis commutative:

Gϕ

H

g

ExpG

dϕh.

ExpH

Proof. Let X ∈ g. We have to prove that

Exp H(dϕ(X)) = ϕ(Exp G(X)

). (2.38)

We set for brevity x := Exp H(dϕ(X)). By definition, we have

x = expH

dϕ(X)(1),

where expH

dϕ(X)(t) is the unique integral curve of dϕ(X) on H passing through theidentity eH of H at time zero (see Theorem 2.1.56). We set y := ϕ(Exp G(X)), i.e.

y = ϕ(expG

X(1)),

where expG

X(t) is the unique integral curve of X on G passing through the identityeG of G at time zero. We want to show that

expH

dϕ(X)(t) = ϕ(expG

X(t)) ∀ t ∈ R. (2.39)

When t = 1, this gives x = y, which is the claimed (2.38).


To this end, we show that μ(t) := ϕ(expG

X(t)) is an integral curve of dϕ(X) on H

passing through eH at time zero. By uniqueness, this will yield μ(t) = expH

dϕ(X)(t),i.e. (2.39). Since ϕ is a homomorphism, we have

μ(0) = ϕ(expG

X(0)) = ϕ(eG) = eH.

Finally, since ϕ and expG

X are Lie-group homomorphisms, μ is a homomorphism.Thus, we have

μ(t) = dtμ

(d

d r

∣∣∣∣r=t

)=(

dμ

(d

d t

))

μ(t)

=(

d(ϕ ◦ expG

X

)( d

d t

))

μ(t)

=((

dϕ ◦ dexpG

X

)( d

dt

))

μ(t)

= (dϕ(X)

)μ(t)

.

In the first equality, we used (2.18); in the second, (2.33); in the third, we exploitedthe very definition of μ; the fourth follows from Remark 2.1.52-(iii); in the fifthequality, we used (2.37). This ends the proof. �Remark 2.1.60 (Some notable commutative diagrams). Theorem 2.1.59 has some im-portant consequences: suppose that Exp G and Exp H are diffeomorphisms with in-verse functions Log G and Log H, respectively. Let us analyze the following commu-tative diagram:

Gϕ

H

Log H

gdϕ

Exp G

h.

We deduce that, given the Lie group homomorphism ϕ, the map

Log H ◦ ϕ ◦ Exp G : g −→ h

coincides with the differential of ϕ, which is a Lie algebra homomorphism (in par-ticular, a linear map!).

Under the same hypotheses on Exp G and Exp H, suppose we are given a Liealgebra homomorphism ψ : g −→ h. Let us consider the map

Exp H ◦ ψ ◦ Log G : G −→ H.

If such a map is a Lie group homomorphism,6 then the differential of this map coin-cides with ψ . Indeed, the following diagram is commutative

GExp H◦ψ◦Log G

Log G

H

gψ

h.

Exp H

6 We shall see that Exp H ◦ψ ◦Log G is always a Lie group homomorphism whenever G andH are connected and simply connected nilpotent Lie groups.

2.2 Carnot Groups 121

Finally, given a Lie group homomorphism ϕ : G −→ H, by the commutative dia-gram

Gϕ

Log G

H

gdϕ

h

Exp H

the mapExp H ◦ dϕ ◦ Log G : G −→ H

coincides with ϕ and is thus a Lie group homomorphism.(For other related topics7 not employed in this book, see, e.g. [War83].)

2.2 Carnot Groups

We give two definitions of Carnot groups: the first one (which we already introducedin Chapter 1, Section 1.4, page 56) is the most convenient for our purposes andit seems very natural in an Analysis context; the second one is the classical onefrom Lie group theory. Then, we compare the two definitions showing that, up toisomorphism, they are equivalent. For reader’s convenience, we recall the definitionof homogeneous Carnot group.

Definition 2.2.1 (Homogeneous Carnot group). Let RN be equipped with a Lie

group structure by the composition law ◦. Let also RN be equipped with a homo-

geneous group structure by a family {δλ}λ>0 of automorphisms of (RN, ◦) (calleddilations) of the following form

δλ(x(1), x(2), . . . , x(r)) = (λx(1), λ2x(2), . . . , λrx(r)). (2.40)

Here x(i) ∈ RNi for i = 1, . . . , r and N1 +N2 +· · ·+Nr = N . Let g be the algebra

of the group (RN, ◦). For i = 1, . . . , N1, let Zi be the (unique) vector field of g

agreeing with ∂/∂ xi at the origin.If the following assumption holds

(H1) the Lie algebra generated by Z1, . . . , ZN1 is the whole g,

then G = (RN, ◦, δλ) is called a homogeneous Carnot group.

7 For example, the following results hold.

Theorem 2.1.61. Let G and H be Lie groups with related algebras g and h.

1. Suppose that G is connected. Let ϕ, ψ : G −→ H be Lie group homomorphisms. Thenϕ and ψ coincide if and only if the Lie algebra homomorphisms dϕ and dψ from g to h

coincide.2. Suppose that G is simply connected. Let ψ : g −→ h be a Lie algebra homomorphism.

Then there exists a unique Lie group homomorphism ϕ : G −→ H such that dϕ = ψ .3. Suppose that G and H are simply connected. If g and h are isomorphic Lie algebras, G

and H are isomorphic Lie groups.


Given a Lie algebra h, for any two subsets V , W of h, we set

[V,W ] = span{[v,w] ∣∣v ∈ V,w ∈ W

}.

Remark 2.2.2. Let m ∈ N, and let X1, . . . , Xm ∈ h. Then, for every k ∈ N, any Liebracket of height k of X1, . . . , Xm is a linear combination of nested brackets of theform (see Proposition 1.1.7, page 12)

[Xi1, [Xi2, . . . [Xik−1, Xik ] . . .]], i1, i2, . . . , ik ∈ {1, . . . , m}.Definition 2.2.3 (Stratified Lie group). A stratified group (or Carnot group) H is aconnected and simply connected Lie group whose Lie algebra h admits a stratifica-tion, i.e. a direct sum decomposition

h = V1 ⊕ V2 ⊕ · · · ⊕ Vr such that

{[V1, Vi−1] = Vi if 2 ≤ i ≤ r,

[V1, Vr ] = {0}. (2.41)

Convention. In the rest of the chapter, we shall use the term “stratified group” to re-fer to a Carnot group according to the above abstract, classical Definition 2.2.3. Weshall instead use the prefix “homogeneous” for Carnot groups as in Definition 2.2.1.From Chapter 3 onwards, after we have proved that (see Theorem 2.2.18) every strat-ified group is isomorphic to a homogeneous Carnot group, we shall use the term“Carnot” group almost indifferently. Yet, whenever the homogeneous structure willhave to be stressed, or whenever it will be important to distinguish stratified versushomogeneous Carnot groups, we shall re-invoke the prefix homogeneous.

Remark 2.2.4. Our Definition 2.2.1 is very natural to deal with in an analytic context.However, its only inconvenient property is that it is not invariant under isomorphismof Lie groups, since the homogeneity property heavily depends on the choice of coor-dinates on the underlying manifold R

N . For example, consider the following grouplaw in R

3:

x ◦ y = (arcsinh(sinh(x1) + sinh(y1)), x2 + y2 + sinh(x1)y3, x3 + y3

).

It is an easy exercise to prove that (R3, ◦) is a homogeneous Carnot group8 accordingto Definition 2.2.3: indeed, the stratification is given by

span{(cosh(x1))−1 ∂x1 , ∂x3 + sinh(x1)∂x2} ⊕ span{∂x2}.

It is also evident that (R3, ◦) is not a homogeneous group (if it were so, the com-position law should have polynomial component functions!, see Theorem 1.3.15).However, (R3, ◦) is isomorphic to the homogeneous Carnot group H

1 = (R3, ∗, δλ),being

8 We remark that ((cosh(x1))−1∂1

)2 +(∂3 + sinh(x1)∂2

)2

is the canonical sub-Laplacian on this group.


x ∗ y = (x1 + y1, x2 + y2, x3 + y3 + 2x2y1 − 2x1y2),

andδλ(x) := (λx1, λx2, λ

2x3).

The main goal of this section is to prove that in every equivalence class of isomor-phic stratified groups there is one group which is homogeneous, according to Defin-ition 2.2.1.

Example 2.2.5. Consider the open subset of R3

Ω := (0,∞) ×(

−π

2,π

2

)× R.

Then Ω is equipped with a structure of (non-homogeneous) Carnot group by thecomposition (ξ = (ξ1, ξ2, ξ3) ∈ Ω and analogously for η ∈ Ω)

ξ ∗ η =⎛

⎝ξ1 η1

arctan(ξ1 + η1 + tan ξ2 + tan η2 − ξ1η1)

ξ3 + η3 + 2(ξ1 ln η1 + tan ξ2 ln η1 − η1 ln ξ1 − tan η2 ln ξ1)

⎞

⎠ .

This can be seen, for example, by remarking that G = (Ω, ∗) is isomorphic to theHeisenberg–Weyl group H

1 (see Example 1.2.2) via the isomorphism

ϕ : Ω → H1, ϕ(ξ1, ξ2, ξ3) = (ln ξ1, ξ1 + tan ξ2, ξ3).

Indeed, recalling that the composition law on H1 is given by

(x1, x2, x3) ◦ (y1, y2, y3) = (x1 + y1, x2 + y2, x3 + y3 + 2 (x2 y1 − x1 y2)

),

and the inverse map of ϕ is

ϕ−1 : H1 → Ω, ϕ−1(x1, x2, x3) = (

exp(x1), arctan(x2 − exp(x1)), x3),

we notice thatξ ∗ η = ϕ−1(ϕ(ξ) ◦ ϕ(η)) ∀ ξ, η ∈ G.

This suffices to prove that G is a Carnot group. Equivalently, we show the stratifica-tion condition for g, the algebra of G: first we find the ∗-left-invariant vector fieldswhich coincide with the partial derivatives ∂ξ1 , ∂ξ2 , ∂ξ3 at the identity of G, say

eG = ϕ−1(0, 0, 0) =(

1,−π

4, 0

).

These vector fields Z1, Z2, Z3 can be found via the following formula (here f ∈C∞(Ω, R))

(Zif )(ξ) = ∂

∂ ηi

∣∣∣∣η=eG

(f (ξ ∗ η)) ∀ ξ ∈ G.

A direct computation then shows that


Z1 = ξ1 ∂ξ1 + 1 − ξ1

1 + tan2(ξ2)∂ξ2 + (2ξ1 + 2 tan ξ2 − 2 ln ξ1) ∂ξ3 ,

Z2 = 2

1 + tan2(ξ2)∂ξ2 − 4 ln ξ1 ∂ξ3 ,

Z3 = ∂ξ3 .

Equivalently, the Zi’s are the vector fields whose column-vector of the coefficientsis given by the columns of the Jacobian matrix of the map τξ (η) = ξ ∗ η at eG,

Jτξ (eG) =⎛

⎝ξ1 0 0

1−ξ11+tan2(ξ2)

21+tan2(ξ2)

0

2ξ1 + 2 tan ξ2 − 2 ln ξ1 −4 ln ξ1 1

⎞

⎠ .

As a consequence, the stratification of g is given by

g = span{Z1, Z2} ⊕ span{[Z1, Z2]},and all commutators of Z1, Z2 with length > 2 vanish identically. Indeed, anotherdirect computation gives

[Z1, Z2] = −8 ∂ξ3 .

We can now directly check the validity of Theorem 2.1.50-(2), i.e. that dϕ is analgebra-isomorphism from g to h1, the algebra of H

1. To this end, we remark that dϕ

is the map that to a vector field Z ∈ g assigns the vector field in h1 whose vector ofthe coefficients (at x ∈ H

1) is given by

(dϕ(Z))I (x) = Jϕ(ϕ−1(x)) · (ZI)(ϕ−1(x)).

Now, we have

Jϕ(ξ) =⎛

⎝ξ−1 0 0

1 1 + tan2(ξ2) 00 0 1

⎞

⎠ ,

Jϕ(ϕ−1(x)) =⎛

⎝e−x1 0 0

1 1 + (x2 − ex1 )2 00 0 1

⎞

⎠ .

Consequently, it holds

(dϕ(Z1))I (x) =⎛

⎝e−x1 0 0

1 1 + (x2 − ex1)2 00 0 1

⎞

⎠

⎛

⎝ex1

1−ex1

1+(x2−ex1 )2

2x2 − 2x1

⎞

⎠ =⎛

⎝11

2x2 − 2x1

⎞

⎠ .

This means that

dϕ(Z1) = ∂x1 + ∂x2 + (2x2 − 2x1) ∂x3 = X1 + X2,


where, as usual, X1 = ∂x1 + 2x2∂x3 , X2 = ∂x2 − 2x1∂x3 are the generators of thealgebra of H

1. Analogously, we have

(dϕ(Z2))I (x) =⎛

⎝e−x1 0 0

1 1 + (x2 − ex1)2 00 0 1

⎞

⎠

⎛

⎝02

1+(x2−ex1 )2

−4x1

⎞

⎠ =⎛

⎝02

−4x1

⎞

⎠ ,

i.e.dϕ(Z2) = 2∂x2 − 4x1 ∂x3 = 2X2.

Moreover,

(dϕ([Z1, Z2]))I (x) =⎛

⎝e−x1 0 0

1 1 + (x2 − ex1)2 00 0 1

⎞

⎠

⎛

⎝00

−8

⎞

⎠ =⎛

⎝00

−8

⎞

⎠ ,

i.e.dϕ([Z1, Z2]) = −8 ∂x3 = 2[X1, X2].

Finally, we can straightforwardly check the algebra-isomorphism condition for dϕ

(see Theorem 2.1.50-(2)), namely

dϕ([Z1, Z2]) = 2[X1, X2] = [X1 + X2, 2X2] = [dϕ(Z1), dϕ(Z2)].This completes our example. �

2.2.1 Some Properties of the Stratification of a Carnot Group

If H is a Carnot group, its Lie algebra admits at least a stratification, but it can havemore than one. For example, if H = H

1 is the Heisenberg–Weyl group on R3 (see

Example 1.2.2 and its notation), its Lie algebra admits the stratifications

span{X1, X2} ⊕ span{[X1, X2]},span{X1 − 3 [X1, X2], X2} ⊕ span{[X1, X2]},span{X1 + X2, 3X1 + [X1, X2]} ⊕ span{[X1, X2]}.

Definition 2.2.6 (Basis adapted to the stratification). Let H be a Carnot group. LetV = (V1, . . . , Vr ) be a fixed stratification of the Lie algebra h of H as in (2.41). Wesay that a basis B of h is adapted to V if

B =(

E(1)1 , . . . , E

(1)N1

; . . . ; E(r)1 , . . . , E

(r)Nr

), (2.42)

where, for i = 1, . . . , r , we have Ni := dim Vi , and

(E

(i)1 , . . . , E

(i)Ni

)is a basis for Vi .

Obviously, every Carnot group admits an adapted basis to any of its stratifica-tions.


Definition 2.2.7 (Step and number of generators). Let H be a stratified group. Let(V1, . . . , Vr ) be any stratification of the algebra of H, as in Definition 2.2.3. We saythat H as step (of nilpotency) r and has m generators, where m := dim(V1).

The following proposition shows that the above definitions are well-posed, i.e.they do not depend on the particular stratification of H.

Proposition 2.2.8. Let H be a stratified group. Suppose that (V1, . . . , Vr ) and(V1, . . . , Vr ) be any two stratifications of the algebra of H, as in Definition 2.2.3.Then r = r and dim(Vi) = dim(Vi) for every i = 1, . . . , r . Moreover, the algebra ofH is a nilpotent Lie algebra of step r .

Hence, the natural number

Q :=r∑

i=1

idim(Vi)

depends only on the stratified nature of H and not on the particular stratification. Q

is called the homogeneous dimension of H.

Proof. From the very Definition 2.2.3, we see that if (V1, . . . , Vr ) is a stratificationof h, the algebra of H, then h is a nilpotent9 Lie algebra of step r . Hence r dependsonly on h and not on the stratification.

The fact that dim(Vi) = dim(Vi) for every i = 1, . . . , r is not trivial and followsfrom Lemma 2.2.9 below. �Lemma 2.2.9 (The two-stratification lemma). Let H be a stratified group with Liealgebra h. Suppose V := (V1, . . . , Vr ) and W := (W1, . . . , Wr) are two stratifica-tions of h.

Then, for every couple of bases V and W of h respectively adapted to the strati-fications V and W , the transition matrix between the two bases is non-singular andhas the block-triangular form

⎛

⎜⎜⎜⎜⎝

M(1) 0 · · · 0

� M(2) . . ....

.... . .

. . . 0� · · · � M(r)

⎞

⎟⎟⎟⎟⎠,

9 We recall that, given an abstract Lie algebra (g, [·, ·]), g is called nilpotent of step r , r ∈ N,if for every X1, . . . , Xr+1 ∈ g,

[X1, [X2, . . . [Xr, Xr+1] . . . ]] = 0,

and there exist Y1, . . . , Yr ∈ g such that

[Y1, [Y2, . . . [Yr−1, Yr ] . . . ]] �= 0.


where, for every i = 1, . . . , r , the block M(i) is a Ni × Ni non-singular matrix(Ni being the common value of dim(Vi) = dim(Wi)).

Proof. Let the notation in the assertion be fixed. We also set, for every i = 1, . . . , r ,Ni = dim(Vi) and Mi = dim(Wi). From V1 ⊆ W1 ⊕ · · · ⊕ Wr and from thestratification condition for V and for W , we infer that

Vi = [V1, · · · [V1, V1] · · ·]︸︷︷︸i times

⊆[{W1 ⊕ · · ·}, · · · [{W1 ⊕ · · ·}, {W1 ⊕ · · ·}] · · ·

]

⊆ Wi ⊕ Wi+1 ⊕ · · · ⊕ Wr.

Hence (the second column obtained by reversing the rôles of V and W ),

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

V1 ⊆ W1 ⊕ W2 ⊕ · · · ⊕ Wr, W1 ⊆ V1 ⊕ V2 ⊕ · · · ⊕ Vr,

......

Vr−1 ⊆ Wr−1 ⊕ Wr, Wr−1 ⊆ Vr−1 ⊕ Vr,

Vr ⊆ Wr, Wr ⊆ Vr .

All this proves that

( )

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

W1 ⊕ W2 ⊕ · · · ⊕ Wr = V1 ⊕ V2 ⊕ · · · ⊕ Vr,

...

Wr−1 ⊕ Wr = Vr−1 ⊕ Vr,

Wr = Vr .

In particular (recalling that dim(A ⊕ B) = dim(A) + dim(B)), this yields

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

M1 + M2 + · · · + Mr = N1 + N2 + · · · + Nr,

...

Mr−1 + Mr = Nr−1 + Nr,

Mr = Nr,

whence, Ni = Mi , i.e. dim(Vi) = dim(Wi) for every i = 1, . . . , r .From ( ) it immediately follows the block-form for the matrix of the change of

basis between V and W . �The following proposition shows that “to be a Carnot group” is an invariant under

isomorphism of Lie groups.

Proposition 2.2.10. Let H be a stratified group. Suppose G is a Lie group isomor-phic to H. Then G is a stratified group too. Moreover, H and G have the same step,the same number of generators and even the dimensions of the layers of the relevant


stratifications are preserved. Also, H and G have the same homogeneous dimen-sion Q.

More precisely, suppose ϕ : H → G is a Lie group isomorphism and that(V1, . . . , Vr ) is a stratification of h, the algebra of H, as in Definition 2.2.3. Then, ifg is the algebra of G, a stratification for g is given by (dϕ(V1), . . . , dϕ(Vr)), wheredϕ is the differential of ϕ (see Definition 2.1.49) which is an isomorphism of Liealgebras (and of vector spaces).

Proof. We follow the notation in the assertion. Set Wi := dϕ(Vi) for every i =1, . . . , r . The proposition will be proved if we demonstrate that

g = W1 ⊕ W2 ⊕ · · · ⊕ Wr

and {[W1,Wi−1] = Wi if 2 ≤ i ≤ r,

[W1,Wr ] = {0}.Now, from the linearity and the invertibility of dϕ, it holds

g = dϕ(h) = dϕ(V1 ⊕ · · · ⊕ Vr) = dϕ(V1) ⊕ · · · ⊕ dϕ(Vr) = W1 ⊕ · · · ⊕ Wr.

Finally, dϕ being a Lie algebra homomorphism (see Theorem 2.1.50-(ii)) it is easyto see that [W1,Wi−1] equals

[dϕ(V1), dϕ(Vi−1)] = d([V1, Vi−1]

) ={

dϕ(Vi) = Wi, 2 ≤ i ≤ r,

dϕ({0}) = {0}.This ends the proof. �

2.2.2 The Campbell–Hausdorff Formula and Some General Results onNilpotent Lie Groups

Before proving that, up to isomorphism, Carnot groups and homogeneous Carnotgroups provide equivalent notions, we recall some results about the so-called Camp-bell–Hausdorff formula.

Definition 2.2.11. Let h be a nilpotent Lie algebra. For X, Y ∈ h, we set10

X � Y :=∑

n≥1

(−1)n+1

n

∑

pi+qi≥1

1≤i≤n

(ad X)p1(ad Y)q1 · · · (ad X)pn(ad Y)qn−1Y

(∑n

j=1(pj + qj )) p1! q1! · · · pn! qn!

= X + Y + 1

2[X, Y ] +

10 We use the notation (ad A)B = [A, B]. Moreover, if qn = 0, the term in the sum (2.43) isby convention · · · (ad X)pn−1(ad Y )qn−1 (ad X)pn−1X. Clearly, if qn > 1, or qn = 0 andpn > 1, the term is zero.


+ 1

12[X, [X, Y ]] − 1

12[Y, [X, Y ]]

− 1

48[Y, [X, [X, Y ]]] − 1

48[X, [Y, [X, Y ]]]

+ {brackets of height ≥ 5}. (2.43)

The sum over n actually runs on {1, . . . , r}, where r < ∞ is the step of nilpotencyof h. The same is true for the sum over the pi’s and qi’s, for which it is left unsaidthat

∑ni=1(pi + qi) ≤ r . We shall refer to the operation defined in (2.43) as the

Campbell–Hausdorff operation on h.

The inner sum in (2.43) can be reordered so that the brackets appear with increas-ing height (here N0 = N ∪ {0}),

X � Y =r∑

n=1

(−1)n+1

n

×r∑

h=1

∑

(p1,q1),...,(pn,qn)∈ N0×N0

(p1,q1),...,(pn,qn) �=(0,0)

(p1+q1)+···+(pn+qn)=h

× (adX)p1(ad Y)q1 · · · (adX)pn(ad Y)qn−1Y

(∑n

j=1(pj + qj )) p1! q1! · · · pn! qn! .

Remark 2.2.12. Since h is nilpotent, (2.43) is a finite sum, and � defines a binaryoperation in h. A striking fact is that this operation is actually associative! Indeed,much more holds: (h,�) is a Lie group. We explicitly remark that the inverse ofX ∈ h w.r.t. � is simply −X.

We shall prove this fact in Corollary 2.2.15 below, by making use of two abstractremarkable results whose proofs are out of our scope here (we refer the reader to themonographs [CG90] and [Var84] for more references).

First of all we recall the following result (see [CG90, p. 13]), which also givesthe well known Campbell–Hausdorff formula (2.44).

Theorem 2.2.13 (Corwin and Greenleaf [CG90, Theorem 1.2.1]). Let (H, ∗) bea connected and simply connected Lie group. Suppose that the Lie algebra h of H

is nilpotent. Then � defines a Lie group structure on h (h being equipped with themanifold structure resulting from its finite-dimensional vector space structure) andExp : (h,�) → (H, ∗) is a group-isomorphism. In particular, we have

Exp (X) ∗ Exp (Y ) = Exp (X � Y) ∀ X, Y ∈ h. (2.44)

We now recall the third fundamental theorem of Lie (see [Var84, Theorem3.15.1]). This is a very deep result in Lie group theory.


Theorem 2.2.14 (The third fundamental theorem of Lie). Let h be a finite-dimens-ional Lie algebra. Then there exists a connected and simply connected Lie groupwhose Lie algebra is isomorphic to h.

Collecting the above two theorems, we obtain the following result.

Corollary 2.2.15. Let h be a finite-dimensional nilpotent Lie algebra. Then � definesa Lie group structure on h.

Moreover, the Lie algebra associated to the Lie group (h,�) is isomorphic to thealgebra h.

Proof. Since h is a finite-dimensional Lie algebra, by Theorem 2.2.14 there exists aconnected and simply connected Lie group (say, G) whose Lie algebra g is isomor-phic to h. Let ϕ : g → h be a Lie algebra isomorphism.

Since h is nilpotent by hypothesis, so is g (since g is isomorphic to h). Hence G

satisfies the hypotheses of Theorem 2.2.13. As a consequence, g equipped with theoperation � defined in (2.43) is a Lie group.

We denote by �g such a composition law � on the Lie algebra g. Analogously,we denote by �h a similar operation on h. Now, the isomorphism ϕ between the twoLie algebras g and h transfers the operation �g into �h. More precisely, for everyX, Y ∈ g, we have

ϕ(X �g Y)

= ϕ

(X + Y + 1

2[X, Y ]g + 1

12[X, [X, Y ]g]g + · · ·

)

(since ϕ is a Lie algebra homomorphism)

= ϕ(X) + ϕ(Y ) + 1

2[ϕ(X), ϕ(Y )]h + 1

12[ϕ(X), [ϕ(X), ϕ(Y )]h]h + · · ·

= ϕ(X) �h ϕ(Y ). (2.45)

We explicitly remark that in the last equality we used, as a crucial tool, the “uni-versal” way in which � is defined on a Lie algebra. In particular, the associativityof �g on g directly implies the associativity of �h on h. Thus, the first part of theassertion of the corollary is proved. The second part follows from Ex. 2 at the end ofthe chapter. �Remark 2.2.16. If h and g are finite-dimensional nilpotent Lie algebras and ϕ : h →g is an algebra-homomorphism, then ϕ is also a group-homomorphism between(h,�) and (g,�). This directly follows from (2.45).

2.2.3 Abstract and Homogeneous Carnot Groups

We now aim to prove that, up to isomorphism, the definitions of classical and homo-geneous Carnot group are equivalent. To begin with, we prove the following simplefact:


Proposition 2.2.17 ((Homogeneous ⇒ stratified) Carnot). A homogeneous Carnotgroup is a stratified group.

Proof. Let G = (RN, ◦, δλ) be a homogeneous Carnot group. Clearly, G is con-nected and simply connected. Let g be the algebra of G.

For i = 1, . . . , r and j = 1, . . . , Ni , let Z(i)j be the vector field of g agreeing

with ∂/∂ x(i)j at the origin. We set

Vi := span{Z(i)1 , . . . , Z

(i)Ni

}.Remark 1.4.8 proves that (V1, . . . , Vr) is a stratification of g, as in Definition 2.2.3.This ends the proof. �

We are now in the position to prove the main result of this section. The proof isa detailed argument of what is commonly used (without comments) in literature, i.e.the identification of the group with its algebra.

Theorem 2.2.18 ((Stratifiedisom.�⇒ homogeneous) Carnot). Let H be a stratified

group, according to Definition 2.2.3. Then there exists a homogeneous Carnotgroup H

∗ (according to our Definition 2.2.1) which is isomorphic to H.We can choose as H

∗ the Lie algebra h of H (identified to RN by a suitable

choice of an adapted basis of h) equipped with the composition law � defined bythe Campbell–Hausdorff operation (2.43) in Definition 2.2.11. In this case, a group-isomorphism from H

∗ to H is the exponential map

Exp : (h,�) → (H, ∗).

Proof. Let (H, ∗) be as in Definition 2.2.3. Let h be the algebra of H. Let h =V1 ⊕ · · · ⊕ Vr be a fixed stratification of h as in (2.41). By Proposition 2.2.8, h isnilpotent of step r .

Then Theorem 2.2.13 yields that

Exp : (h,�) → (H, ∗) is a Lie-group isomorphism, (2.46)

where � is as in (2.43). We now prove that (a coordinate version of) (h,�) is ahomogeneous Carnot group according to Definition 2.2.1.

We fix a basis for h adapted to its stratification (see Definition 2.2.6): for i =1, . . . , r , set Ni := dim Vi , and let (E

(i)1 , . . . , E

(i)Ni

) be a basis for Vi . Then considerthe basis for h given by

E =(E

(1)1 , . . . , E

(1)N1

; . . . ; E(r)1 , . . . , E

(r)Nr

).

By means of this basis, we fix a coordinate system on h, and we identify h with RN ,

where N := N1 + · · · + Nr . More precisely, we consider the map11

11 See also the map introduced in (1.35) of Remark 1.2.20, page 21. Incidentally, we noticethat (h, πE ) is a coordinate system for the whole h, determining its differentiable structure.Obviously, along with πE , we have other coordinate maps (for example those arising by adifferent choice of a linear basis of h, adapted or not) and the differentiable structure of h

does not depend on E .


Fig. 2.1. Turning a stratified Carnot group into a homogeneous one

πE : h → RN, E · ξ :=

r∑

i=1

Ni∑

j=1

ξ(i)j E

(i)j �→ (ξ (1), . . . , ξ (r)),

where ξ (i) = (ξ(i)1 , . . . , ξ

(i)Ni

) ∈ RNi for every i = 1, . . . , r . Next, we set

Ψ := Exp ◦ (πE )−1 : RN → H, Ψ (ξ) = (Exp (E · ξ)).

(See also Fig. 2.1.) Notice that, more explicitly,

Ψ (ξ) = Exp

(r∑

i=1

Ni∑

j=1

ξ(i)j E

(i)j

)∀ ξ ∈ R

N. (2.47)

Finally, we equip RN with the composition law �E defined by

ξ �E η := Ψ −1(Ψ (ξ) ∗ Ψ (η)), ξ, η ∈ R

N, (2.48)

We define a family of dilations {Δλ}λ>0 on the Lie algebra h as follows:

Δλ : h → h, Δλ

(r∑

i=1

Xi

):=

r∑

i=1

λiXi, where Xi ∈ Vi. (2.49a)

Obviously,Δλ is a vector-space automorphism of h. (2.49b)

(Note that N , the Ni’s and the form of Δλ do not depend neither on the choiceof the stratification of h nor on the adapted basis; see, for instance, Lemma 2.2.9.)Obviously, Δλ turns into a family of dilations {δλ}λ>0 on R

N via Ψ by setting

δλ := πE ◦ Δλ ◦ π−1E . (2.49c)

We claim thatH

∗ := (RN,�E , δλ) is a homogeneous Carnot group(of step r and N1 generators)

isomorphic to (H, ∗) via the Lie group isomorphism Ψ .

To prove the claim, we split the proof in steps.


(I). By the very definition of �E and Ψ , we have

Ψ (ξ �E η) = Ψ (ξ) ∗ Ψ (η) ∀ ξ, η ∈ RN, (2.50a)

which, in turn, is equivalent to (exploit (2.46) and recall that Ψ = Exp ◦ π−1E )

π−1E (ξ �E η) = π−1

E (ξ) � π−1E (η) ∀ ξ, η ∈ R

N, (2.50b)

or, equivalently,

πE (X � Y) = πE (X) �E πE (Y ) ∀ X, Y ∈ h. (2.50c)

Now, we recognize that (2.50a) and (2.50b) mean that

(RN,�E ), (h,�), (H, ∗)

are isomorphic Lie groups via the Lie-group isomorphisms

(RN,�E )π−1E−→ (h,�)

Exp−→ (H, ∗).

In particular,

Ψ = Exp ◦ π−1E : (RN,�E ) → (H, ∗) is a Lie-group isomorphism. (2.51)

(II). We now investigate the dilation δλ. The stratified notation

h � E · ξ =r∑

i=1

Ni∑

j=1

ξ(i)j E

(i)j

for an arbitrary vector of h and the fact that

πE (E · ξ) = ξ (2.52)

suggests the notationR

N � ξ = (ξ (1), . . . , ξ (r))

for the points in RN . We claim that, with the above notation, δλ introduced in (2.49c)

has the form in (2.40), i.e.

δλ(ξ(1), ξ (2), . . . , ξ (r)) = (λξ (1), λ2ξ (2), . . . , λrξ (r)). (2.53)

Indeed,

δλ(ξ) = (see (2.49c))(πE ◦ Δλ ◦ π−1

E)(ξ)

(see (2.52)) = πE(Δλ(E · ξ)

)

= πE

(Δλ

(r∑

i=1

Ni∑

j=1

ξ(i)j E

(i)j

))


(see (2.49b)) = πE

((r∑

i=1

Ni∑

j=1

ξ(i)j Δλ(E

(i)j )

))

(see (2.49a)) = πE

((r∑

i=1

Ni∑

j=1

ξ(i)j λiE

(i)j

))

= πE(E · (λξ(1), . . . , λrξ (r)

))

(see (2.52)) = (λξ(1), . . . , λrξ (r)

).

Next, we proceed by showing that Δλ is an automorphism of the Lie-group (h,�),i.e.

Δλ(X � Y) = Δλ(X) � Δλ(Y ) ∀ X, Y ∈ h, ∀ λ > 0. (2.54)

Recalling Remark 2.2.16, it is enough to prove that

Δλ

([X, Y ]) = [Δλ(X),Δλ(Y )] for every X, Y ∈ h.

If X = ∑ri=1 Xi and Y = ∑r

i=1 Yi , where Xi , Yi ∈ Vi , we have [Xi, Yj ] ∈ Vi+j (bythe stratification condition), whence

Δλ

([X, Y ]) =r∑

i,j=1

Δλ

([Xi, Yj ]) =

r∑

i,j=1

λi+j [Xi, Yj ]

=r∑

i,j=1

[λiXi, λjYj ] =

r∑

i,j=1

[Δλ(Xi),Δλ(Yj )] = [Δλ(X),Δλ(Y )].

Now, a joint application of (2.50b), (2.50c) and (2.54) prove that δλ is a Lie-groupautomorphism of (RN,�E ), i.e.

δλ(ξ �E η) = δλ(ξ) �E δλ(η) ∀ ξ, η ∈ RN, ∀ λ > 0.

(III). Thus, H∗ := (RN,�E , δλ) is a homogeneous Lie group on R

N , as definedin Definition 1.3.1, page 31. Let now h∗ be the Lie algebra of H

∗. Dealing with aLie group on R

N (and the fixed Cartesian coordinates ξ ’s on RN ), the Jacobian basis

related to the composition �E is well-posed. We denote by

Z =(Z

(1)1 , . . . , Z

(1)N1

; . . . ; Z(r)1 , . . . , Z

(r)Nr

)

this Jacobian basis, i.e. Z(i)k is the vector field in h∗ agreeing at the origin with

∂/∂ ξ(i)k . The proof is complete if we show that the Lie algebra generated by

Z1, . . . , ZN1 coincides with the whole h∗.

To this end, we first observe that, thanks to (2.51), d Ψ : h∗ → h is an algebra-isomorphism (see Theorem 2.1.50). Furthermore, from Remark (2.1.52)-(iii), wehave


dΨ = (d Exp ) ◦ (d(π−1E )

).

Moreover, since E(1)1 , . . . , E

(1)N1

is a system of Lie-generators for h (by the verydefinition of stratification!), it is enough to prove that

d Ψ (Z(i)k ) = E

(i)k for every i = 1, . . . , r and every k = 1, . . . , Ni . (2.55)

In order to prove (2.55), we recall that a left-invariant vector field is determined byits value at the identity. Hence, (2.55) will follow if we show that

(d Ψ (Z

(i)k ))e

= (E(i)k )e.

For every f ∈ C∞(H, R), we have

(d Ψ (Z

(i)k ))e(f ) = (

d0Ψ (Z(i)k )0

)(f ) = (Z

(i)k )0(f ◦ Ψ )

= (∂/∂ ξ(i)k )|ξ=0f

(Exp

(r∑

i=1

Ni∑

j=1

ξ(i)j E

(i)j

))

= d

dt

∣∣∣∣t=0

f(Exp (t E

(i)k ))

= d

dt

∣∣∣∣t=0

f(

expE

(i)k

(t))

= (E(i)k )e(f ).

In the first equality, we used the very definition of the differential of a homomor-phism (see Definition 2.1.49); in the second one, we used the definition of the dif-ferential at a point; in the third equality, we exploited the very definition of Z

(i)k and

that of Ψ (see (2.47)); the fourth equality is a triviality from calculus; the fifth andthe sixth equalities are the core of the computation: they follow, respectively, fromProposition 2.1.58-(1) and from (2.36b). The theorem is thus completely proved. �Remark 2.2.19. From Theorem 2.2.18 and Remark 2.2.12, it follows that any strati-fied group is isomorphic to a homogeneous Carnot group in which the group inver-sion law is simply given by x−1 = −x. This is commonly assumed in great partof the literature involving Carnot groups, without further comments. See Proposi-tion 2.2.22 for more details.

Remark 2.2.20 (Change of adapted basis). We give some more information on Theo-rem 2.2.18. Let H be a fixed stratified group with Lie algebra h. Let h = V1⊕· · ·⊕Vr

be a given stratification of h, as in (2.41). We know that r and every Ni := dim(Vi)

(for i = 1, . . . , r) are invariants of the stratified group. So is N := N1 + · · · + Nr

(= dim(h)).We fix any arbitrary basis E for h adapted to its stratification. We use the notation

in the proof of Theorem 2.2.18. We therein demonstrated that RN is a homogeneous

Carnot group isomorphic to (H, ∗) if RN is equipped with the composition law �E

defined by (2.48) and the dilation (2.53).


The map Ψ is a Lie-group isomorphism from (RN,�E ) to (H, ∗). Another wayto write (2.48) is obviously

Exp(E · (ξ �E η)

) = (Exp (E · ξ)) ∗ (Exp (E · η)). (2.56a)

Suppose now that we choose another basis E (adapted to the same stratification),with analogous notation as for E . R

N can be equipped with an another composition�E characterized by


) = (Exp (E · ξ )) ∗ (Exp (E · η)). (2.56b)

For every i ∈ {1, . . . , r} and every j ∈ {1, . . . , Ni}, there exist scalars c(i)h,j ’s such

thatE

(i)j = c

(i)1,j E

(i)1 + · · · + c

(i)Ni ,j

E(i)Ni

.

Let us introduce the notation

C(i) := (c(i)h,j

)1≤h≤Ni, 1≤j≤Ni

and C to denote the N × N matrix whose diagonal blocks are C(1), . . . , C(r),

C =⎛

⎝C(1) · · · 0

.... . .

...

0 · · · C(r)

⎞

⎠ ,

and, finally, let us use once again the notation C to denote the linear map

C : RN → R

N, x �→ Cx.

We note that if a vector field in h has coordinates ξ ∈ RN with respect to the basis

E , then12 it has coordinates Cξ with respect to the basis E , i.e.

E · ξ = E · (Cξ) for every ξ ∈ RN. (2.57)

12 Indeed, it holds

E · ξ =r∑

i=1

Ni∑

j=1

ξ(i)j

E(i)j

=r∑

i=1

Ni∑

j=1

ξ(i)j

Ni∑

h=1

c(i)h,j

E(i)h

=r∑

i=1

Ni∑

h=1

⎛

⎝Ni∑

j=1

c(i)h,j

ξ(i)j

⎞

⎠ E(i)h

=r∑

i=1

Ni∑

h=1

(C(i) · ξ(i)

)

hE

(i)h

=r∑

i=1

Ni∑

h=1

(C · ξ)(i)h

E(i)h

= E · (Cξ).


We now claim that the same happens for the relevant group structures

G := (RN,�E ), G := (RN,�E ),

i.e. the linear map C : RN → R

N is a group isomorphism from G to G. Roughlyspeaking, the same change of basis in h from the basis E to the basis E acts as aLie-group isomorphism from (RN,�E ) to (RN,�E ).

Indeed, from (2.56a), (2.56b) and (2.57) we have


) = (Exp (E · ξ)) ∗ (Exp (E · η))

= Exp(E · (Cξ)

) ∗ Exp(E · (Cη)

)

= Exp(E · ((Cξ) �E (Cη))

)

= Exp(E · (C−1{(Cξ) �E (Cη)

})),

which impliesξ �E η = C−1{(Cξ) �E (Cη)

}, (2.58a)

i.e.C(ξ �E η) = (Cξ) �E (Cη) ∀ ξ, η ∈ R

N. (2.58b)

Now, (2.58b) is equivalent to say that

C : (RN,�E ) → (RN,�E ), x �→ Cξ

is a Lie-group isomorphism. �Example 2.2.21 (From “stratified” to “homogeneous”). We consider once again theLie group introduced in Examples 1.2.18 and 1.2.31. To be consistent with the nota-tion in Theorem 2.2.18, we denote this group by H. We explicitly remark that H is aCarnot group which is not homogeneous (w.r.t. the coordinates initially assigned toit).

We find an explicit isomorphism of Lie groups turning H into a homogeneousCarnot group: the existence of such an isomorphism is indeed ensured by Theo-rem 2.2.18, whereas an explicit way to construct it comes from the very proof ofTheorem 2.2.18 (see also Remark 2.2.20).

We have H = (R3, ∗), where

x ∗ y =(

arcsinh(sinh(x1) + sinh(y1)), x2 + y2 + sinh(x1)y3, x3 + y3

).

A stratification of the algebra h of H is V1 ⊕ V2, where V1 = span{E(1)1 , E

(1)2 },

V2 = span{E(2)1 }, being

E(1)1 = 1

cosh(x1)∂x1 , E

(1)2 = ∂x3 + sinh(x1) ∂x2; E

(2)1 = ∂x2 .

The relevant dilation on h is given by


δλ : h −→ h,

δλ(ξ1 E(1)1 + ξ2 E

(1)2 + ξ3 E

(2)1 ) := ξ1 λ E

(1)1 + ξ2 λ E

(1)2 + ξ3 λ2 E

(2)1 .

Since h is a nilpotent algebra of step two, the Campbell–Hausdorff operation � on h

is

X � Y = X + Y + 1

2[X, Y ].

Now, since [E(1)1 , E

(1)2 ] = E

(2)1 , a direct computation gives

(ξ1 E(1)1 + ξ2 E

(1)2 + ξ3 E

(2)1 ) � (η1 E

(1)1 + η2 E

(1)2 + η3 E

(2)1 )

= (ξ1 + η1) E(1)1 + (ξ2 + η2) E

(1)2 +

(ξ3 + η3 + 1

2(ξ1 η2 − ξ2 η1)

)E

(2)1 .

Via the choice of the basis E = {E(1)1 , E

(1)2 , E

(2)1 }, the Lie group (h,�) can be iden-

tified to H∗ = (R3,�E ), where

ξ �E η =(

ξ1 + η1, ξ2 + η2, ξ3 + η3 + 1

2(ξ1 η2 − ξ2 η1)

).

The Lie group isomorphism between (H∗,�E ) and (H, ∗) is the “exponential-typemap”

H∗ � (ξ1, ξ2, ξ3) �→ Exp (ξ1 E

(1)1 + ξ2 E

(1)2 + ξ3 E

(2)1 ) ∈ H.

Now, from the computation13 in Example 1.2.31 (page 28) we derive that this mapis given by

(ξ1, ξ2, ξ3) �→ Ψ (ξ1, ξ2, ξ3) =(

arcsinh(ξ1), ξ3 + 1

2ξ1ξ2, ξ2

).

The map Ψ is the isomorphism of Lie groups turning the homogeneous Carnotgroup H

∗ = (R3,�E , δλ) (where δλ(ξ1, ξ2, ξ3) = (λξ1, λξ2, λ2ξ3)) into the non-

homogeneous Carnot group H. We recognize that

x ∗ y = Ψ (Ψ −1(x) �E Ψ −1(y)) ∀ x, y ∈ H. �

2.2.4 More Properties of the Lie Algebra of a Stratified Group

In the proof of Theorem 2.2.18, we described how to identify a stratified groupwith a homogeneous Carnot group on R

N . More precisely, given a stratified groupH, the equivalence class of the Lie groups which are isomorphic to H contains atleast one (in fact, infinite) homogeneous Carnot group H

∗ on RN , according to De-

finition 2.2.1. Namely, H∗ is a “coordinate-copy” of (h,�), the Lie algebra of H

equipped with the Campbell–Hausdorff operation.

13 We warn the reader that a slight change of notation here is needed, if compared to thenotation in Example 1.2.31, interchanging ξ2 and ξ3.


Since, in the specialized literature, one often meets with phrases such as “it is notrestrictive to suppose that. . . ” H has some distinguished properties, it is advisable tolook more closely to the properties of (h,�). We furnish some of these properties inthe following proposition which collect several already proved facts.

Proposition 2.2.22. Let H be a stratified group with Lie algebra h and exponentialmap Exp H : h → H. Let also � be the Campbell–Hausdorff operation on h definedin (2.43).

Let V1 ⊕ · · · ⊕ Vr be a stratification of h, as in (2.41). Let E be any basis for h

adapted to the stratification, as in Definition (2.2.6). Set N := dim(h), consider themap πE : h → R

N , where, for every X ∈ h, πE (X) is the N -tuple of the coordinatesof X w.r.t. E .

Then the binary operation on RN defined by

x �E y = πE((

π−1E (x)

) � (π−1E (y)

)) ∀ x, y ∈ RN

has the following properties:

(1) G := (RN,�E ) is a Lie group on RN ; G is isomorphic to H via the map Ψ =

Exp H◦π−1E and to (h,�) via πE , whence (G,�E ) and (h,�) are stratified groups.

(2) Let Z = {Z1, . . . , ZN } be the Jacobian basis related to G; then, denoting theadapted basis by E = {E1, . . . , EN }, we have

dΨ (Zi) = Ei for every i = 1, . . . , N ,

or, equivalently,Zi(f ◦ Ψ ) ≡ Ei(f ) ◦ Ψ on G

for every f ∈ C∞(H, R). Moreover, if g is the algebra of G, the exponential mapExp G : g → G is a linear map and it sends Zi in the i-th element of the standardbasis of G ≡ R

N , whence

Exp G

((x1, . . . , xN)Z

) = (x1, . . . , xN),

being (x1, . . . , xN)Z = x1Z1 + · · · + xNZN .(3) The inversion on G is the Euclidean inversion −x.(4) For every i ∈ {1, . . . , N}, we have

(x �E y)i = xi + yi + Ri (x, y),

where Ri (x, y) is a polynomial function depending on the xk’s and yk’s withk < i, and Ri (x, y) can be written as a sum of polynomials each containing afactor of the following type

xhyk − xkyh with h �= k and h, k < i.

(5) Let Δλ be the linear map on h such that, for every i = 1, . . . , r ,

Δλ(X) = λi X whenever X ∈ Vi .

Let δλ := πE ◦ Δλ ◦ π−1E . Then (RN,�E , δλ) is a homogeneous Carnot group of

the same step and number of generators as H.


Proof. (1). The proof of (1) is contained in the proof of Theorem 2.2.18.(2). In the proof of Theorem 2.2.18, we demonstrated that (see (2.55))

d Ψ (Z(i)k ) = E

(i)k ∀ i ∈ {1, . . . , r}, k ∈ {1, . . . , Ni} (2.59)

if E =(E

(1)1 , . . . , E

(1)N1

; . . . ; E(r)1 , . . . , E

(r)Nr

)

and Z =(Z

(1)1 , . . . , Z

(1)N1

; . . . ; Z(r)1 , . . . , Z

(r)Nr

)

are, respectively, the usual notation for the adapted basis E and the notation for theJacobian basis resulting from the coordinates induced by πE on R

N . This proves thefirst part of (2).

As to the first part, consider the following diagram (h∗ denotes the algebra of theLie group (h,�) and Exp h : h∗ → h the related exponential map)

(G,�E )π−1E

(h,�)Exp H

(H, ∗)

gd π−1

E

Exp G

h∗ d Exp H

dExp h

h.

Exp H

From the commutativity result in Theorem 2.1.59 it holds

Exp G = πE ◦ Exp h ◦ d π−1E = πE ◦ {(Exp H)−1 ◦ Exp H ◦ d Exp H

} ◦ d π−1E

= πE ◦ {d Exp H} ◦ d π−1E ,

whenceExp G = πE ◦ d Exp H ◦ d π−1

E . (2.60)

Hence, Exp G is a linear map since πE , d Exp H and d π−1E are linear.

We now prove that

Exp G(Z(i)k ) = (

0(1), . . . , e(i)k , . . . , 0(r)

) ∀ i ≤ r, k ≤ Ni, (2.61)

where e(i)k denotes the k-th element of the standard basis of R

Ni (for k ∈ {1, . . . , Ni}).To this end, first notice that

πE (E(i)k ) = (

0(1), . . . , e(i)k , . . . , 0(r)

) ∀ i ≤ r, k ≤ Ni. (2.62)

Then, by applying (2.60), we infer

Exp G(Z(i)k ) =

(πE ◦ d Exp H ◦ d π−1

E

)(Z

(i)k ) = πE

(d Ψ (Z

(i)k ))

= πE (E(i)k ) = (

0(1), . . . , e(i)k , . . . , 0(r)

).

In the second equality, we invoked the definition of Ψ and Remark (2.1.52)-(iii),whereas in the third we exploited (2.59); finally, the last equality is (2.62).


Now, the linearity of Exp G and (2.61) produce

Exp G

((x1, . . . , xN)Z

)

= Exp G

(r∑

i=1

Ni∑

j=1

x(i)j Z

(i)j

)

=r∑

i=1

Ni∑

j=1

x(i)j Exp G(Z

(i)j ) =

r∑

i=1

Ni∑

j=1

x(i)j

(0(1), . . . , e

(i)j , . . . , 0(r)

)

=r∑

i=1

(0(1), . . . ,

Ni∑

j=1

x(i)j e

(i)j , . . . , 0(r)

)=

r∑

i=1

(0(1), . . . , x(i), . . . , 0(r)

)

= (x(1), . . . , x(r)).

(3). Since in (1) we proved that (G,�E ) is isomorphic to (h,�) and πE providesa Lie-group isomorphism between them, which is also a linear map of vector spaces,we infer that, for any x ∈ G,

(x)−1 = πE((

π−1E (x)

)−1)

= πE(− π−1

E (x)) = −πE

(π−1E (x)

) = −x.

Here we used the fact that the inversion in (h,�) is X �→ −X, for, by the Campbell–Hausdorff operation, we have

X � (−X) = X − X + 1

2[X,−X] + 1

12[X, [X,−X]] + · · · = 0.

(4). To begin with, we know that, by definition,

x �E y = πE((

π−1E (x)

) � (π−1E (y)

)) ∀ x, y ∈ RN.

Writing x = (x1, . . . , xN), we have X := π−1E (x) = ∑N

i=1 xiEi . Analogously,

Y := π−1E (y) = ∑N

j=1 yjEj . Let us consider the Campbell–Hausdorff operationX � Y . Considering the form of the operation in (2.43), we see that, besides thesummands

X + Y =N∑

i=1

xiEi +N∑

j=1

yjEj =N∑

i=1

(xi + yi)Ei,

any other summand has the following form: it is given by a rational number times

( ) (ad X)α1(ad Y)β1 · · · (ad X)αn(ad Y)βn([X, Y ]),

or an analogous term with [X, Y ] replaced by [Y,X]. Let us analyze the case[X, Y ] (the other being analogous). Replacing X and Y (but in the final [X, Y ]) by∑N

i=1 xiEi and∑N

j=1 yjEj , respectively, ( ) is a finite sum of summands of thefollowing type:


( ) p(x, y) [Ek1 , · · · [EkM, [X, Y ]]],

where p(x, y) is a polynomial, M ∈ N and i1, . . . , iM ∈ {1, . . . , N}. Now, ( ) isequal to

p(x, y)[Ek1 , · · · [EkM, [X, Y ]]]

=N∑

i,j=1

p(x, y) xi yj [Ek1, · · · [EkM, [Ei,Ej ]]]

=∑

i �=j

{ · · ·} =∑

i<j

{ · · ·} +∑

i>j

{ · · ·}

=∑

1≤h<k≤N

p(x, y) (xh yk − xk yh) [Ek1 , · · · [EkM, [Eh,Ek]]].

In the last equality, we suitably renamed the dummy variables, and we exploitedanti-commutativity [Ei,Ej ] = −[Ej ,Ei]. All these facts together prove that

x �E y = πE (X � Y) = πE

(N∑

i=1

(xi + yi)Ei + {finite sum of terms like

p(x, y) (xh yk − xk yh) [Ek1 , · · · [EkM, [Eh,Ek]]]

})

.

Finally, the linearity of πE proves that, for every i = 1, . . . , N , it holds

(x �E y)i = xi + yi + Ri (x, y),

where Ri (x, y) is the sum of polynomials each containing a factor of the type xhyk −xkyh with h �= k. The fact that Ri (x, y) does depend only on the xk’s and yk’s withk < i easily follows from (5) of the assertion and the results on homogeneous groupsfrom Section 1.3.2.

(5). The proof of (5) is contained in the proof of Theorem 2.2.18 (see also (1)and Proposition 2.2.8). �

A natural question arises from the proof of Theorem 2.2.18. In that proof, wehandled with a Lie group H, its Lie algebra h which (equipped with the Campbell–Hausdorff operation �) becomes a Lie group itself and, finally, we considered the Liealgebra h∗ of the Lie group (h,�) (roughly speaking, “the algebra of the algebra”).We now prove the very natural fact that h∗ is “essentially” h itself. Moreover, theexponential map Exp h : h∗ → h is essentially the “identity map”.

Example 2.2.23. Before entering the details, we give an example. Let us consider thehomogeneous Lie group (H, ∗) on R

3 with the composition law

x ∗ y =⎛

⎝x1 + y1x2 + y2

x3 + y3 + x1 y2

⎞

⎠ .


The Jacobian basis for h (the algebra of H) is

Z1 = ∂1, Z2 = ∂2 + x1 ∂3, Z3 = [Z1, Z2] = ∂3.

Then a natural Lie group structure (h,�) is obtained by considering on h theCampbell–Hausdorff law � and, at the same time, by identifying h to R

3 via co-ordinates w.r.t. the Jacobian basis. This gives

ξ � η =⎛

⎝ξ1 + η1ξ2 + η2

ξ3 + η3 + 12 (ξ1η2 − ξ2η1)

⎞

⎠ .

The next Jacobian basis related to (h,�) is

Z1 = ∂1 − ξ2

2∂3, Z2 = ∂2 + ξ1

2∂3, Z3 = [Z1, Z2] = ∂3.

Finally, consider the Lie algebra h∗ of (h,�). It is easily checked that the exponentialmap

Exp h : h∗ → h

maps Zi into Zi for i = 1, 2, 3. In other words, by an abuse of language, h∗ coincideswith h. �

A general formulation of the above facts is given by the following proposition,which highlights once more the relevance of the Jacobian basis. Notice that no strat-ification condition is supposed in the first part of the following statement.

Proposition 2.2.24 (Algebra of the algebra). Let (H, ∗) be a connected and simplyconnected nilpotent Lie group with Lie algebra h. Let � be the Campbell–Hausdorffoperation on h defined in Definition 2.2.11, and let (h,�) be equipped with its naturalLie group structure (see Theorem 2.2.13). Finally, let h∗ denote the Lie algebra ofthe Lie group (h,�), and let

Exp h : h∗ → h

denote the related exponential map.Then, Exp h is a linear map of vector spaces.For example, if H is a homogeneous Carnot group, and we equip the above Lie

algebras h and h∗ with the relevant Jacobian bases, then Exp h is the linear maprelated to the identity matrix.

Proof. We follow the notation of the assertion, and we denote by Exp H : h → H therelevant exponential map. We know that the following diagram is commutative (seeTheorem 2.1.59 jointly with Theorem 2.2.13):

(h,�)Exp H

(H, ∗)

h∗ d Exp H

Exp h

h.

Exp H


As a consequence, we have

Exp h = (Exp H)−1 ◦ Exp H ◦ (d Exp H) = d Exp H,

so that Exp h is a linear map (here, the vector space structures on h and h∗ are con-sidered).

Let now (H, ∗) be a homogeneous Carnot group on RN . We consider on h the

Jacobian basis Z1, . . . , ZN (related to the operation ∗). We equip (h,�) with thenatural system of coordinates given by

h �N∑

j=1

ξj Zj �→ (ξ1, . . . , ξN ) ∈ RN,

thus identifying h to RN . On h ≡ R

N , we have the Lie group law �: we then considerthe Jacobian basis Z∗

1 , . . . , Z∗N of h∗ (related to the operation �).

We have to show that d Exp H sends Z∗k into Zk . To this end, given f ∈

C∞(H, R), we have

(d Exp H(Z∗k ))0(f ) = (Z∗

k )0(f ◦ Exp H)

= (∂/∂ξk)|ξ=0f

(Exp H

(N∑

j=1

ξj Zj

))= d

d t

∣∣∣∣0f(Exp (t Zk)

) = (Zk)0(f ).


2.2.5 Sub-Laplacians of a Stratified Group

We begin with a central definition.

Definition 2.2.25 (Sub-Laplacian of a stratified group). Let H be a stratified groupwith Lie algebra h. The second order differential operator

L =m∑

j=1

X2j (2.63)

is referred to as a sub-Laplacian on H, if there exists a stratification for h

h = V1 ⊕ · · · ⊕ Vr,

as in Definition 2.2.3, such that

X1, . . . , Xm is a (linear) basis of V1.

With the above notation, we also say that L is a sub-Laplacian on H with related (or,generating) stratification (V1, . . . , Vr).


We shall treat a sub-Laplacian L as an operator on smooth functions on the opensubsets of H. More precisely, if Ω ⊆ H is open and f ∈ C∞(Ω, R), by L(f ) wemean the function on Ω defined by

Ω � x �→m∑

j=1

(Xj )x

(Ω � m �→ (Xj )m(f )

).

Note that if L is as in (2.63), then m = dim(V1) is the number of generators of H

according to Definition 2.2.7. Observe also that if∑m

j=1 X2j is a sub-Laplacian on

H with related stratification (V1, . . . , Vr ) and if {Y1, . . . , Ym} is another basis of V1,then

∑mj=1 Y 2

j is a sub-Laplacian on H.Since stratified groups and homogeneous Carnot’s are deeply related, a compar-

ison of the above notion of sub-Laplacian and that of sub-Laplacian on a homoge-neous Carnot group is in order.

Remark 2.2.26. Let H be a stratified group with an algebra h. Let L = ∑mj=1 X2

j bea sub-Laplacian on H, and let V = (V1, . . . , Vr ) be the stratification of h related to Laccording to Definition 2.2.25. Let also E be a basis for h adapted to the stratificationV and such that X1, . . . , Xm are the first m elements of E . (The existence of such abasis E is evident.)

By Proposition 2.2.22 (whose notation we presently follow), there exists a homo-geneous Carnot group G on R

N which is isomorphic to H. Let g be the Lie algebraof G, and let Z1, . . . , ZN be the Jacobian basis of g (the Jacobian basis is well-posedfor G is a homogeneous Carnot group on R

N ). Let Ψ : G → H be the isomorphism,as in Proposition 2.2.22-(1). By Proposition 2.2.22-(2), we know, in particular, that

dΨ (Zi) = Xi for every i = 1, . . . , m, (2.64)

or, equivalently, Zi(f ◦ Ψ ) = Xi(f ) ◦ Ψ on G for every f ∈ C∞(H, R). Set

ΔG :=m∑

i=1

Z2i (which is the canonical sub-Laplacian on G).

We infer that, for every f ∈ C∞(H, R),

ΔG(f ◦ Ψ ) ≡ (L(f )

) ◦ Ψ, i.e. L(f ) ≡ (ΔG(f ◦ Ψ )

) ◦ Ψ −1. (2.65a)

Equivalently, since we have (with obvious meaning of the notation)

C∞(H, R) ◦ Ψ ≡ C∞(G, R), C∞(G, R) ◦ Ψ −1 ≡ C∞(H, R), (2.65b)

for every given g ∈ C∞(G, R), if we set f := g ◦ Ψ −1, (2.65a) rewrites as

ΔGg ≡ (L(g ◦ Ψ −1)

) ◦ Ψ ∀ g ∈ C∞(G, R). (2.65c)

Roughly speaking, (2.65a) says that the sub-Laplacian L of H is turned by Ψ −1 intothe canonical sub-Laplacian ΔG of G, or, equivalently, L and ΔG are Ψ -related.


This fact ensures that the theory of sub-Laplacians on a stratified group can be recastin its homogeneous-Carnot-group dual version.

Moreover, let L1 and L2 be two sub-Laplacians on H with the same related strat-ification V . Suppose we perform the above identification of L1 with the canonicalsub-Laplacian of G. What is L2 Ψ -related to on G? Obviously, to a (possibly non-canonical) sub-Laplacian on G. Indeed, let

L1 =m∑

i=1

X2i , L2 =

m∑

i=1

Y 2i .

We have that the Xi’s and the Yi’s form two bases for V1. Hence, there exists anon-singular m × m matrix (ai,j ) such that

Yi =m∑

j=1

ai,j Xi for every i = 1, . . . , m.

Thus we have (see (2.64))

Yi := dΨ −1(Yi) =m∑

j=1

ai,j Zi, (2.66)

so that (being dΨ (Yi) = Yi)(

m∑

j=1

Y 2i

)(f ◦ Ψ ) ≡

(m∑

j=1

Y 2i (f )

)◦ Ψ

for every f ∈ C∞(H, R). This proves the claimed fact that L2 = ∑mj=1 Y 2

i is Ψ -

related to∑m

j=1 Y 2i . In turn,

∑mj=1 Y 2

i is a sub-Laplacian on G, in force of (2.66),

which grants that Y1, . . . , Ym is a basis for span{Z1, . . . , Zm}. �We introduce another central definition.

Definition 2.2.27 (L-harmonic function on a stratified group). Let H be a strati-fied group, and let L be a fixed sub-Laplacian on H, according to Definition 2.2.25.Let Ω ⊆ H be an open set. A real-valued function f on Ω is called L-harmonic ifand only if it holds

f ∈ C∞(Ω, R) and Lf = 0 on Ω . (2.67)

We denote by HL(Ω) the vector space of the L-harmonic functions on Ω .

Remark 2.2.28. (i) Let H be a stratified group, and let L = ∑mj=1 X2

j be a sub-Laplacian on H with generating stratification V = (V1, . . . , Vr ). Let E be any basisof the algebra of H adapted to the stratification V (but not necessarily containingX1, . . . , Xm as its first elements).


Let Ψ = Exp H ◦ π−1E be as in Proposition 2.2.22. Then, according to the results

in Remark 2.2.26, L is Ψ -related to a sub-Laplacian on G, say L. This means that

Lf = (L(f ◦ Ψ )

) ◦ Ψ −1, Lf = (L(f ◦ Ψ −1)

) ◦ Ψ (2.68)

for every f smooth on H and every f smooth on G.(ii) Let now Ω �= ∅ be an open subset of H. Then Ω := Ψ −1(Ω) is an open

subset of G. We claim that

HL(Ω) = HL(Ω) ◦ Ψ, HL(Ω) = HL(Ω) ◦ Ψ −1. (2.69)

More precisely,

f ∈ HL(Ω) ⇐⇒ f := f ◦ Ψ ∈ HL(Ω),

f ∈ HL(Ω) ⇐⇒ f := f ◦ Ψ −1 ∈ HL(Ω).

Indeed, (2.69) follows at once from (2.68).All the above facts ensure that the theory of the L-harmonic functions on (the

open subsets of) a stratified group can be recast in its homogeneous-Carnot-groupdual version.

This is what it will be done in the foregoing chapters.

Bibliographical Notes. In this chapter, we intended to provide only the topics ofdifferential geometry and Lie group theory which are strictly necessary to read thisbook. The reader is referred to, e.g. M. Hausner, J.T. Schwartz [HS68], V.S. Varadara-jan [Var84] and F.W. Warner [War83], for exhaustive introductions to the differentialgeometry and the theory of Lie groups and Lie algebras. See also L.J. Corwin andF.P. Greenleaf [CG90] for the theory of representation of nilpotent Lie groups.


Ex. 1) Following the lines of Example 2.2.21 (page 137), find an explicit isomor-phism of Lie groups between a suitable homogeneous Carnot group and the(non-homogeneous) stratified group G = (Ω, ∗), where

Ω := R ×(

−π

2,π

2

)× (0,∞)

and

ξ ∗ η =(

ξ1 + η1 + 2(ξ3 ln η3 + tan ξ2 ln η3 − η3 ln ξ3 − tan η2 ln ξ3)

arctan(ξ3 + η3 + tan ξ2 + tan η2 − ξ3η3)

ξ3 η3

).


Ex. 2) Let h be a nilpotent Lie algebra. Denote by � the Campbell–Hausdorff mul-tiplication on h (see Definition 2.2.11, page 128). Then (h,�) is a Lie group(see the first part of Corollary 2.2.15). Denote by h∗ the Lie algebra of thisgroup. Prove that h and h∗ are isomorphic as Lie algebras.

Hint: Let (G, ∗) be a Lie group whose Lie algebra g is isomorphic to h (seethe third fundamental theorem of Lie, Theorem 2.2.14). Let ϕ : h → g

be a Lie-algebra isomorphism. Denote by Exp G : g → G the exponen-tial map related to G. Prove that Exp G ◦ ϕ is a Lie-group isomorphismfrom (h,�) to (G, ∗), using the universality of the operation �. Derivethat d(Exp G ◦ ϕ) : h∗ → g is a Lie-algebra isomorphism. Consequently,ϕ−1 ◦ d(Exp G ◦ ϕ) is a Lie-algebra isomorphism from h∗ to h. See also thefollowing diagram

(h,�)ϕ−→ (g,�)

Exp G−→ (G, ∗)

↑ ↑h∗ −→ −→ −→ g

ϕ−1

−→ h.

d(Exp G ◦ ϕ)

Ex. 3) Let h be a nilpotent Lie algebra. Denote by � the Campbell–Hausdorff mul-tiplication on h (see Definition 2.2.11, page 128). Let (G, ∗) be a Lie groupisomorphic to the Lie group (h,�). Prove that the Lie algebra g of G isisomorphic to h as Lie algebras. (Hint: Use the preceding exercise: g isisomorphic to h∗, which is isomorphic to h.)

Ex. 4) Prove that the following is another equivalent definition of stratified group:

(H∗) A stratified group is a connected and simply connected Lie group G

whose Lie algebra g admits a (vector space) decomposition of the type

g = V1 ⊕ · · · ⊕ Vr,

where {[Vi, Vj ] ⊆ Vi+j ∀ i, j : i + j ≤ r,

[Vi, Vj ] = 0 ∀ i, j : i + j > r,(2.70)

and V1 generates (by iterated commutators) all g.

In fact, in (2.70), it holds [Vi, Vj ] = Vi+j if i + j ≤ r . This is yet anotherequivalent definition of Carnot group, frequently adopted in literature.

Hint: Use the following facts: If G is a stratified group according to Defini-tion 2.2.3, then (setting Vi := {0} if i > r) it holds

Vi = [V1, · · · [V1, V1]]︸︷︷︸i times

for every i ∈ N.

In particular, V1 Lie-generates all the Vi’s (whence it generates also g =V1 ⊕ · · · ⊕ Vr ). This also gives (using Proposition 1.1.7, page 12)


[Vi, Vj ] =[[V1, · · · [V1, V1]]︸︷︷︸

i times

, [V1, · · · [V1, V1]]︸︷︷︸j times

]

⊆ [V1, · · · [V1, V1]]︸︷︷︸i + j times

= Vi+j .

In particular, (2.70) holds. Vice versa, let G satisfy the above hypothesis(H∗). Set W1 := V1 and

Wi := [W1,Wi−1] = [V1, · · · [V1, V1]]︸︷︷︸i times

for i ≥ 2.

Prove that condition (2.70) implies that Wi ⊆ Vi for every 1 ≤ i ≤ r andWi = {0} for every i > r . Moreover, the second hypothesis in (H∗) (i.e.V1 Lie-generates g) ensures that g = W1 + · · · + Wr . Now, a simple linearalgebra argument shows that the following conditions

W1 + · · · + Wr = g = V1 ⊕ · · · ⊕ Vr, Wi ⊆ Vi ∀ i ≤ r

are sufficient to derive that Wi = Vi for every 1 ≤ i ≤ r . As a consequence,we have [V1, Vj ] = [W1,Wj ] = Wj+1 = Vj+1 whenever 1 + j ≤ r , and[V1, Vj ] = [W1,Wj ] = {0} whenever 1+j > r , so that G is a Carnot groupaccording to Definition 2.2.3.

Ex. 5) Let g be a Lie algebra. Consider the so-called lower central series of g, i.e.the sequence of subspaces defined by

g(1) := g, g

(j) := [g, g(j−1)], j ≥ 1.

In other words, for every j ≥ 2, we have

g(j) = [g, [g, · · · [g, g]]]︸︷︷︸

j times

.

Prove the following facts:• It holds g(1) ⊇ g(2) ⊇ · · · ⊇ g(j) ⊇ g(j+1) ⊇ · · · for every j ∈ N;• if there exists r ≥ 1 such that g(r) = g(r+1), then g(r) = g(j) for every

j ≥ r;• g is nilpotent of step r iff g(r+1) = {0}, but g(r) �= {0},• g is nilpotent of step r iff

g(1)

� g(2)

� · · · � g(r)

� g(r+1) = {0}.

Ex. 6) Prove the following fact:

A (finite dimensional) nilpotent Lie algebra g

of step two is necessarily stratified.


Indeed, let us set V2 = [g, g] and choose any V1 such that g = V1 ⊕ V2:then it also holds [V1, V1] = V2 and [V1, V2] = {0}.Hint: for every X, Y ∈ g, write X = X1 + X2 and Y = Y1 + Y2, whereXi, Yi ∈ Vi , and observe that [g, g] � [X, Y ] = [X1, Y1] ∈ [V1, V1].

Ex. 7) Consider the Carnot group on R4 (whose points are denoted by (x, y) with

x ∈ R, y = (y1, y2, y3) ∈ R3) with the composition law

(x, y) ◦ (ξ, η) =⎛

⎜⎝

x + ξ

y1 + η1y2 + η2 + 1

2 (xη1 − ξy1)

y3 + η3 + 12 (xη2 − ξy2) + 1

12 (x − ξ)(xη1 − ξy1)

⎞

⎟⎠ .

Its Lie algebra g is spanned by

X = ∂x − 1

2y1 ∂y2 − 1

2y2 ∂y3 − 1

12xy1 ∂y3 ,

Y1 = ∂y1 + 1

2x ∂y2 + 1

12x2 ∂y3 ,

Y2 = ∂y2 + 1

2x ∂y3 ,

Y3 = ∂y3 ,

and the following commutator relations hold:

[X, Y1] = Y2, [X, Y2] = Y3, [X, Y3] = 0, [Yi, Yj ] = 0,

i, j ∈ {1, 2, 3}.Prove that the following are three different stratifications of g:

g = span{X, Y1} ⊕ span{Y2} ⊕ span{Y3},g = span{X, Y1 + Y2} ⊕ span{Y2 + Y3} ⊕ span{Y3},g = span{X, Y1 + Y2 + Y3} ⊕ span{Y2 + Y3} ⊕ span{Y3}.

Ex. 8) Let g be a (finite dimensional) stratified Lie algebra, i.e. suppose we haveg = V (1) ⊕ V (2) ⊕ · · · ⊕ V (r) with (set V (i) := {0} whenever i > r)[V (1), V (i)] = V (i+1) for every i ∈ N. Prove the following facts:

• g is nilpotent of step r;• V (r) = g(r), the r-th element of the lower central series for g (see Ex. 5);

(Hint: for every X1, · · · , Xr ∈ g, write Xi = X(1)i + · · · + X

(r)i , where

X(1)i ∈ V (1), · · · , X(r)

i ∈ V (r), and observe that [X1, · · · [Xr−1, Xr ]] =[X(1)

1 , · · · [X(1)r−1, X

(1)r ]].)

• Suppose V (1) ⊕· · ·⊕V (r) and W(1) ⊕· · ·⊕W(r) are two stratificationsof g. Prove that


W(r) = V (r),

W(r−1) = V (r−1) ⊕ U(r)r−1 (where U

(r)r−1 is a subspace of V (r)),

W(r−2) = V (r−2) ⊕ U(r−1)r−2 ⊕ U

(r)r−2

(where U(r−1)r−2 , U

(r)r−2 are subspaces of V (r−1), V (r), respectively).

In other words, we have

W(r) = V (r),

W(r−1) = V (r−1) (modulo V (r)),

W(r−2) = V (r−2) (modulo V (r−1) + V (r)),

...

W(1) = V (1) (modulo V (2) + · · · + V (r−1) + V (r)).

Precisely, prove the following facts:(i) dim(V (i)) = dim(W(i)) =: Ni for every i = 1, . . . , r . Choosetwo bases V and W of g adapted respectively to the stratificationswith the V (i)’s and with the W(i)’s. Set V = {V1, . . . , VN } and W ={W1, . . . ,WN } and prove the existence of non-singular matrices M(i)’sof order Ni × Ni such that (with clear meaning of the notation)

⎛

⎝W1...

WN

⎞

⎠ =

⎛

⎜⎜⎜⎝

M(1) � · · · �0 M(2) . . .

......

. . .. . . �

0 · · · 0 M(r)

⎞

⎟⎟⎟⎠

⎛

⎝V1...

VN

⎞

⎠ .

(Hint: By the preceding part of the exercise, we have W(r) = V (r) =g(r). Now consider the quotient g/g(r) . It holds

W(1)

/W(r) ⊕ · · · ⊕ W(r−1)

/W(r) = g/g(r) = V(1)

/V (r) ⊕ · · · ⊕ V(r−1)

/V (r) ,

and these are stratifications of the Lie algebra g/g(r) too. Consequently,

W(r−1)

/W(r) = V(r−1)

/V (r) . Proceed inductively. Alternatively, argue as in thetwo-stratification Lemma 2.2.9.)

Ex. 9) Let g be a stratified Lie algebra, i.e. g = V (1) ⊕ V (2) ⊕ · · · ⊕ V (r) with (setV (i) := {0} whenever i > r) [V (1), V (i)] = V (i+1) for every i ∈ N. Provethat the lower central series of g (see Ex. 5 above) is given by

g(k) =

r⊕

i=k

V (i) ∀ k = 1, . . . , r, g(k) = {0} ∀ k > r.

(Hint: First prove that g(k) ⊆ ⊕ri=k V (i) arguing inductively as follows:

g(2) = [g, g] = [V (1) + · · · + V (r), V (1) + · · · + V (r)] ⊆ ⊕ri=2 V (i);


g(3) = [g, [g, g]] ⊆ [⊕ri=1 V (i),

⊕ri=2 V (i)] ⊆ ⊕r

i=3 V (i), etc. Vice versa,it holds

V (i) = span{[X1, · · · [Xi−1, Xi] · · ·] : X1, . . . , Xi ∈ V (1)}⊆ [[g, · · · [g, g] · · ·]]︸︷︷︸

i times

= g(i),

then derive⊕r

i=k V (i) = V (k) ⊕ · · · ⊕ V (r) ⊆ g(k) ⊕ · · · ⊕ g(r) = g(k).)

Ex. 10) The natural identification between the Lie algebra g of Lie group G withthe tangent space Ge to G at the identity must be “handled with care”, asthe following example shows. Suppose G and H are Lie groups with Liealgebras g, h, respectively. We denote by e both the identity of G and thatof H. Suppose F : G → H is a C∞-map such that F(e) = e. Obviously,we have deF : Ge → He. But Ge is identified with g and He is identifiedwith h. Consequently, F induces a natural map, say f , between g and h. Thefollowing question arises: as in the case when F is a group homomorphism,does f represent pointwise the differential of F ? Id est (see (2.33)), does ithold (for every X ∈ g and every x ∈ G)

(f (X))F(x)?= dxF (Xx).

The answer is in general negative. Indeed, with the above notation (seealso (2.26)) the map f : g → h is defined in the following way: for everyX ∈ g and every y ∈ H, we have

(f (X))y = deτy(deF (Xe)),

i.e. f (X) ∈ h is the vector field on H defined by

H � y �→ de(τy ◦ F)(Xe).

We ask whether

(∗1) dxF (Xx)?= de(τF(x) ◦ F)(Xe) ∀ X ∈ g, x ∈ G.

Since X is left-invariant, it holds Xx = deτx(Xe), so that dxF (Xx) =dxF (deτx(Xe)) = de(F ◦ τx)Xe. As a consequence, (∗1) holds iff (recallthat Ge = {Xe : X ∈ g} by Theorem 2.1.43-(1))

(∗2) de(F ◦ τx)?= de(τF(x) ◦ F) ∀ x ∈ G.

(Note that this certainly holds if F is a group homomorphism!) In turn, thisis equivalent to

(∗3) dxF?= dx(τF(x) ◦ F ◦ τx−1) ∀ x ∈ G.


Note that (∗3) is a “differential equation”{

dxF = deτF(x) ◦ deF ◦ dxτx−1 for every x ∈ G,

F (e) = e.

For example, prove that the above system holds for G = H = (RN,+)

iff F is a linear map, i.e. F is an automorphism of the Lie group (RN,+).Consequently, the map F : (R,+) → (R,+), F(x) = x2 furnishes acounterexample.

Ex. 11) Suppose that G = (RN, ◦) and H = (Rn, ∗) are two Lie groups on RN

and Rn, respectively. Let g and h denote the relevant Lie algebras. Finally,

let ϕ : G → H be a homomorphism of Lie groups. If dϕ : g → h is the mapdefined in Definition 2.1.49, prove that it operates in the following way: forevery X ∈ g, dϕ(X) is the left invariant vector field on H whose columnvector of the coefficients at the point y ∈ R

n is given by

(dϕ(X)

)I (y) = Jτy◦ϕ(0) · XI (0),

or, equivalently,(dϕ(X)

)I (y) = Jτy (0) · Jϕ(0) · XI (0).

Observe that (since Jτ0(0) is the identity matrix, and since the coordinatesof a vector field Y w.r.t. the Jacobian basis are given by YI (0)) the matrixrepresenting the linear map dϕ with respect to the relevant Jacobian baseson g and h, respectively, is simply

Jϕ(0).

Ex. 12) Provide the ODE’s details for what is stated in the first paragraph of theproof of Proposition 2.1.53, page 116.

Ex. 13) Suppose G = (RN, ◦) and H = (Rn, ∗) are two Lie groups on RN and R

n,respectively (according to the definition given in Chapter 1). Let g and h

be the relevant Lie algebras. Let ϕ : G → H be a homomorphism of Liegroups. Show that the differential of ϕ, as defined in Definition 2.1.49, isthe map dϕ : g → h such that, for every X ∈ g, dϕ(X) is the only vectorfield in h such that

(dϕ(X)I

)(ϕ(x)) = Jϕ(x) (XI)(x). (2.71)

Ex. 14) The following is an example to the “change of adapted basis”, described inRemark 2.2.20. We follow the therein notation.

Example 2.3.1 (Stratified groups of step two). Let (H, ∗) be a homogeneousCarnot group of step two on R

N . We shall prove in Section 3.2, page 158,that if Z denotes the Jacobian basis related to H, then it holds


(ξ, τ ) �Z (ξ ′, τ ′) =(

ξj + ξ ′j (j = 1, . . . , m)

τi + τ ′i + 1

2 〈B(i)ξ, ξ ′〉 (i = 1, . . . , n)

),

where m is the number of generators of H, n = N − m and if

{X1, . . . , Xm; T1, . . . , Tn}re-denotes the Jacobian basis, then B(i) is the m × m matrix

B(i) = (b

(i)h,k

)1≤h,k≤m

defined by[Xk,Xh] = ∑n

k=1 b(i)h,kTi .

If Z is another basis of the algebra of g adapted to its stratification, say

Z = {X1, . . . , Xm; T1, . . . , Tn},there exist two non-singular matrices U (of order m × m) and V (of ordern × n) such that

(X1I · · · XmI

) = (X1I · · ·XmI

) · U,(T1I · · · TnI

) = (T1I · · · TnI

) · V.

A simple computation shows that the new composition law �Z is given by(using, for example, (2.58a))

(ξ, τ ) �Z (ξ ′, τ ′) =(

ξj + ξ ′j (j = 1, . . . , m)

τi + τ ′i + 1

2 〈B(i)ξ, ξ ′〉 (i = 1, . . . , n)

),

where, if V −1 = (wi,j )i,j≤n,

B(i) = UT ·(

n∑

j=1

wi,j B(j)

)· U.

The test of this fact is left to the reader. �

3

Carnot Groups of Step Two

The aim of this chapter is to collect some results and many explicit examples ofCarnot groups of step two. Some examples are well known in literature, some arenew. To begin with, we present the most studied (and by far one of the most impor-tant) among Carnot groups, the Heisenberg–Weyl group. Then, we turn our attentionto general homogeneous Carnot groups of step two and m generators, m ≥ 2. Inparticular, we show that they are naturally given with the data on R

m+n of n skew-symmetric matrices of order m.

The set of examples that we provide here contains the free step-two homogeneousgroups, the prototype groups of Heisenberg-type (which will be widely studied inChapter 18) and the H-groups in the sense of Métivier.

3.1 The Heisenberg–Weyl Group

Let us consider in Cn × R (whose points we denote by (z, t) with t ∈ R and z =

(z1, . . . , zn) ∈ Cn) the following composition law

(z, t) ◦ (z′, t ′) = (z + z′, t + t ′ + 2 Im(z · z′)). (3.1)

In (3.1), we have set (i obviously denotes the imaginary unit) Im(x + iy) = y

(x, y ∈ R), whereas z · z′ denotes the usual Hermitian inner product in Cn,

z · z′ =n∑

j=1

(xj + iyj )(x′j − iy′

j ).

Hereafter we agree to identify Cn with R

2n and to use the following notation1 todenote the points of C

n × R ≡ R2n+1:

1 Someone may say that the notation (z, t) ≡ (x1, y1, . . . , xn, yn, t) would be more appro-priate, but the other notation is so deeply entrenched that we have no choice but to go alongwith it.

156 3 Carnot Groups of Step Two

(z, t) ≡ (x, y, t) = (x1, . . . , xn, y1, . . . , yn, t)

with z = (z1, . . . , zn), zj = xj + iyj and xj , yj , t ∈ R. Then, the composition law◦ can be explicitly written as

(x, y, t) ◦ (x′, y′, t ′) = (x + x′, y + y′, t + t ′ + 2〈y, x′〉 − 2〈x, y′〉), (3.2)

where 〈·, ·〉 denotes the usual inner product in Rn. It is quite easy to verify that

(R2n+1, ◦) is a Lie group whose identity is the origin and where the inverse is givenby (z, t)−1 = (−z,−t). Let us now consider the dilations

δλ : R2n+1 → R

2n+1, δλ(z, t) = (λz, λ2t).

A trivial computation shows that δλ is an automorphism of (R2n+1, ◦) for everyλ > 0. Then H

n = (R2n+1, ◦, δλ) is a homogeneous group. It is called theHeisenberg–Weyl group in R

2n+1. For example, when n = 1, the Heisenberg–Weylgroup H

1 in R3 is equipped with the composition law

(x, y, t) ◦ (x′, y′, t ′) = (x + x′, y + y′, t + t ′ + 2 (yx′ − xy′)),

while, when n = 2, the Heisenberg–Weyl group H2 in R

5 is equipped with thecomposition law

(x1, x2, y1, y2, t) ◦ (x′1, x

′2, y

′1, y

′2, t

′)= (x1 + x′

1, x2 + x′2, y1 + y′

1, y2 + y′2, t + t ′ + 2 (y1x

′1 + y2x

′2 − x1y

′1 − x2y

′2)).

The Jacobian matrix at the origin of the left translation τ(z,t) is the following blockmatrix

Jτ(z,t)(0, 0) =

⎛

⎝In 0 00 In 0

2 yT −2 xT 1

⎞

⎠ ,

where In denotes the n×n identity matrix, while 2 yT and −2 xT stand for the 1 ×n

matrices (2y1 · · · 2yn) and (−2x1 · · · − 2xn), respectively. Then, the Jacobian basisof hn, the Lie algebra of H

n, is given by

Xj = ∂xj+ 2yj ∂t , Yj = ∂yj

− 2xj ∂t , j = 1, . . . , n, T = ∂t .

Since [Xj , Yj ] = −4 ∂t , we have

rank(Lie{X1, . . . , Xn, Y1, . . . , Yn}(0, 0)

)

= dim(span{∂x1 , . . . , ∂xn , ∂y1 , . . . , ∂yn ,−4∂t }

) = 2n + 1.

This shows that Hn is a Carnot group with the following stratification2

hn = span{X1, . . . , Xn, Y1, . . . , Yn} ⊕ span{∂t }. (3.3)

2 Obviously, there exist other possible stratifications, but the above one is the most frequentlyadopted and the one we shall refer to.

3.1 The Heisenberg–Weyl Group 157

The step of (Hn, ◦) is r = 2 and its Jacobian generators are the vector fields Xj ,Yj (j = 1, . . . , n). The canonical sub-Laplacian on H

n (also referred to as KohnLaplacian) is then given by

ΔHn =n∑

j=1

(X2

j + Y 2j

).

An explicit formula for ΔHn can be found in Ex. 1, at the end of this chapter. Finally,we exhibit the explicit form of the exponential map for H

n. It is given by3

Exp ((ξ, η, τ ) · Z) = (ξ, η, τ ). (3.4)

Here we have set (ξ, η, τ ) · Z = ∑nj=1(ξjXj + ηjYj ) + τT .

We now want to perform a change of variables in Hn, inspired by the change of

basis in hn, which turns the Jacobian basis into the new basis

X∗j := Xj , Y ∗

j := Yj (j = 1, . . . , n), T ∗ := [X1, Y1] = −4 T .

As above, we set (ξ, η, τ ) · Z∗ = ∑nj=1(ξjX

∗j + ηjY

∗j ) + τT ∗. In other words, we

have chosen another basis of hn adapted to the stratification (3.3).Then, what we aim to do (see the proof of Theorem 2.2.18, page 131) is to equip

hn with a Lie-group structure isomorphic (via Exp ) to that of (Hn, ◦); we identify hn

with R2n+1 via the cited basis Z∗: this amounts to equip R

2n+1 with the composition∗ such that

Log((Exp ((ξ, η, τ ) · Z∗)) ◦ (Exp ((ξ ′, η′, τ ′) · Z∗))

)

= ((ξ, η, τ ) ∗ (ξ ′, η′, τ ′)

) · Z∗.

The explicit expression for ∗ is easily found,4

(ξ, η, τ ) ∗ (ξ ′, η′, τ ′)

=(

ξ + ξ ′, η + η′, τ + τ ′ − 1

2〈η, ξ ′〉 + 1

2〈ξ, η′〉

), (3.5a)

or, equivalently,

(ζ, τ ) ∗ (ζ ′, τ ′) =(

ζ + ζ ′, τ + τ ′ + 1

2〈Bζ, ζ ′〉

)(3.5b)

with

B =(

0 −In

In 0

).

The natural isomorphism ϕ : (R2n+1, ∗) → (Hn, ◦) is given by

ϕ(ξ, η, τ ) = Exp ((ξ, η, τ ) · Z∗) = Exp ((ξ, η,−4τ) · Z) = (ξ, η,−4τ).

3 See (3.11a) in Section 3.5, page 167, and Remark 3.2.4, page 163.4 For all the details, see Section 4.3.2, page 200.


3.2 Homogeneous Carnot Groups of Step Two

Let m, n ∈ N. Set RN := R

m × Rn and denote its points by z = (x, t) with x ∈ R

m

and t ∈ Rn. Given an n-tuple B(1), . . . , B(n) of m × m matrices with real entries, let

(x, t) ◦ (ξ, τ ) =(

x + ξ, t + τ + 1

2〈Bx, ξ 〉

). (3.6)

Here 〈Bx, ξ 〉 denotes the n-tuple

(〈B(1)x, ξ 〉, . . . , 〈B(n)x, ξ 〉)(

also written asm∑

i,j=1

Bi,j xj ξi

)

and 〈·, ·〉 stands for the inner product in Rm. One can easily verify that (RN, ◦) is a

Lie group whose identity is the origin and where the inverse is given by

(x, t)−1 = (−x,−t + 〈Bx, x〉).We highlight that the inverse map is the usual −(x, t) if and only if, for every k =1, . . . , n, it holds

〈B(k)x, x〉 = 0 ∀ x ∈ Rm,

i.e. iff the matrices B(k)’s are skew-symmetric. It is also quite easy to recognize thatthe dilation

δλ : RN → R

N, δλ(x, t) = (λx, λ2t) (3.7)

is an automorphism of (RN, ◦) for any λ > 0. Then G = (RN, ◦, δλ) is a homoge-neous Lie group.

We explicitly remark that the composition law of any Lie group in Rm × R

n,homogeneous w.r.t. the dilations {δλ}λ as in (3.7), takes the form (3.6) (see Theo-rem 1.3.15, page 39).

The Jacobian matrix at (0, 0) of the left translation τ(x,t) takes the followingblock form

Jτ(x,t)(0, 0) =

⎛

⎝Im 0

12 B x In

⎞

⎠ ,

where, if B(k) = (b(k)i,j )i,j≤m for k = 1, . . . , n, Bx denotes the matrix

(m∑

j=1

b(k)i,j xj

)

k≤n, i≤m

.

More explicitly, we have

Jτ(x,t) (0, 0) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

Im 0m×n

12

∑mj=1 b

(1)1,j xj · · · 1

2

∑mj=1 b

(1)m,j xj

... · · · ...12

∑mj=1 b

(n)1,j xj · · · 1

2

∑mj=1 b

(n)m,j xj

In

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

3.2 Homogeneous Carnot Groups of Step Two 159

Then the Jacobian basis of g, the Lie algebra of G, is given by

Xi = (∂/∂xi) + 1

2

n∑

k=1

(m∑

l=1

b(k)i,l xl

)(∂/∂tk)

= (∂/∂xi) + 1

2〈(Bx)i,∇t 〉, i = 1, . . . , m, (3.8)

Tk = ∂/∂tk, k = 1, . . . , n.

Here, we briefly denoted by (Bx)i the vector of Rn

((B(1)x)i , . . . , (B

(n)x)i),

where (B(k)x)i is the i-th component of B(k)x. An easy computation shows that

[Xj ,Xi] =n∑

k=1

1

2

(b

(k)i,j − b

(k)j,i

)∂tk =:

n∑

k=1

c(k)i,j ∂tk .

We have denoted by C(k) = (c(k)i,j )i,j≤m the skew-symmetric part of B(k), i.e.

C(k) = 1

2

(B(k) − (B(k))T

).

Let us now assume that C(1), . . . , C(n) are linearly independent. This implies thatthe m2 × n matrix ⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

C(1)1,1 · · · C

(n)1,1

C(1)1,2 · · · C

(n)1,2

... · · · ...

C(1)1,m · · · C

(n)1,m

C(1)2,1 · · · C

(n)2,1

... · · · ...

(proceed analogously up to)... · · · ...

C(1)m,m · · · C

(n)m,m

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

has rank equal to n. As a consequence,

span{[Xj ,Xi] | i, j = 1, . . . , m} = span{∂t1 , . . . , ∂tn}.Therefore,

rank(Lie{X1, . . . , Xm}(0, 0)

)

= dim(span{∂x1 , . . . , ∂xm, ∂t1 , . . . , ∂tn}

) = m + n.

This shows that G is a Carnot group of step two and Jacobian generators X1, . . . , Xm.


We explicitly remark that the linear independence of the matrices

C(1), . . . , C(n)

is also necessary for G to be a Carnot group. Then, we have proved the followingproposition.

Proposition 3.2.1 (Characterization. I). Every homogeneous Lie group G on RN ,

homogeneous with respect to the dilation

δλ : RN → R

N, δλ(x, t) = (λx, λ2t)

(where x ∈ Rm, t ∈ R

n and N = m + n), is equipped with the composition law

(x, t) ◦ (ξ, τ ) =(

x + ξ, t1 + τ1 + 1

2〈B(1)x, ξ 〉, . . . , tn + τn + 1

2〈B(n)x, ξ 〉

)

for n suitable m × m matrices B(1), . . . , B(n).Moreover, a characterization of homogeneous Carnot groups of step two and m

generators is given by the above G = (Rm+n, ◦, δλ), where the skew-symmetric partsof the B(k)’s are linearly independent.

We remark that the above arguments show that there exist Carnot groups of anydimension m ∈ N of the first layer and any dimension

n ≤ m(m − 1)/2

of the second layer: it suffices to choose n linearly independent matrices B(1),

. . . , B(n) in the vector space of the skew-symmetric m × m matrices (which hasdimension m(m − 1)/2) and then define the composition law as in (3.6). Finally, bymeans of the general results on stratified groups in Chapter 2, we obtain the followingtheorem.

Theorem 3.2.2 (Characterization. II). Every N -dimensional (not necessarily ho-mogeneous) stratified group of step two and m generators is naturally isomorphic toa homogeneous Carnot group (RN, ◦, δλ) with Lie group law as in (3.6) for somem×m linearly independent skew-symmetric matrices B(k)’s. The group of dilationsis given by (3.7), and the inverse of x is −x.

By (3.8), we can write explicitly the canonical sub-Laplacian of the Lie groupG = (RN, ◦) with ◦ as in (3.6). It is given by

ΔG = Δx + 1

4

n∑

h,k=1

〈B(h)x, B(k)x〉 ∂thtk

+m∑

k=1

〈B(k)x,∇x〉 ∂tk + 1

2

n∑

k=1

trace(B(k)) ∂tk . (3.9)

3.2 Homogeneous Carnot Groups of Step Two 161

Here, we denoted

Δx =m∑

i=1

∂xi ,xiand ∇x = (∂x1 , . . . , ∂xm).

We recognize that ΔG contains partial differential terms of second order only iftrace(B(k)) = 0 for every k = 1, . . . , n. This happens, for example, if the B(k)’sare skew-symmetric, i.e. if the inverse map on G is x �→ −x.

Example 3.2.3. Following all the above notation, let us take m = 3, n = 2 and

B(1) =⎛

⎝1 1 0

−1 0 00 0 0

⎞

⎠ , B(2) =⎛

⎝0 0 −10 1 01 0 0

⎞

⎠ .

Then the composition law on R5 = R

3 × R2 as in (3.6) becomes (denoting (x, t) =

(x1, x2, x3, t1, t2) and analogously for (ξ, τ ))

(x, t) ◦ (ξ, τ ) =

⎛

⎜⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3

t1 + τ1 + 12 (x1ξ1 + ξ1x2 − ξ2x1)

t2 + τ2 + 12 (x2ξ2 − ξ1x3 − ξ3x1)

⎞

⎟⎟⎟⎟⎠,

and the dilation is

δλ(x1, x2, x3, t1, t2) = (λx1, λx2, λx3, λ2t1, λ

2t2).

Then G = (R5, ◦, δλ) is a homogeneous Carnot group, for the skew-symmetric partsof B(1) and B(2) are linearly independent,

1

2

(B(1) − (B(1))T

) =⎛

⎝0 1 0

−1 0 00 0 0

⎞

⎠ ,1

2

(B(2) − (B(2))T

) =⎛

⎝0 0 −10 0 01 0 0

⎞

⎠ .

In fact, we can compute the first three vector fields of the Jacobian basis and verifythat they are Lie-generators for the whole Lie algebra,

X1 = ∂x1 + 1

2(x1 + x2)∂t1 − 1

2x3∂t2 ,

X2 = ∂x2 − 1

2x1∂t1 + 1

2x2∂t2 ,

X3 = ∂x3 + 1

2x1∂t2 ,

[X1, X2] = −∂t1,

[X1, X3] = ∂t2 ,

[X2, X3] = 1

2∂t2 .


The related canonical sub-Laplacian is

ΔG = ∂x1,x1 + ∂x2,x2 + ∂x3,x3

+ 1

4

{{(x1 + x2)

2 + (−x1)2} ∂t1,t1 + {

(−x3)2 + (x2)

2 + (x1)2} ∂t2,t2

+ 2{(x1 + x2)(−x3) + (−x1)(x2)

}∂t1,t2

}

+ {(x1 + x2) ∂x1 − x1 ∂x2} ∂t1 + {−x3 ∂x1 + x2 ∂x2 + x1 ∂x3} ∂t2

+ 1

2∂t1 + 1

2∂t2 .

ΔG contains first order terms, for trace(B(1)) �= 0 �= trace(B(2)).On the contrary, if

B(1) =⎛

⎝1 1 0

−1 0 00 0 0

⎞

⎠ , B(2) =⎛

⎝0 −2 02 1 00 0 0

⎞

⎠ ,

then the composition law on R5 given by

(x, t) ◦ (ξ, τ ) =

⎛

⎜⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3

t1 + τ1 + 12 (x1ξ1 + ξ1x2 − ξ2x1)

t2 + τ2 + 12 (x2ξ2 − 2ξ1x2 + 2ξ2x1)

⎞

⎟⎟⎟⎟⎠

does not define a homogeneous Carnot group, because the skew-symmetric parts ofB(1) and B(2) are linearly dependent,

1

2

(B(1) − (B(1))T

) =⎛

⎝0 1 0

−1 0 00 0 0

⎞

⎠ ,1

2

(B(2) − (B(2))T

) =⎛

⎝0 −2 02 0 00 0 0

⎞

⎠ .

In fact, the only admissible dilation would be

δλ(x, t) = (λ x1, λ x2, λ x3, λ2 t1, λ

2 t2),

but the first three vector fields of the related Jacobian basis are not Lie-generators forthe whole Lie algebra, since

X1 = ∂x1 + 1

2(x1 + x2)∂t1 − x2∂t2 ,

X2 = ∂x2 − 1

2x1∂t1 +

(1

2x2 + x1

)∂t2,

X3 = ∂x3 ,

[X1, X2] = −∂t1 + 2∂t2 ,

[X1, X3] = [X2, X3] = 0.

3.3 Free Step-two Homogeneous Groups 163

Remark 3.2.4. The composition law on the Heisenberg–Weyl group Hk on R

2k+1,which is given by

(x, y, t) ◦ (x′, y′, t ′) = (x + x′, y + y′, t + t ′ + 2 (〈y, x′〉 − 〈x, y′〉)),

is of the form (3.6) with m = 2k, n = 1 and

B(1) = 4

(0 Ik

−Ik 0

).

3.3 Free Step-two Homogeneous Groups

In this section, following the notation of the previous one, we fix a particular setof matrices B(k)’s and consider the relevant homogeneous Carnot group (Fm,2, �),which will serve as prototype for what we shall call free Carnot group of step twoand m generators. Throughout the section, m ≥ 2 is a fixed integer.

Let i, j ∈ {1, . . . , m} be fixed with i > j , and let S(i,j) be the m × m skew-symmetric matrix whose entries are −1 in the position (i, j), +1 in the position(j, i) and 0 elsewhere. For example, if m = 3, we have

S(2,1) =⎛

⎝0 1 0

−1 0 00 0 0

⎞

⎠ , S(3,1) =⎛

⎝0 0 10 0 0

−1 0 0

⎞

⎠ ,

S(3,2) =⎛

⎝0 0 00 0 10 −1 0

⎞

⎠ .

Then, we agree to denote by (Fm,2, �) the Carnot group on RN associated to these

m(m − 1)/2 matrices according to (3.6) of the previous section. We set

n := m(m − 1)/2, N = m + n = m(m + 1)/2,

I := {(i, j) | 1 ≤ j < i ≤ m}.We observe that the set I has exactly n elements.

In the sequel of this section, we shall use the following notation, different fromthe one used in the previous section: instead of using the notation t for the coordinatein the “second layer” of the group, we denote the points of Fm,2 by (x, γ ), wherex = (x1, . . . , xm) ∈ R

m, γ ∈ Rn, and the coordinates of γ are denoted by

γi,j where (i, j) ∈ I.

Here we have ordered I in an arbitrary (henceforth) fixed way. Then, the composi-tion law � is given by

(x, γ ) � (x′, γ ′) =(

xh + x′h, h = 1, . . . , m

γi,j + γ ′i,j + 1

2 (xi x′j − xj x′

i ), (i, j) ∈ I

).


For example, when m = 3, we have

(x, γ ) ◦ (x′, γ ′) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

x1 + x′1

x2 + x′2

x3 + x′3

γ2,1 + γ ′2,1 + 1

2 (x2x′1 − x1x

′2)

γ3,1 + γ ′3,1 + 1

2 (x3x′1 − x1x

′3)

γ3,2 + γ ′3,2 + 1

2 (x3x′2 − x2x

′3)

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

.

By (3.8), we can compute the Jacobian basis

Xh, h = 1, . . . , m, Γi,j , (i, j) ∈ I,

of fm,2, the Lie algebra of Fm,2: it holds

Xh = (∂/∂xh) + 1

2

∑

1≤j<i≤m

(m∑

l=1

S(i,j)h,l xl

)(∂/∂γi,j )

=

⎧⎪⎪⎨

⎪⎪⎩

∂x1 + 12

∑1<i≤m xi ∂γi,1 if h = 1,

∂xh+ 1

2

∑h<i≤m xi ∂γi,h − 1

2

∑1≤j<m xj ∂γh,j if 1 < h < m,

∂xm + 12

∑1≤j<m xj ∂γm,j if h = m,

Γi,j = ∂/∂γi,j , (i, j) ∈ I.

Moreover, for every (i, j) ∈ I, we have the commutator identities

[Xj ,Xi] =∑

1≤k<h≤m

S(h,k)i,j ∂γh,k = ∂γj,i − ∂γi,j ,

whence we recognize that the algebra fm,2 is “the most non-Abelian as possible” (asit is allowed for an algebra with m generators and step two).

This is the reason why we shall refer to (any algebra isomorphic to) fm,2 as a freeLie algebra with m generators and step two (see Chapter 14 for more details on freeLie algebras). For example, when m = 3, we have

X1 = ∂x1 + 1

2(x2 ∂γ2,1 + x3 ∂γ3,1),

X2 = ∂x2 + 1

2(x3 ∂γ3,2 − x1 ∂γ2,1),

X3 = ∂x3 − 1

2(x1 ∂γ3,1 − x2 ∂γ3,2)

Γ2,1 = ∂γ2,1, Γ3,1 = ∂γ3,1, Γ3,2 = ∂γ3,2.

From (3.9), we derive the explicit expression for the canonical sub-Laplacian of F3,2,

3.4 Change of Basis 165

ΔF3,2 = (∂x1)2 + (∂x2)

2 + (∂x3)2

+ 1

4

{(x2

2 + x21)(∂γ2,1)

2 + (x23 + x2

1)(∂γ3,1)2 + (x2

3 + x22)(∂γ3,2)

2}

+ 1

2x2x3 (∂γ2,1 ∂γ3,1) − 1

2x1x3 (∂γ2,1 ∂γ3,2) + 1

2x1x2 (∂γ3,1 ∂γ3,2)

+ (x2∂x1 − x1∂x2)∂γ2,1 + (x3∂x1 − x1∂x3)∂γ3,1

+ (x3∂x2 − x2∂x3)∂γ3,2. (3.10)

3.4 Change of Basis

In this section we consider “stratified” changes of basis on a homogeneous Carnotgroup of step two, according to the definition in Remark 1.4.10, page 61. We use theusual notation, as in the beginning of Section 3.2.

Let C,D be two fixed non-singular matrices, with C of dimension m × m and D

of dimension n × n. We denote by L the following N × N matrix

L =(

C 00 D

).

Finally, we denote again by L the relevant linear change of basis on RN , i.e., the

linear map (· denotes row-times-column matrix product)

L : RN → R

N, L(x, t) = L ·(

x

t

)=(

C · x

D · t

).

We define on RN a new composition law ∗ obtained by writing ◦ in the new coordi-

nates defined by (ξ, τ ) = L(x, t). More precisely, we have

(ξ, τ ) ∗ (ξ ′, τ ′) := L((L−1(ξ, τ )) ◦ (L−1(ξ ′, τ ′))

).

An explicit computation gives (here D = (dh,k)h,k≤n)

(ξ, τ ) ∗ (ξ ′, τ ′)

=(

ξi + ξ ′i , i = 1, . . . , m

τh + τ ′h + 1

2

⟨(C−1)T · (

∑nk=1 dh,k B(k)) · C−1ξ, ξ ′

⟩, h = 1, . . . , n

),

i.e. setting

B(h) := (C−1)T ·(

n∑

k=1

dh,k B(k)

)· C−1 for every h = 1, . . . , n,

∗ can be written as follows:

(ξ, τ ) ∗ (ξ ′, τ ′) =(

ξi + ξ ′i , i = 1, . . . , m

τh + τ ′h + 1

2

⟨B(h)ξ, ξ ′⟩, h = 1, . . . , n

).


We explicitly remark that, if the matrices B(h)’s are skew-symmetric, the same istrue for the matrices B(h)’s. It can easily be proved that H = (RN, ∗, δλ) is a homo-geneous Carnot group isomorphic to G = (RN, ◦, δλ) via L (see also Section 16.3,page 637).

It is immediately seen that the canonical basis of H is

(∂/∂ξi) + 1

2

n∑

h=1

(m∑

j=1

B(h)i,j xj

)(∂/∂τh), i = 1, . . . , m,

∂/∂τk, k = 1, . . . , n.

Moreover, it is not difficult to see that the expression Xi of the i-th vector Xi of thecanonical basis of G in the new coordinates (ξ, τ ) on H (i.e. the image via dL of Xi)coincides at the origin with

L · XiI (0) = (C · ei, 0),

where (e1, . . . , em) denotes the canonical basis of Rm. In particular, given an arbi-

trary sub-Laplacian L = ∑mj=1 Y 2

j on G, it is always possible to perform a linear

change of basis on RN so that L is turned into the canonical sub-Laplacian on a

homogeneous Carnot group isomorphic to G.

3.5 The Exponential Map of a Step-two Homogeneous Group

We now turn to compute the exponential map of a homogeneous group of step two.As usual, if ξ ∈ R

m and τ ∈ Rn, we write

(ξ, τ ) · Z =m∑

j=1

ξjXj +n∑

i=1

τiTi,

where X1, . . . , Xm, T1, . . . , Tn is the Jacobian basis. More explicitly, by (3.8), wehave

((ξ, τ ) · Z

)I =

⎛

⎜⎜⎜⎜⎝

ξ12

∑mi,j=1 b

(1)i,j ξixj

...12

∑mi,j=1 b

(n)i,j ξixj

⎞

⎟⎟⎟⎟⎠.

By the definition of exponential map, we have Exp ((ξ, τ ) ·Z) = (x(1), t (1)), where(x(s), t (s)) solves

(x(s), t (s)) = ((ξ, τ ) · Z)I (x(s), t (s)), (x(0), t (0)) = (0, 0),

i.e. more explicitly,

3.5 The Exponential Map of a Step-two Homogeneous Group 167{

x(s) = ξ, x(0) = 0,

t(s) = τ + 12

∑mi,j=1 Bi,j ξi xj (s), t (0) = 0.

Here, we have set for brevity

Bi,j :=

⎛

⎜⎜⎝

b(1)i,j

...

b(n)i,j

⎞

⎟⎟⎠ .

A straightforward computation gives

Exp ((ξ, τ ) · Z) =(

ξ, τ + 1

4

m∑

i,j=1

Bi,j ξiξj

), (3.11a)

(here we again set Bi,j = (B(1)i,j , . . . , B

(n)i,j )), whence

Log (x, t) =(

x, t − 1

4

m∑

i,j=1

Bi,j xixj

)· Z. (3.11b)

We explicitly remark that, by (3.11a), the exponential map acts like a sort of “identitymap” (provided g is equipped with coordinates w.r.t. the Jacobian basis) if and onlyif the matrices B(i)’s are skew-symmetric.

We now operate a change of coordinates in RN inspired by the exponential map.

In other words (see the proof of Theorem 2.2.18, page 131), we equip the algebra g

of G by a Lie-group structure isomorphic via Exp to that of (G, ◦), and we identifyg with R

N via the Jacobian basis.This amounts to equip R

N = Rm × R

n with the composition law �Z (the reasonfor this notation will become clearer in further sections) such that

Log((Exp ((ξ, τ ) · Z)) ◦ (Exp ((ξ ′, τ ′) · Z))

)= (

(ξ, τ ) �Z (ξ ′, τ ′)) · Z.

We find an explicit expression for �Z . It holds

Log((Exp ((ξ, τ ) · Z)) ◦ (Exp ((ξ ′, τ ′) · Z))

)

= Log

((ξ, τ + 1

4

m∑

i,j=1

Bi,j ξiξj

)◦(

ξ ′, τ ′ + 1

4

m∑

i,j=1

Bi,j ξ ′i ξ

′j

))

= Log

(ξ + ξ ′, τ + τ ′ + 1

4

m∑

i,j=1

Bi,j (ξiξj + ξ ′i ξ

′j ) + 1

2

m∑

i,j=1

Bi,j ξj ξ′i

)

=(

ξ + ξ ′, τ + τ ′ + 1

4

m∑

i,j=1

Bi,j (ξiξj + ξ ′i ξ

′j ) + 1

2

m∑

i,j=1

Bi,j ξj ξ′i


− 1

4

m∑

i,j=1

Bi,j (ξi + ξ ′i )(ξj + ξ ′

j )

)· Z

=(

ξ + ξ ′, τ + τ ′ + 1

4

m∑

i,j=1

Bi,j ξj ξ′i − 1

4

m∑

i,j=1

Bi,j ξiξ′j )

)· Z

=(

ξ + ξ ′, τ + τ ′ + 1

2

m∑

i,j=1

Bi,j − Bj,i

2ξj ξ

′i

)· Z.

As a consequence, the composition law ∗ on RN has the form

(ξ, τ ) �Z (ξ ′, τ ′) =(

ξ + ξ ′, τ + τ ′ + 1

2

m∑

i,j=1

Bi,j − Bj,i

2ξj ξ

′i

), (3.12)

i.e. it has the form as in (3.6) with the matrices B(k)’s replaced by the skew-symmetricmatrices

B(k) = B(k) − (B(k))T

2, k = 1, . . . , n,

the skew-symmetric parts of the B(k)’s. The inverse map on (RN,�Z) is the usual(−x,−t). The homogeneous Lie group (RN,�Z, δλ) is isomorphic to (RN, ◦, δλ)

via the Lie-group isomorphism

Ψ : (RN,�Z) → (RN, ◦), Ψ (ξ, τ ) :=(

ξ, τ + 1

4

m∑

i,j=1

Bi,j ξiξj

),

which “essentially” equals the exponential map (written in the Jacobian coordinates).Since the differential of Ψ is associated to the matrix

JΨ (ξ, τ ) =⎛

⎝Im 0

(∑mj=1

B(k)i,j +B

(k)j,i

2 ξj

)

k≤n,i≤mIn

⎞

⎠ ,

we see that the Jacobian basis of (RN,�Z) is turned into the Jacobian basis of(RN, ◦) by the change of coordinates Ψ (see, for instance, (2.17b), page 101, or(2.71), page 153).

For future references, we summarize what we have just proved in the followingproposition.

Proposition 3.5.1. Let RN = R

m × Rn be equipped with a homogeneous Lie group

structure by the composition law

(x, t) ◦ (ξ, τ ) =(

x + ξ, t1 + τ1 + 1

2〈B(1)x, ξ 〉, . . . , tn + τn + 1

2〈B(n)x, ξ 〉

)

and the dilation δλ(x, t) = (λx, λ2t), where B(1), . . . , B(n) are fixed m × m ma-trices. Then G = (RN, ◦, δλ) is isomorphic to the homogeneous Lie group G =(RN,�Z, δλ), where:

3.6 Prototype Groups of Heisenberg Type 169

1) δλ is the same dilation as above;2) �Z is defined by

(x, t) �Z (ξ, τ ) =(

x + ξ, t1 + τ1 + 1

2〈B(1)x, ξ 〉, . . . , tn + τn + 1

2〈B(n)x, ξ 〉

)

where B(i) is the skew-symmetric part of B(i) for every i = 1, . . . , n; the inversemap on G is (−x,−t);

3) the Lie-group isomorphism is Ψ : G → G with

Ψ (ξ, τ ) =(

ξ, τ1 + 1

4〈B(1) ξ, ξ 〉, . . . , τn + 1

4〈B(n) ξ, ξ 〉

),

so that Ψ is the identity map iff all the B(i)’s are skew-symmetric;4) the Jacobian basis of G corresponds (via dΨ ) to the Jacobian basis of G, in

particular the canonical sub-Laplacian of G corresponds to that of G;5) the exponential map of the group G is the linear map sending the Jacobian basis

of the algebra of G into the canonical basis of G;6) if G is a homogeneous Carnot group, then the same is true for G.

3.6 Prototype Groups of Heisenberg Type

Definition 3.6.1 ((Prototype) Heisenberg-type group). Consider the homogeneousLie group

H = (Rm+n, ◦, δλ)

with composition law as in (3.6) (page 158), i.e.

(x, t) ◦ (ξ, τ ) =(

x + ξ, t1 + τ1 + 1

2〈B(1)x, ξ 〉, . . . , tn + τn + 1

2〈B(n)x, ξ 〉

),

where B(1), . . . , B(n) are fixed m × m matrices, and dilations as in (3.7). Let us alsoassume that the matrices B(1), . . . , B(n) have the following properties:

1) B(j) is an m × m skew-symmetric and orthogonal matrix for every j ≤ n;2) B(i) B(j) = −B(j) B(i) for every i, j ∈ {1, . . . , n} with i �= j .

If all these conditions are satisfied, H is called a ( prototype) group of Heisen-berg-type, in short, a ( prototype) H-type group.

A H-type group is a Carnot group, since conditions 1) and 2) imply the linearindependence of B(1), . . . , B(n).

Indeed, if α = (α1, . . . , αn) ∈ Rn \ {0}, then

1

|α|n∑

s=1

αs B(s)


is orthogonal (hence non-vanishing), as the following computation shows,

(1

|α|n∑

s=1

αsB(s)

)·(

1

|α|n∑

s=1

αsB(s)

)T

= − 1

|α|2∑

r,s≤n

αrαs B(r)B(s)

= − 1

|α|2∑

r≤n

α2r (B(r))2 − 1

|α|2∑

r,s≤n, r �=s

αrαs B(r)B(s)

= Im.

Here we used the following facts: (B(r))2 = −Im, since B(r) is skew-symmetric andorthogonal; B(r)B(s) = −B(s)B(r) according to condition 2).

The generators of H are the vector fields (see (3.8), page 159)

Xi = ∂xi+ 1

2

n∑

k=1

(m∑

l=1

b(k)i,l xl

)∂tk , i = 1, . . . , m. (3.13a)

Moreover, if we setTk := ∂/∂tk, k = 1, . . . , n, (3.13b)

then (again from (3.8)) we know that

{X1, . . . , Xm; T1, . . . , Tn} (3.13c)

is the Jacobian basis for H.A direct computation shows that the canonical sub-Laplacian ΔH = ∑m

i=1 X2i

can be written as follows (see (3.9))

ΔH = Δx + 1

4

n∑

h,k=1

〈B(h)x, B(k)x〉 ∂thtk

+m∑

k=1

〈B(k)x,∇x〉 ∂tk + 1

2

n∑

k=1

trace(B(k)) ∂tk .

On the other hand, by conditions 1) and 2),

〈B(h)x, B(h)x〉 = |x|2,while, for h �= k,

〈B(h)x, B(k)x〉 = 0

since

〈B(h)x, B(k)x〉 = −〈B(k) B(h)x, x〉 = 〈B(h) B(k)x, x〉= −〈B(k)x, B(h)x〉.

3.6 Prototype Groups of Heisenberg Type 171

We also have trace(B(k)) = 0, since B(k) is skew-symmetric. Then ΔH takes thevery compact form

ΔH = Δx + 1

4|x|2Δt +

n∑

k=1

〈B(k)x,∇x〉 ∂tk . (3.14)

By the computations made in Section 3.5 in finding the exponential map of a gen-eral homogeneous group of step two, we infer that (see precisely (3.11a), page 167)if {X1, . . . , Xm; T1, . . . , Tn} is the Jacobian basis for H (where the Xi’s and the Tk’sare as in (3.13a), (3.13b)) then the exponential map for the ( prototype) H-type groupH is

Exp : h → H, Exp (x1X1 + · · · + xmXm + t1T1 + · · · + tnTn) = (x, t). (3.15)

Remark 3.6.2. From (3.8) one obtains

|∇Hu|2 = |∇xu|2 + 1

4

m∑

i=1

〈(Bx)i,∇t u〉2

+m∑

i=1

〈(Bx)i,∇t u〉 ∂xiu, u ∈ C∞.

On the other hand,

m∑

i=1

〈(Bx)i,∇t u〉2 =n∑

h,k=1

〈B(h)x, B(k)x〉∂thu ∂tku

= |x|2 |∇t u|2

andm∑

i=1

〈(Bx)i,∇t u〉 ∂xiu =

n∑

k=1

〈B(k)x,∇xu〉 ∂tku.

Thus, for every smooth real-valued function u, it holds

|∇Hu|2 = |∇xu|2 + 1

4|x|2 |∇t u|2 +

n∑

k=1

〈B(k)x,∇xu〉 ∂tku. (3.16)

Remark 3.6.3. The first layer of a H-type group has even dimension m. Indeed, ifB is a m × m skew-symmetric orthogonal matrix, we have Im = B · BT = −B2,whence 1 = (−1)m(det B)2.

Remark 3.6.4. With the previous notation, if H = (Rm+n, ◦, δλ) is a H -type group,then

z = {(0, t) | t ∈ Rn}

is the center of H. Indeed, let (y, t) ∈ H be such that


(x, s) ◦ (y, t) = (y, t) ◦ (x, s) for every (x, s) ∈ H.

This holds iff〈B(k)x, y〉 = 〈B(k)y, x〉

for any x ∈ Rm and any k ∈ {1, . . . , n}. Then, since (B(k))T = −B(k),

〈B(k)y, x〉 = 0 ∀ x ∈ Rm ∀ k ∈ {1, . . . , n},

so that y = 0 because B(k) is orthogonal (hence non-singular).

Remark 3.6.5. The classical Heisenberg–Weyl group Hk on R

2k+1 is canonically iso-morphic to a prototype H-type group. Precisely (see (3.5b), page 157), it is isomor-phic to the prototype H-type group (H, ∗) corresponding to the case m = 2k, n = 1and

B(1) =(

0 −Ik

Ik 0

).

The isomorphism ϕ : (R2k+1, ∗) → (Hk, ◦) is given by

ϕ(ξ, η, τ ) = (ξ, η,−4τ).

Moreover, H is in its turn isomorphic to the prototype H-type group with m = 2k,n = 1 and

B(1) = diag

{(0 −11 0

), . . . ,

(0 −11 0

)}, the block occurring k times.

This type of prototype Heisenberg-group is the only (up to isomorphism) H-typegroup with one-dimensional center (see Chapter 18).

Remark 3.6.6. Groups of Heisenberg type with center of dimension n ≥ 2 do exist.For example, the following two matrices

B(1) =

⎛

⎜⎜⎝

0 −1 0 01 0 0 00 0 0 −10 0 1 0

⎞

⎟⎟⎠ , B(2) =

⎛

⎜⎜⎝

0 0 1 00 0 0 −1

−1 0 0 00 1 0 0

⎞

⎟⎟⎠

satisfy conditions 1)–2) and hence they define in R6 = R

4 × R2 a H-type group

whose center has dimension 2. The composition law is

(x, t) ◦ (ξ, τ ) =

⎛

⎜⎜⎜⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3x4 + ξ4

t1 + τ1 + 12 (−x2ξ1 + x1ξ2 − x4ξ3 + x3ξ4)

t2 + τ2 + 12 (x3ξ1 − x4ξ2 − x1ξ3 + x2ξ4)

⎞

⎟⎟⎟⎟⎟⎟⎠.

3.7 H-groups (in the Sense of Métivier) 173

The above matrices B(1) and B(2), together with

B(3) =

⎛

⎜⎜⎝

0 0 0 10 0 1 00 −1 0 0

−1 0 0 0

⎞

⎟⎟⎠ ,

define in R7 = R

4 × R3 a H-type group whose center has dimension 3. The compo-

sition law is

(x, t) ◦ (ξ, τ ) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3x4 + ξ4

t1 + τ1 + 12 (−x2ξ1 + x1ξ2 − x4ξ3 + x3ξ4)

t2 + τ2 + 12 (x3ξ1 − x4ξ2 − x1ξ3 + x2ξ4)

t3 + τ3 + 12 (x4ξ1 + x3ξ2 − x2ξ3 − x1ξ4)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Remark 3.6.7. The following result holds (see [Kap80, Corollary 1]). Let m, n betwo positive integers. Then there exists a H-type group of dimension m + n whosecenter has dimension n if and only if it holds n < ρ(m), where ρ is the so-calledHurwitz–Radon function, i.e.

ρ : N → N, ρ(m) := 8p + q, where m = (odd) · 24p+q, 0 ≤ q ≤ 3.

We explicitly remark that if m is odd, then ρ(m) = 0, whence the first layer of anyH-type group has even dimension (as we already proved in Remark 3.6.3).

Remark 3.6.8. The groups of Heisenberg-type were introduced by A. Kaplan in[Kap80]. Kaplan’s definition of H-type groups is more abstract than the one givenhere. In Chapter 18, we shall show that, up to an isomorphism, the two definitionsare equivalent.

3.7 H-groups (in the Sense of Métivier)Following G. Métivier [Met80], we give the following definition.

Definition 3.7.1 (H-group (in the sense of Métivier)). Let g be a (finite-dimensionalreal) Lie algebra, and let us denote by z its center. We say that g is of H-type (in thesense of Métivier) if it admits a vector space decomposition

g = g1 ⊕ g2

{[g1, g1] ⊆ g2,

g2 ⊆ z

with the following additional property: for every η ∈ g∗2 (the dual space of g2), the

skew-symmetric bilinear form on g1 defined by

Bη : g1 × g1 → R, Bη(X,X′) := η([X,X′])is non-degenerate5 whenever η �= 0.

5 We recall that a bilinear map on a finite-dimensional vector space is non-degenerate if any(or equivalently, if one) of the matrices representing it w.r.t. a fixed basis is non-singular.


We say that a Lie group is a H-group (in the sense of Métivier), or a HM-groupin short, if its Lie algebra is of H-type (in the sense of Métivier).

Remark 3.7.2. In this remark, we follow the notation of Definition 3.7.1.First, a (Métivier-)H-type algebra is obviously nilpotent of step two. Moreover,

we have

[g, g] = [g1 + g2, g1 + g2] ⊆ [g1, g1] (since g2 ⊆ z),

[g1, g1] ⊆ [g, g] (since g1 ⊆ g).

Consequently, it holds[g, g] = [g1, g1]. (3.17a)

Finally, we claim thatg2 = [g, g]. (3.17b)

Indeed, from (3.17a) we first derive that [g, g] = [g1, g1] ⊆ g2 (by the very definitionof (Métivier-)H-type algebra). We are left to show that g2 ⊆ [g, g]. Suppose to thecontrary that there exists Z ∈ g2 such that Z /∈ [g, g]. This implies, in particular, thatZ �= {0}. Moreover, since both Z ∈ g2 and [g, g] ⊆ g2, there certainly exists η ∈ g∗

2such that g2(Z) �= 0 (whence η �= 0) and η vanishes identically on [g, g] (here, weare using the fact that Z /∈ [g, g]). But this implies that, for every X,X′ ∈ g1, wehave

Bη(X,X′) = η([X,X′]) = 0,

for[X,X′] ∈ [g1, g1] = [g, g] and η|[g,g] ≡ 0.

This is in contradiction with the non-degeneracy of Bη.Collecting together (3.17a) and (3.17b), we see that a (Métivier-)H-type algebra

is stratified: indeed we have

g = g1 ⊕ g2 with [g1, g1] = g2 and [g1, g2] = {0}.As a consequence, a HM-group is a Carnot group.

Moreover, if g1 is any other complement of g2 = [g, g] in g, it is easy to provethat also the decomposition g = g1 ⊕ g2 satisfies Definition 3.7.1.

Collecting the above results, we have proved the following proposition.

Proposition 3.7.3 (Characterization. I). A HM-group is a Carnot group G of steptwo such that if

g = g1 ⊕ g2 ([g1, g1] = g2, [g1, g2] = {0})is any stratification of the Lie algebra g of G, then the following property holds: Forevery non-vanishing linear map η from g2 to R, the (skew-symmetric) bilinear formBη on g1 defined by

Equivalently, a bilinear map B on the vector space V is non-degenerate if, for every v ∈V \ {0}, there exists w ∈ V such that B(v,w) �= 0.

3.7 H-groups (in the Sense of Métivier) 175

Bη(X,X′) := η([X,X′]), X, X′ ∈ g1,

is non-degenerate.

When G is expressed in its logarithmic coordinates (making it a homogeneousCarnot group), the above definition is easily re-written as follows. We consider ahomogeneous Lie group of step two G = (Rm+n, ◦, δλ) with the composition law asin (3.6) (page 158), i.e.

(x, t) ◦ (ξ, τ ) =(

x + ξ, t1 + τ1 + 1

2〈B(1)x, ξ 〉, . . . , tn + τn + 1

2〈B(n)x, ξ 〉

),

where B(1), . . . , B(n) are fixed m × m matrices, and the group of dilations isδλ(x, t) = (λx, λ2t). For the sake of simplicity, we may also suppose that the matri-ces B(k)’s are skew-symmetric (see, for instance, Proposition 3.5.1). We make use ofthe results and the notation of Section 3.2.

Now, if η is a linear map from g2 to R, there exist n scalars η1, . . . , ηn ∈ R suchthat

η : g2 → R, η(∂ti ) = ηi for all i = 1, . . . , n.

In particular, the map Bη, as in Proposition 3.7.3, can be explicitly written (after asimple calculation which we leave to the reader) as follows6

if X = ∑mi=1 vi Xi and X′ = ∑m

i=1 v′i Xi ,

then Bη(X,X′) =n∑

i,j=1

(−

n∑

k=1

ηk B(k)i,j

)vi v′

j .

In other words, the matrix representing the (skew-symmetric) bilinear map Bη w.r.t.the basis X1, . . . , Xm of g1 is the matrix

η1B(1) + · · · + ηnB

(n).

Hence, to ask for Bη to be non-degenerate (for every η �= 0) is equivalent to askthat any linear combination of the matrices B(k)’s is non-singular, unless it is thenull matrix (recall that the B(k)’s are linearly independent). We have thus proved thefollowing proposition.

6 Recall that, by (3.8), page 159, a basis for g1 is {X1, . . . , Xm} with

Xi = (∂/∂xi) + 1

2

n∑

k=1

⎛

⎝m∑

l=1

B(k)i,l

xl

⎞

⎠(∂/∂tk),

where we have setB(k) =

(B

(k)i,l

)

i,l≤m.


Proposition 3.7.4 (Characterization. II). Let G = (Rm+n, ◦) be a homogeneousCarnot group of step two, with the composition law

(x, t) ◦ (ξ, τ ) =(

x + ξ, t1 + τ1 + 1

2〈B(1)x, ξ 〉, . . . , tn + τn + 1

2〈B(n)x, ξ 〉

),

where B(1), . . . , B(n) are m×m skew-symmetric linearly independent matrices. ThenG is a HM-group if and only if every non-vanishing linear combination of the matri-ces B(k)’s is non-singular.

In particular, if the above G is a HM-group, then the B(k)’s are all non-singularm × m matrices, but since the B(k)’s are also skew-symmetric, this implies that m isnecessarily even.

Remark 3.7.5 (Any H-type group is a HM-group). Any prototype H-type group (ac-cording to Definition 3.6.1, page 169) is a HM-group. Indeed, as it can be seenfrom the computations on page 169, in a prototype H-type group, for every η =(η1, . . . , ηn) ∈ R

n, η �= 0, we proved that∑n

k=1 ηkB(k) is |η| times an orthogonal

matrix, hence (in particular)∑n

k=1 ηkB(k) is non-singular. The converse is not true.

For example, consider the group on R5 (the points are denoted by (x, t), x ∈ R

4,t ∈ R) with the composition law

(x, t) ◦ (ξ, τ ) =(

x + ξ, t + τ + 1

2〈Bx, ξ 〉

),

where

B =

⎛

⎜⎜⎝

0 1 0 0−1 0 0 00 0 0 20 0 −2 0

⎞

⎟⎟⎠ .

Then G is obviously a HM-group, for B is a non-singular skew-symmetric matrix.But G is not a prototype H-type group, for B is not orthogonal. But more is true: aswe shall prove in Chapter 18 (see Remark 18.2.7 and Corollary 18.2.8, page 695) G

is not even isomorphic to any prototype H-type group.

Bibliographical Notes. The explicitness of the composition law of a homogeneousCarnot group of step two allows to face a great variety of problems. See, e.g. somerecent results concerning with step-two Carnot groups: [Dai00] for mappings withbounded distortion, [FSS03b] for geometric measure theory results, [Mil88] for amicrolocal version of some results concerning hypoellipticity, [Ric06] and [SY06]for results on the convex functions, [Tha94] for theorems of Paley–Wiener type.

For an introduction to harmonic analysis on the Heisenberg group, see the mono-graph by S. Thangavelu [Tha98], and for a survey see R. Howe [How80].



Ex. 1) With reference to Section 3.1, recognize that ΔH1 equals

(∂x1)2 + (∂x2)

2 + 4(x21 + x2

2) (∂x3)2 + 4 x2 ∂x1,x3 − 4 x1 ∂x2,x3 .

Prove the analogous (general) formula for ΔHn ,

ΔHn =n∑

j=1

(∂2xj

+ ∂2yj

) + 4

(N∑

j=1

(x2j + y2

j )

)∂2t + 4

N∑

j=1

(yj ∂xj− xj ∂yj

) ∂t .

Recognize that the quadratic form related to ΔHn at (x, y, t) is the quadraticform q related to the symmetric matrix

A =⎛

⎝ In2 y

−2 x

2 yT −2 xT 4 (|x|2 + |y|2)

⎞

⎠ .

For every fixed (x, y, t) ∈ Hn, find all vectors (ξ, η, τ ) ∈ R

2n+1 such that

(ξ, η, τ ) · A · (ξ, η, τ )T = 0.

Compare this to what was shown in (A4), page 65.Ex. 2) Prove (3.9), page 160.Ex. 3) Making use of (3.9), derive an explicit expression for the canonical sub-

Laplacian of Fm,2 generalizing (3.10) (page 165).Ex. 4) Consider the Lie group on R

4 (the points are denoted by x = (x1, x2, x3, x4))with the composition law given by

x ◦ y =

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + x1y2 − x2y1x4 + y3 + x1y2 − x2y1

⎞

⎟⎟⎠ .

Prove that (R4, ◦) is a Lie group nilpotent of step two. According to Propo-sition 3.2.1, this is not a homogeneous Carnot group (for any dilation!) since(we use the usual notation of (3.6), page 158) the relevant matrices

B(1) = B(2) =(

0 1−1 0

)

have the same skew-symmetric parts (whence the latter are not linearly inde-pendent). However, according to Ex. 6 in Chapter 2, page 149, since (R4, ◦)

is nilpotent of step two, its algebra g is necessarily stratified! Indeed, setting

X1 = ∂x1 − x2 ∂x3 − x2 ∂x4 ,

X2 = ∂x2 + x1 ∂x3 + x1 ∂x4 ,

X3 = ∂x3 , X4 = ∂x4 , (3.18)


we have[X1, X2] = 2 ∂x3 + 2 ∂x4 = 2(X3 + X4),

whenceg = span{X1, X2, X3} ⊕ span{X3 + X4},

and this is a stratification, as in Definition 2.2.3 (page 122). Consequently,this time thanks to Theorem 3.2.2, (R4, ◦) is isomorphic to a homogeneousCarnot group. Indeed, retracing the arguments in the proof of Theorem 2.2.18(page 131, see also the notation therein), prove that (R4, ◦) is isomorphic to(R4,�, δλ), where

δλ(ξ1, ξ2, ξ3, ξ4) = (λξ1, λξ2, λξ3, λ2ξ4),

and � is obtained by the Campbell–Hausdorff multiplication on g equippedwith the basis {X1, X2, X3, X4 + X3}, which, in turn, is given by

ξ ◦ η =

⎛

⎜⎜⎝

ξ1 + η1ξ2 + η2ξ3 + η3

ξ4 + η4 + ξ1η2 − ξ2η1

⎞

⎟⎟⎠ .

A Lie group isomorphism between (R4, ◦) and (R4,�) can be calculatednoting that (R4,�) is essentially the Lie algebra of (R4, ◦): we thus computethe exponential map via the usual system of ODE’s as follows. Since

{ξ1 X1 + ξ2 X2 + ξ3 X3 + ξ4 (X4 + X3)

}I (x) =

⎛

⎜⎜⎝

ξ1ξ2

ξ3 + ξ4 − ξ1 x2 + ξ2 x1ξ4 − ξ1 x2 + ξ2 x1

⎞

⎟⎟⎠

(denote this field by YI (x)), the solution γ to γ (t) = YI (γ (t)), γ (0) = 0 is

γ (t) = (ξ1 t, ξ2 t, (ξ3 + ξ4)t, ξ4 t

),

and we have γ (1) = (ξ1, ξ2, ξ3 + ξ4, ξ4). Define the map

Φ : (R4,�) → (R4, ◦), Φ(ξ) := (ξ1, ξ2, ξ3 + ξ4, ξ4)

and prove that it is a Lie group isomorphism, which turns the vector fieldsin (3.18), respectively, into

X1 = ∂ξ1 − ξ2 ∂x4 , X2 = ∂ξ2 + ξ3 ∂x4 ,

X3 = ∂ξ3 , X4 = ∂ξ4 − ∂ξ3 .

Ex. 5) Prove in details the last statement in Remark 3.7.2, page 174.


Ex. 6) Consider the following homogeneous Carnot group of step two (which willbe called of Kolmogorov type, see Section 4.1.4): this is R

5 (the points are,by our usual notation, (x, t) = (x1, x2, x3, t1, t2)) with the composition law

(x, t) ◦ (ξ, τ ) =

⎛

⎜⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3

t1 + τ1 − x2 ξ1t2 + τ2 − x3 ξ1

⎞

⎟⎟⎟⎟⎠.

The relevant matrices are

B(1) =⎛

⎝0 −2 00 0 00 0 0

⎞

⎠ , B(2) =⎛

⎝0 0 −20 0 00 0 0

⎞

⎠ .

Prove that the related canonical sub-Laplacian is

(∂x1 − x2 ∂t1 − x3 ∂t2

)2 + (∂x2)2 + (∂x3)

2.

Now, with a slight change of notation, consider on R1+2n (whose points are

denoted by (t, x, y), t ∈ R, x, y ∈ Rn) the operator

L := Δx + (∂ − ⟨

x,∇y

⟩)2,

where Δx = ∑ni=1(∂xi

)2 and ∇y = (∂y1 , . . . , ∂yn). Find a homogeneousCarnot group structure on R

1+2n such that L is the related canonical sub-Laplacian.

Ex. 7) Operate a change of variables on R5 in such a way that the group (R5, ◦)

defined in the previous exercise is turned into a homogeneous Carnot groupwhose related matrices B(k)’s are skew-symmetric. (Do the same for the gen-eral case of R

1+2n.) Prove that the new composition law and the new canon-ical sub-Laplacian are, respectively,

(x, t) ◦ (ξ, τ ) =

⎛

⎜⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3

t1 + τ1 + 12 (x1 ξ2 − x2 ξ1)

t2 + τ2 + 12 (x1 ξ3 − x3 ξ1)

⎞

⎟⎟⎟⎟⎠

and(

∂x1 − 1

2x2 ∂t1 − 1

2x3 ∂t2

)2

+(

∂x2 + 1

2x1 ∂t1

)2

+(

∂x3 + 1

2x1 ∂t2

)2

.

Note that the relevant matrices are


B(1) = 1

2

(B(1) − (B(1))T

) =⎛

⎝0 −1 01 0 00 0 0

⎞

⎠ ,

B(2) = 1

2

(B(2) − (B(2))T

) =⎛

⎝0 0 −10 0 01 0 0

⎞

⎠ .

Ex. 8) (The polarized Heisenberg group). Let us denote by (x, y, t) the points ofR

n × Rn × R ≡ R

2n+1. Consider the composition law

(x, y, t) ◦ (x′, y′, t ′) := (x + x′, y + y′, t + t ′ + 〈x′, y〉)

(where 〈·, ·〉 denotes the Euclidean inner product) and the dilation

δλ(x, y, t) := (λx, λy, λ2t).

Show that Hpoln = (R2n+1, ◦, δλ) is a homogeneous Carnot group and write

down its canonical sub-Laplacian.Note. H

poln is called the polarized Heisenberg group.

(Matrix representation of HHHpoln ). Given x, y ∈ R

n and t ∈ R, define the(n + 2) × (n + 2) matrix (x, y are being considered as column vectors)

m(x, y, t) := In+2 +⎛

⎝0 yT t

0 0 x

0 0 0

⎞

⎠ ,

where In+2 is the identity matrix of order n + 2. Show that

Mn := {m(x, y, t) | (x, y, t) ∈ R

2n+1 }

is a group with respect to the matrix multiplication. Show also that

m : Hpoln −→ Mn, (x, y, t) �→ m(x, y, t)

is a group isomorphism.

(A unitary representation of HHHpoln ). Given x, y ∈ R

n and t ∈ R, define theoperators

e(x), τ (y), χ(t)

from L2(Rn, C) into itself as follows:(e(x)f

)(ξ) := ei〈x,ξ〉 f (ξ), ξ ∈ R

n,(τ(y)f

)(ξ) := f (ξ + y), ξ ∈ R

n,(χ(t)f

)(ξ) := ei t f (ξ), ξ ∈ R

n.

(Again, 〈·, ·〉 denotes the Euclidean inner product.) Show that the set of oper-ators


Un := {e(x) τ (y) χ(t) | (x, y, t) ∈ R

2n+1 }

is a group with respect to the composition of operators. Show also that

π : Hpoln −→ Un, (x, y, t) �→ e(x) τ (y) χ(t)

is a group isomorphism.

Note. e(x) and τ(y) are the unitary operators generated by the position andthe momentum operators in quantum mechanics (see [Tha98] for more de-tails).

4

Examples of Carnot Groups

The aim of this chapter is to provide a wide number of explicit examples of Carnotgroups of step greater than two, thus enlarging the set of examples given in Chapter 3.Some are already known in literature (for instance, here we collect some results onthe filiform groups), some are new. Among the latter, there are what we call theB-groups, the Kolmogorov-type groups (or K-type groups), the Bony-type groups.

Moreover, in Section 4.2, we furnish a criterion to recognize when a given oper-ator sum of squares of vector fields in R

N is a sub-Laplacian on some homogeneousCarnot group. Precisely, given a set of linearly independent vector fields X1, . . . , Xm,homogeneous of degree one with respect to a family of dilations {δλ}λ>0 in R

N andsatisfying a suitable “rank-type” condition, we show how to explicitly define a com-position law ◦ on R

N making G = (RN, ◦, δλ) a Carnot group whose generators areX1, . . . , Xm. Our construction rests on the solvability of relevant systems of ODE’s.

In Section 4.3, we illustrate how this method can be applied to produce newexamples of Carnot groups. In Chapter 17, Section 17.4, we shall use the results ofthe present chapter to produce “lifted” groups.

4.1 A Primer of Examples of Carnot Groups

4.1.1 Euclidean Group

The additive group (RN,+) is a homogeneous group with respect to the dilations

δλ(x) = λ x, λ > 0.

We call E = (RN,+, δλ) the Euclidean group. E is a Carnot group of step 1. Itsgenerators are ∂x1 , . . . , ∂xN

. Thus, the canonical sub-Laplacian on E is the classicalLaplace operator

Δ =N∑

j=1

∂2xj

.

184 4 Examples of Carnot Groups

Moreover, any sub-Laplacian on E has the following form

L =N∑

j=1

Y 2j =

N∑

j=1

(N∑

i=1

bj,i ∂xi

)2

,

where B = (bi,j )i,j≤N is a non-singular constant matrix. Hence, L can also bewritten as

L =N∑

i,k=1

(N∑

j=1

bj,i bj,k

)∂xi ,xk

=N∑

i,k=1

(BT · B)i,k ∂xi ,xk,

so that L is a second order constant-coefficient strictly elliptic operator.Vice versa, if L =∑N

i,k=1 ai,k ∂xi ,xk, where A = (ai,k)i,k≤N is a positive-definite

symmetric matrix, then L is a sub-Laplacian on E. Indeed, this immediately followsby writing A = B2 (where B is a non-singular symmetric square-root of A), so thatL =∑N

j=1 Y 2j , where Yj :=∑N

i=1 Bj,i ∂xi.

We want to stress that

E is the only Carnot groupof step 1 (and N generators).

4.1.2 Carnot Groups with Homogeneous Dimension Q ≤ 3

Let G = (RN, ◦, δλ) be a Carnot group with homogeneous dimension Q ≤ 3. Withreference to the results and notation in Remark 1.4.8, we recall that Q =∑r

j=1 j Nj ,where r and N1, . . . , Nr are, respectively, the step of G and the dimensions of thelayers W(1), . . . ,W(r) of g.

Obviously, the group is not the Euclidean group in RN iff W(2) �= {0}, i.e.

r ≥ 2. In this case, the first layer W(1) must be at least two-dimensional since[W(1),W(1)] = W(2) �= {0}.

This shows that any non-Euclidean Carnot group has homogeneous dimensionQ ≥ 4. Thus, if Q ≤ 3, then G is the Euclidean group in R

N , i.e. ◦ = + andδλ(x) = λx. The sub-Laplacians on G are the second order elliptic operators withconstant coefficients. The canonical sub-Laplacians are d2/dx2

1 in R (Q = N = 1),∂2x1

+ ∂2x2

in R2 (Q = N = 2), and ∂2

x1+ ∂2

x2+ ∂2

x3in R

3 (Q = N = 3).

4.1.3 B-groups

Let us consider an N × N matrix B with real entries bi,j (with i, j = 1, . . . , N ). Letus put

E(t) := exp(t B), t ∈ R.

In R1+N , whose points will be denoted by z = (t, x), t ∈ R, x ∈ R

N , let us introducethe following composition law

4.1 A Primer of Examples of Carnot Groups 185

(t, x) ◦ (t ′, x′) = (t + t ′, x′ + E(t ′)x).

One easily verifies that B = (R1+N, ◦) is a Lie group whose identity is the origin(0, 0), and where the inverse is given by

(t, x)−1 = (−t,−E(−t)x).

The Jacobian matrix at the origin of the left translation τ(t,x) is the following blockmatrix

Jτ(t,x)(0, 0) =

(1 0b IN

),

where b stands for the N × 1 column-matrix

d

ds

∣∣∣∣s=0

E(s)x = BE(s)x∣∣s=0 = Bx.

Then the Jacobian basis of b, the Lie algebra of B, is given by

Y = ∂t + ∇x · Bx, ∂x1 , . . . , ∂xN. (4.1)

We explicitly remark that, for a general matrix B, the group (B, ◦) may not be nilpo-tent. Indeed, an easy computation shows that, for any j ∈ {1, . . . , N},

[[· · · [∂xjY ] · · ·]Y, Y︸︷︷︸

k times

] =N∑

i=1

(Bk)i,j ∂xi.

Hence, (B, ◦) is a nilpotent group iff B is a nilpotent matrix. For example, if N = 1and B = (1), the composition law is

(t, x) ◦ (t ′, x′) = (t + t ′, x′ + x exp(t ′)), (t, x), (t ′, x′) ∈ R

2,

and the Jacobian basis is ∂t +x∂x , ∂x . Then, R2 equipped with the above composition

law is not a stratified group (least of all a homogeneous Carnot group!) since it is notnilpotent. In particular, the second order differential operator on R

2 defined by

L = ∂2x + (∂t + x ∂x)

2 = (1 + x2) ∂2x + ∂2

t + 2x ∂x∂t + x ∂x

is a sum of squares of left-invariant vector fields on (R2, ◦), it satisfies Hörmander’shypoellipticity condition

rank(Lie{∂x, ∂t + x∂x}(t, x)

) = 2 ∀ (t, x) ∈ R2,

but L is not a sub-Laplacian on any Carnot group. Indeed, L is elliptic at any point(but B is not Euclidean!).


4.1.4 K-type Groups

Let us now suppose that the matrix B in the previous example takes the followingspecial block-form

B =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 · · · 0 0B1 0 · · · 0 0

0 B2. . .

......

......

. . . 0 00 0 · · · Br 0

⎞

⎟⎟⎟⎟⎟⎟⎠, (4.2a)

where Bj is a pj × pj−1 block with rank equal to pj , for every j = 1, 2, . . . , r .Moreover, p0 ≥ p1 ≥ · · · ≥ pr and p0 + p1 + · · · + pr = N . Finally, the 0 blocksin (4.2a) are suitably chosen in such a way that B has dimension N × N . We wantto show that the group B related to this matrix is a Carnot group. It will be called agroup of Kolmogorov type or, in short, a K-type group.

Let us split RN as follows

RN = R

p0 × Rp1 × · · · × R

pr

and define, for every λ > 0,

Dλx = Dλ(x(0), x(1), . . . , x(r)) = (λx(0), λ2x(1), . . . , λr+1x(r)), (4.2b)

where x(i) ∈ Rpi for 0 ≤ i ≤ r . We also put δλ(t, x) = (λt,Dλx).

Claim 1. For every λ > 0, the dilation δλ is an automorphism of B.

To prove this claim we need the following lemma.

Lemma 4.1.1. For every t ∈ R and λ > 0, we have

E(λt)Dλ = Dλ E(t) (4.2c)

where E(t) = exp(tB), B is as in (4.2a) and Dλ is the dilation in (4.2b).

Proof. Since Bk = 0 for every k ≥ r + 1, one has

E(t) =r∑

k=0

tk Bk/k!,

and (4.2c) holds for every t ∈ R and λ > 0 iff

λk Bk Dλ = Dλ Bk ∀ k ≥ 0, ∀ λ > 0. (4.2d)

This identity holds true when k = 0. An easy direct computation shows that it alsoholds true for k = 1. As a consequence,

λ2B2Dλ = λB(λBDλ) = λB(DλB)

= (λBDλ)B = (DλB)B = DλB2.

Then (4.2d) holds true for k = 2. An iteration of this argument shows (4.2d) fork ≥ 2. �


From this lemma the Claim 1 easily follows. Indeed, for every z = (t, x), z′ =(t ′, x′) ∈ R

1+N , we have

(δλz) ◦ (δλz′) = (λt,Dλx) ◦ (λt ′,Dλx

′)= (λt + λt ′,Dλx

′ + E(λt ′)Dλx)

(by Lemma 4.1.1) = (λ(t + t ′),Dλx′ + DλE(t ′)x)

= δλ(t + t ′, x′ + E(t ′)x)

= δλ(z ◦ z′). �Thus, B = (R1+N, ◦, δλ) is a homogeneous group whose first layer is

R × Rp0 = {(t, x(0)) | t ∈ R, x(0) ∈ R

p0}.Moreover, the vector fields in the Jacobian basis related to this first layer are givenby

Y = ∂t + 〈Bx,∇x〉, ∂x1 , . . . , ∂xp0. (4.2e)

Claim 2. We have rank(Lie{Y, ∂x1 , . . . , ∂xp0}(0, 0)) = 1 + N .

Once this claim is proved, it will follow that (R1+N, ◦, δλ) is a Carnot group ofstep r + 1 with 1 + p0 generators, which are the vector fields in (4.2e). Thus therelated canonical sub-Laplacian is given by

ΔB = Y 2 + ΔRp0 , where ΔR

p0 =p0∑

j=1

∂2xj

. (4.2f)

This sub-Laplacian will be said of Kolmogorov type. To prove Claim 2, the followinglemma will be useful.

Lemma 4.1.2. In Rp × R

q , let us consider the vector field

Z = Ay · (∇z)T , where A is a q × p matrix, y ∈ R

p and z ∈ Rq .

Suppose rank(A) = q ≤ p. Then

span{[∂yi

, Z] | i = 1, . . . , p} = span{∂z1 , . . . , ∂zq }. (4.2g)

Proof. Let A = (ai,j )i≤q, j≤p . Then

[∂yi, Z] =

q∑

j=1

aj,i ∂zj, i = 1, . . . , p,

so that, since rank(A) = q,

dim(span

{[∂yi, Z] | i = 1, . . . , p

}) = q.

This implies (4.2g). �


We now prove Claim 2. Since B has the form (4.2a), we can write

Y = ∂t +r∑

i=1

Bix(i−1) · (∇x(i) )

T .

Then, by applying Lemma 4.1.2, we get

span{[∂xi

, Y ] | i = 1, . . . , p0}

= span{[∂xi

, B1x(0) · (∇x(1) )

T ] | i = 1, . . . , p0}

= span{∂x

(1)i

| i = 1, . . . , p1}.

Another application of Lemma 4.1.2 gives

span{[∂

x(1)i

, Y ] | i = 1, . . . , p1} = span

{∂x

(2)i

| i = 1, . . . , p2}.

Iterating this argument, we get

Lie{Y, ∂x1 , . . . , ∂xp0} = Lie{Y, ∂x1 , . . . , ∂xN

}.This obviously proves the claim. �Note. The groups of Kolmogorov type were introduced by E. Lanconelli and S. Poli-doro [LP94] in studying a class of hypoelliptic ultraparabolic operators including theclassical prototype operators of Kolmogorov–Fokker–Planck. The composition lawin [LP94] was suggested by the structure of the fundamental solution of the operator∂2x1

+ x1 ∂x2 − ∂x3 in R3 given by A.N. Kolmogorov in [Kol34].

Example 4.1.3. With the notation of Section 4.1.4, an example of K-type group isgiven by the choice p0 = p1 = 1, B1 = (1), whence

B =(

0 0(1) 0

), N = p0 + p1 = 2, exp(s B) =

(1 0s 1

).

Hence, our K-type group B is R3 (whose points are denoted by (t, x1, x2)) equipped

with the operation

(t, x1, x2) ◦ (s, y1, y2) = (t + s, x1 + y1, x2 + y2 + s x1)

and the dilationδλ(t, x1, x2) = (λt, λx1, λ

2x2).

This is naturally isomorphic to the Heisenberg–Weyl group H1.

Again following the notation of Section 4.1.4, another example of K-type groupis given by the choice p0 = p1 = p2 = 1, B1 = B2 = (1), whence

B =⎛

⎝0 0 0

(1) 0 00 (1) 0

⎞

⎠ , N = p0 + p1 + p2 = 3, exp(s B) =⎛

⎝1 0 0s 1 0s2

2 s 1

⎞

⎠ .


Hence, our K-type group B is R4 (whose points are denoted by (t, x1, x2, x3))

equipped with the operation

(t, x1, x2, x3) ◦ (s, y1, y2, y3)

=(

t + s, y1 + x1, y2 + x2 + s x1, y3 + x3 + sx2 + s2

2x1

),

and the dilationδλ(t, x1, x2, x3) = (λt, λx1, λ

2x2, λ3x3).

Alternatively, the same non-trivial block(

(1) 00 (1)

)

in the latter matrix B can also be realized by the choice

p0 = p1 = 2, B1 =(

1 00 1

),

whence we have N = p0 + p1 = 4 and

B =(

0 0(1 00 1

)0

), where 0 =

(0 00 0

).

Hence, our K-type group B is R5 (whose points are denoted by (t, x) = (t, x1, x2,

x3, x4)) equipped with the operation

(t, x) ◦ (s, y) = (t + s, y1 + x1, y2 + x2, y3 + x3 + s x1, y4 + x4 + s x2),

and the dilation

δλ(t, x1, x2, x3, x4) = (λt, λx1, λx2, λ2x3, λ

2x4).

Remark 4.1.4. Assume that the matrix B is as in (4.2a). If we define

dλ : R1+N → R

1+N,

dλ(t, x(0), . . . , x(r)) = (λ2t, λx(0), λ3x(1), . . . , λ2r+1x(r)),

then {dλ}λ>0 is a group of automorphisms of B. For a proof of this statement, wedirectly refer to [LP94]. This remark shows that (R1+N, ◦, dλ) is a homogeneousLie group. It can be also easily proved that the ultraparabolic operator

L = Δp0 + Y (4.3)

is left-invariant (w.r.t. ◦) and homogeneous of degree two with respect to {dλ}λ>0.Operator (4.3) generalizes the prototypes of the ones introduced by Kolmogorovin [Kol34].


4.1.5 Sum of Carnot Groups

Suppose we are given two homogeneous Carnot groups G(1) = (RN, ◦(1)), G

(2) =(RM, ◦(2)) with dilations

δ(1)λ (x) = (λ x(1), . . . , λrx(r)), x ∈ G

(1),

δ(2)λ (y) = (λ y(1), . . . , λsy(s)), y ∈ G

(2),

wherex(i) ∈ R

Ni , i ≤ r , N1 + · · · + Nr = N

andy(i) ∈ R

Mi , i ≤ s, M1 + · · · + Ms = M .

Let

ΔG(1) =

N1∑

j=1

X2j and Δ

G(2) =M1∑

j=1

Y 2j

be the canonical sub-Laplacians on G(1) and G

(2), respectively. We define a homo-geneous Carnot group G on R

N+M as follows. Suppose r ≤ s. If (x, y) ∈ RN ×R

M ,we consider the following permutation of the coordinates

R(x, y) = (x(1), y(1), . . . , x(r), y(r), y(r+1), . . . , y(s)).

We then denote the points of G ≡ RN+M by z = R(x, y). We finally define the

group law ◦ and the dilation δλ on G as one can expect: for every z = R(x, y), ζ =R(ξ, η) ∈ G, we set

z ◦ ζ = R(x ◦(1) ξ, y ◦(2) η), δλz = R(δ(1)λ x, δ

(2)λ y).

It is then easily checked that (G, ◦, δλ) is a homogeneous stratified group of step s

and N1 + M1 generators. Moreover, the canonical sub-Laplacian on G is the sum ofthe sub-Laplacians on G

(1) and G(2):

ΔG = ΔG(1) + Δ

G(2) =N1∑

j=1

X2j +

M1∑

j=1

Y 2j .

For example, if G(1) is the ordinary Euclidean group on R

2 and G(2) is the Heisen-

berg–Weyl group on R3, then the “sum” of G

(1) and G(2) is the Carnot group on R

5

(whose points are denoted z = (x, y) = (x1, x2, y1, y2, y3)) with the compositionlaw

(x, y) ◦ (x′, y′) =(

x1 + x′1, x2 + x′

2,

y1 + y′1, y2 + y′

2, y3 + y′3 + 2(y2 y′

1 − y1 y′2)

),

and dilation δλ(x, y) = (λx1, λx2, λy1, λy2, λ2y3).

4.2 From a Set of Vector Fields to a Stratified Group 191

4.2 From a Set of Vector Fields to a Stratified Group

In the analysis of PDE’s, the following problem naturally arises: given a linear secondorder operator L = ∑m

j=1 X2j , where the Xj ’s are smooth vector fields on R

N , does

there exist a Lie group on RN with respect to which L is a sub-Laplacian? And, if the

answer is affirmative, is the group law explicitly expressible? The aim of this sectionis to answer these questions.

First of all, we recall some notation. For every k ∈ N, denote

W(k) = span{XJ | J ∈ {1, . . . , m}k},

where, if J = (j1, . . . , jk),

X(j1,...,jk) = [Xj1, · · · [Xjk−1 , Xjk] · · ·].

Assume the vector fields Xj ’s satisfy the following conditions:

(H0) X1, . . . , Xm are linearly independent and δλ-homogeneous of degree one withrespect to a suitable family of dilations {δλ}λ>0 of the following type

δλ : RN → R

N, δλ(x) = δλ(x(1), . . . , x(r)) := (λx(1), . . . , λrx(r)),

where r ≥ 1 is an integer, x(i) ∈ RNi for i = 1, . . . , r , N1 = m and N1 +

· · · + Nr = N ;(H1) dim(W(k)) = dim{XI (0) : X ∈ W(k)} for every k = 1, . . . , r;(H2) dim(Lie{X1, . . . , Xm}I (0)) = N .

By the results of the previous sections, conditions (H0)–(H1)–(H2) are neces-sary for

∑mj=1 X2

j to be a sub-Laplacian on a suitable homogeneous Carnot group(see Proposition 1.2.13, Remark 1.4.8 and the very definitions of Carnot group andsub-Laplacian). Moreover, the hypotheses (H0)–(H1)–(H2) are independent, as thefollowing examples show:

• The vector fields ∂x1 , ∂x2 on R3 satisfy (H0) with respect to the dilation (λx1, λx2,

λ2x3). Moreover, they satisfy (H1) but not (H2);• The vector fields X1 = ∂x1 + x2∂x4 , X2 = ∂x2 , X3 = ∂x3 + x2∂x4 + x2

2∂x5 inR

5 satisfy (H0) with respect to the dilations (λx1, λx2, λx3, λ2x4, λ

3x5). More-over, since [X1, X2] = −∂x4 , [X1, X3] = 0, [X2, X3] = ∂x4 + 2x2∂x5 and[X2, [X2, X3]] = 2∂x5 , the given vector fields satisfy (H2) but not (H1);

• the vector fields ∂x1 + x1∂x2 , ∂x2 on R2 satisfy (H1) and (H2) but do not satisfy

(H0) with respect to any dilation (λσ1x1, λσ2x2).

(See also Section 4.4.)We are going to show that conditions (H0)–(H1)–(H2) are sufficient for the solv-

ability of our problem.To begin with, we notice that the vector fields in W(k) are δλ-homogeneous of

degree k (see Proposition 1.3.10). Moreover, by Proposition 1.3.9 and hypothesis(H1)


W(i) ∩ W(j) = {0} if i �= j .

By Remark 1.3.7, we also have W(k) = {0} for every k ≥ r + 1. Then, by Proposi-tion 1.1.7,

Lie{X1, . . . , Xm} = W(1) ⊕ · · · ⊕ W(r). (4.4)

Moreover, by hypotheses (H0) and (H2),

dim(W(1)) = dim(span{X1, . . . , Xm}) = m

and∑r

k=1 dimW(k) = N . The following proposition shows a crucial link betweenthe dimension of the W(k)’s and the Nk’s in (H0).

Proposition 4.2.1. If X1, . . . , Xm satisfy hypotheses (H0)–(H1)–(H2), thendim(W(k)) = Nk for any k ∈ {1, . . . , r}.Proof. Let us set Mk := dim(W(k)) and fix a basis {Z(k)

1 , . . . , Z(k)Mk

} of W(k), 1 ≤k ≤ r . Then

{Z(k)1 I (0), . . . , Z

(k)Mk

I (0)}span W(k)I (0), so that, by (H1), it is a basis of

W(k)I (0) := {XI (0) : X ∈ W(k)}.By (4.4), the set of vector fields

Z(1)1 , . . . , Z

(1)M1

, . . . , Z(r)1 , . . . , Z

(r)Mr

(4.5)

is a basis of Lie{X1, . . . , Xm}. Then, by (H2), the column vectors of the matrix

A := (Z(1)1 I (0) · · · Z

(1)M1

I (0) · · · Z(r)1 I (0) · · · Z

(r)Mr

I (0))

span RN . We shall show that they are also linearly independent. By Proposition 1.3.5

and Remark 1.3.7, the vector fields Z(k)j can be written as

Z(k)j =

r∑

s=k

Ns∑

i=1

a(k,j)s,i (∂/∂x

(s)i ),

where a(k,j)s,i is a polynomial function δλ-homogeneous of degree s − k. In particular,

a(k,j)s,i (0) = 0 for every k < s ≤ r . As a consequence, the matrix A takes the form

⎛

⎜⎝A(1) · · · 0

.... . .

...

0 · · · A(r)

⎞

⎟⎠ , where A(k) = (a(k,j)k,i (0)

)1≤i≤Nk, 1≤j≤Mk

.

The block A(k) has dimension Nk × Mk and rank Mk since


{Z(k)1 I (0), . . . , Z

(k)Mk

I (0)}

is a basis for W(k)I (0) and dim(W(k)I (0)) = dim(W(k)) = Mk . It follows thatNk ≥ Mk for 1 ≤ k ≤ r . On the other hand, since the column vectors of A span R

N ,

r∑

k=1

Mk ≥ N =r∑

k=1

Nk.

Then Mk = Nk for any k ∈ 1, . . . , r . �Following the previous proof, we infer that the matrix

(Z

(1)1 I (x) · · · Z

(1)N1

I (x) · · · Z(r)1 I (x) · · · Z

(r)Nr

I (x)), x ∈ R

N,

takes the following form

⎛

⎜⎜⎜⎝

A(1) 0 · · · 0� A(2) · · · 0...

.... . .

...

� � · · · A(r)

⎞

⎟⎟⎟⎠ ,

where A(1), . . . , A(r) are square constant non-singular matrices. As a consequence,if the vector fields X1, . . . , Xm satisfy hypotheses (H0)–(H1)–(H2), then they alsosatisfy

(H1)∗ dim(W(k)I (x)

) = dim(W(k)

) ∀ k ≤ r, ∀ x ∈ RN.

(H2)∗ dim(Lie{X1, . . . , Xm}I (x)

) = N ∀ x ∈ RN.

Condition (H2)∗ is the well-known Hörmander’s hypoellipticity condition for thepartial differential operator

∑mj=1 X2

j .Throughout the remaining part of this section, X1, . . . , Xm will be a given set of

smooth vector fields satisfying hypotheses (H0)–(H1)–(H2). The family {δλ}λ>0 willdenote the family of dilations in (H0). We let

a = Lie{X1, . . . , Xm}. (4.6)

Finally, for every k = 1, . . . , r , Z(k)1 , . . . , Z

(k)Nk

will be a fixed basis for W(k). Weknow that

{Z1, . . . , ZN } := {Z(1)1 , . . . , Z

(1)N1

, . . . , Z(r)1 , . . . , Z

(r)Nr

}is a basis of a. In particular,

dim(a) = N. (4.7)

For every ξ = (ξ1, . . . , ξN ) = (ξ (1), . . . , ξ (r)), we set


ξ · Z =N∑

j=1

ξj Zj =r∑

k=1

Nk∑

j=1

ξ(k)j Z

(k)j .

Obviously, a = {ξ · Z : ξ ∈ RN }. Moreover, from the structure of Z

(k)j (see the

matrix (4.6)) we get

ξ · Z =N∑

k=1

(r∑

j=1

ξj a(k)j (x1, . . . , xk−1)

)∂xk

,

where a(k)j is a suitable polynomial function independent of xk, . . . , xN . Then, by

Remark 1.1.3, the map (x, t) �→ exp(t ξ · Z)(x) is well defined for every x ∈ RN

and t ∈ R. Furthermore,1

Exp : RN −→ R

N, Exp(ξ) := exp(ξ · Z)(0)

is a global diffeomorphism with polynomial component functions. Its inverse func-tion, which we shall denote by Log, has polynomial components too.

We are now ready to define a composition law on RN , suggested by Corol-

lary 1.3.29. The notation “exp” is introduced in Definition 1.1.2, page 8.

Definition 4.2.2. If X1, . . . , Xm satisfy hypotheses (H0)–(H1)–(H2), we set

x, y ∈ RN, x ◦ y := exp(Log(y) · Z)(x). (4.8)

Remark 4.2.3. We shall show that G := (RN, ◦, δλ) is a Carnot group whose Liealgebra g is a in (4.7). Since the group operation depends only on the Lie algebraitself, it will follow, in particular, that the definition of ◦ is independent of the choiceof the basis

{Z(k)j | 1 ≤ k ≤ r, 1 ≤ j ≤ Nk }

of a. The main task of the proof is to show that ◦ is associative. To this end, we shalluse the following result, which is a consequence of the Campbell–Hausdorff–Dynkinformula.

Lemma 4.2.4 (Particular case of Campbell–Hausdorff formula). With the hy-potheses (H0)–(H1)–(H2) on X1, . . . , Xm and set a = Lie{X1, . . . , Xm}, the fol-lowing result holds:

for every X, Y ∈ a, there exists a unique V ∈ a such that

exp(Y )(

exp(X)(x)) = exp(V )(x) for every x ∈ R

N . (4.9)

1 We explicitly note that we are using the notation Exp to denote a map on RN instead of on

an algebra of vector fields.


We postpone the proof of this deep result to Part III of the book (see Chapter 15).We would like to stress that V in (4.9) depends only on X and Y , in particular, it isindependent of x ∈ R

N . Note that no mention of the word “associativity” is made inthe above lemma, yet it will be the turning point in proving the associativity of ◦.

The statement of Lemma 4.2.4 can be rewritten as follows: for every ξ , η ∈ RN ,

there exists a unique ζ ∈ RN , which we shall denote by ξ ∗ η, such that

exp(η · Z)(

exp(ξ · Z)(x)) = exp

((ξ ∗ η) · Z

)(x) (4.10)

for every x ∈ RN . Then, by Definition 4.2.2, for every x, y ∈ R

N , we have

x ◦ y = exp(Log(y) · Z)(x) = exp(Log(y) · Z)(

exp(Log(x) · Z)(0))

= exp((Log(x) ∗ Log(y)) · Z

)(0) = Exp

(Log(x) ∗ Log(y)

).

Then we have the identity

x ◦ y = Exp(Log(x) ∗ Log(y)

) ∀ x, y ∈ RN,

which is equivalent to the following one

Log(x ◦ y) = Log(x) ∗ Log(y), ∀ x, y ∈ RN. (4.11)

With this identity at hand, we easily get the proof of the following theorem.

Theorem 4.2.5. Let ◦ be the composition law introduced in Definition 4.2.2. Then(RN, ◦) is a Lie group.

Proof. 1) IDENTITY ELEMENT. Since Exp(0) = 0 = Log(0), for every x ∈ RN , we

have

x ◦ 0 = exp(Log(0) · Z)(x) = x,

0 ◦ x = exp(Log(x) · Z)(0) = Exp(Logx) = x.

Then 0 is the identity of (RN, ◦).2) INVERSE ELEMENT. For any x ∈ R

N , we have

x ◦ Exp(−Log(x)) = exp(−Log(x) · Z)(x)

= exp(−Log(x) · Z)(

exp(Log(x) · Z)(0)) = 0.

An analogous argument shows that

Exp(−Log(x)) ◦ x = 0.

Then, for every x ∈ RN ,

x−1 = Exp(−Log(x)).

3) ASSOCIATIVITY. For every x, y, z ∈ RN , we have


(x ◦ y) ◦ z = exp(Log(z) · Z)(x ◦ y)

= exp(Log(z) · Z)(

exp(Log(y) · Z)(x))

(by (4.10)) = exp((Log(y) ∗ Log(z)) · Z)(x).

On the other hand, by (4.11),

x ◦ (y ◦ z) = exp(Log(y ◦ z) · Z)(x) = exp((Log(y) ∗ Log(z)) · Z)(x).

This, together with the previous identity, shows the associativity of ◦.To complete the proof of the theorem, we only have to note that the maps

(x, y) �→ x ◦ y = exp(Log(y) · Z)(x), x �→ x−1 = Exp(−Log(x))

are smooth. �In order to prove that {δλ}λ>0 is a family of automorphisms of (RN, ◦), we need

the following lemma.

Lemma 4.2.6. For every x, ξ ∈ RN , we have

δλ

(exp(ξ · Z)(x)

) = exp((δλξ) · Z)(δλ(x)) ∀ λ > 0. (4.12)

Proof. The path R � t �→ γ (t) = exp(tξ · Z)(x) is the solution to the Cauchyproblem

γ (t) =N∑

j=1

ξj Zj I (γ (t)), γ (0) = x.

We have

d

dt(δλ(γ (t))) = δλ(γ (t)) =

N∑

j=1

ξj δλ

(ZjI (γ (t))

)

(by Corollary 1.3.6, page 35) =N∑

j=1

ξj λσj (Zj I )(δλ(γ (t)))

= ((δλξ) · Z)(δλ(γ (t))).

Moreover, δλ(γ (0)) = δλ(x). This shows that μ := δλ(γ ) solves the Cauchy prob-lem

μ(t) = (δλ(ξ) · Z)I (μ(t)), μ(0) = δλ(x).

Thusδλ(γ (t)) = exp(t (δλξ) · Z)(δλ(x)).

By replacing t = 1 in this identity, we obtain (4.12). �Theorem 4.2.7. G := (RN, ◦, δλ) is a homogeneous group.


Proof. By Theorem 4.2.5, we only have to prove that δλ is an automorphism of(RN, ◦). By Lemma 4.2.6, for every ξ ∈ R

N , we have

δλ(Exp(ξ)) = δλ(exp(ξ · Z)(0)) = exp((δλξ) · Z)(0) = Exp(δλ(ξ)),

so that, since Log = Exp−1,

Log(δλ(ξ)) = δλ(Log(ξ)).

Then, for every x, y ∈ RN ,

δλ(x ◦ y) = δλ(exp(Log(y) · Z)(x))

(by Lemma 4.2.6) = exp(δλ(Log(y)) · Z)(δλ(x))

= exp(Log(δλ(y)) · Z)(δλ(x)) = (δλ(x)) ◦ (δλ(y)).

This completes the proof. �By using the associativity property of the composition law ◦, it is easy to show

that the vector fields Z1, . . . , ZN are invariant with respect to the left translations on(RN, ◦, δλ).

Theorem 4.2.8. The vector fields Z1, . . . , ZN are left-invariant on G.

Proof. If ej = (0, . . . , 1, . . . , 0) (1 being the j -th component), then ej ·Z = Zj and

x ◦ Exp(t ej · Z) = exp(tZj )(x)

for every x ∈ RN and t ∈ R. For any fixed α ∈ R

N and u ∈ C∞(RN), let us denoteuα(x) := u(α ◦ x). From (1.15) (page 10) and the associativity of ◦, we obtain

Zj (u(α ◦ x)) = Zj(uα(x)) = d

d t

∣∣∣∣t=0

{uα

(x ◦ Exp(tej · Z)

)}

= d

d t

∣∣∣∣t=0

{u((α ◦ x) ◦ Exp(tej · Z)

)} = (Zju)(α ◦ x),

for every x ∈ RN . This completes the proof. �

Corollary 4.2.9. Let g be the Lie algebra of G. Then g = a, where a = Lie{X1,

. . . , Xm}.Proof. By Theorem 4.2.8, we have a ⊆ g. On the other hand, by (4.7) and Proposi-tion 1.2.7, dim(a) = N = dim(g). Thus a = g. �

Let us now consider the vector fields Y1, . . . , YN1 ∈ g such that Yj (0) = ∂xj,

1 ≤ j ≤ N1. We know that N1 = m. Since Yj is δλ-homogeneous of degree one (seeCorollary 1.3.19), by Corollaries 4.2.9 and 1.3.11 (the latter on page 37) and identity(4.4), we have Yj ∈ span{X1, . . . , Xm}, 1 ≤ j ≤ m. Then, since Y1, . . . , Ym arelinearly independent,


span{Y1, . . . , Ym} = span{X1, . . . , Xm}.This identity obviously implies

Lie{Y1, . . . , Ym} = Lie{X1, . . . , Xm} = a = g.

Thus Y1, . . . , Ym are the generators of g. Then G = (RN, ◦, δλ) is a homogeneousCarnot group, ΔG = ∑m

j=1 Y 2j is its canonical sub-Laplacian and L = ∑m

j=1 X2j is

a sub-Laplacian on G. �We can summarize the results of this section by stating the following theorem.

Theorem 4.2.10. Let X1, . . . , Xm be smooth vector fields in RN satisfying hypothe-

ses (H0)–(H1)–(H2). Let {δλ}λ>0 be the family of dilations defined in (H0). Finally, let◦ be the composition law on R

N introduced in Definition 4.2.2. Then

G = (RN, ◦, δλ)

is a homogeneous Carnot group of step r and with m generators whose Lie algebrag is Lie-generated by X1, . . . , Xm, i.e.

g = Lie{X1, . . . , Xm}.Moreover, the second order partial differential operator L = ∑m

j=1 X2j is a sub-

Laplacian on G.

4.3 Further Examples

In this section, we exhibit some non-trivial examples of homogeneous Carnot groupsconstructed starting from a set of vector fields satisfying hypotheses (H0)–(H1)–(H2)of the previous section. We introduce the notation: given n ∈ N, Bn will denote thefollowing n × n (nilpotent of step n) matrix

Bn :=

⎛

⎜⎜⎜⎝

0 0 · · · 01 0 · · · 0...

. . .. . .

...

0 · · · 1 0

⎞

⎟⎟⎟⎠ . (4.13)

4.3.1 The Vector Fields ∂1, ∂2 + x1∂3

Let us consider in R3 the vector fields

X1 = ∂x1 , X2 = ∂x2 + x1∂x3 .

We denote by x = (x1, x2, x3) the points of R3. It is straightforwardly verified that

{X1, X2} satisfy conditions (H0)–(H1)–(H2) of the previous section with respect tothe dilations

4.3 Further Examples 199

δλ(x1, x2, x3) = (λx1, λx2, λ2x3), λ > 0.

Then, by Theorem 4.2.10, the operator

L = ∂2x1

+ (∂x2 + x1∂x3)2

is a sub-Laplacian (indeed, the canonical one) on a suitable homogeneous Carnotgroup G = (R3, ◦, δλ). We now construct the composition law ◦ by using Defini-tion 4.2.2. With the notation of the previous section, we have

W(1) = span{X1, X2}, W(2) = span{X3},where X3 = [X1, X2] = ∂x3 . For every ξ = (ξ1, ξ2, ξ3) ∈ R

3, we have

ξ · X =3∑

j=1

ξj Xj = ξ1∂x1 + ξ2∂x2 + (ξ2x1 + ξ3)∂x3 ,

so that exp(ξ · X)(x) = γ (1), where γ = (γ1, γ2, γ3) and⎧⎪⎨

⎪⎩

γ1(t) = ξ1, γ1(0) = x1,

γ2(t) = ξ2, γ2(0) = x2,

γ3(t) = ξ2γ1(t) + ξ3, γ3(0) = x3.

An easy computation shows that

exp(ξ · X)(x) =(

x1 + ξ1, x2 + ξ2, x3 + ξ3 + ξ2x1 + 1

2ξ1ξ2

).

As a consequence,

Exp(ξ) = exp(ξ · X)(0) =(

ξ1, ξ2, ξ3 + 1

2ξ1ξ2

),

Log(η) = Exp−1(η) =(

η1, η2, η3 − 1

2η1η2

).

Then, by Definition 4.2.2, the composition law determined by X1, X2, X3 is givenby

x ◦ y = exp(Log(y) · X)(x)

= exp

(y1X1 + y2X2 +

(y3 − 1

2y1y2

)X3

)(x1, x2, x3)

=(

x1 + y1, x2 + y2, x3 +(

y3 − 1

2y1y2

)+ y2x1 + 1

2y1y2

),

i.e.x ◦ y = (x1 + y1, x2 + y2, x3 + y3 + x1y2).


4.3.2 Classical and Kohn Laplacians

In this section, we re-derive the classical Laplace operator and the Kohn Laplacian(already introduced in Sections 4.1.1 and 3.1, respectively) starting from the relevantvector fields.

• The only example of homogeneous Carnot group of step 1 is the usual additivegroup (RN,+). If δλ denotes the usual dilation on R

N

δλ(x1, . . . , xN) = (λ x1, . . . , λ xN),

a family of fields satisfying hypotheses (H0)–(H1)–(H2) is necessarily of the form{Xj }j≤N , where

Xj =N∑

i=1

ai,j ∂i

and A = (ai,j )i,j is a non-singular N×N matrix so that∑N

j=1 X2j is a strictly-elliptic

constant-coefficient operator. Given ξ, x ∈ RN , it holds

exp(ξ · X)(x) = γ (1)

(set ξ · X :=

N∑

j=1

ξjXj

),

where γ (r) = A · ξ and γ (0) = x. Hence

exp(ξ · X)(x) = x + A · ξ, Exp(ξ) = A · ξ, Log(y) = A−1 · y.

We find out x ◦y = exp(Log(y) ·X)(x) = x +A ·Log(y) = x +A ·A−1 ·y = x +y,the usual additive structure of R

N . The canonical sub-Laplacian related to this groupis the ordinary Laplace operator

Δ =N∑

j=1

(∂xj)2.

• Consider now on R2N+1 (whose points are denoted by z = (x, y, t), x, y ∈

RN , t ∈ R) the 2N vector fields

Xj := ∂xj+ 2yj ∂t , Yj := ∂yj

− 2xj ∂t , j = 1, . . . , N .

If R2N+1 is equipped with the dilation δλ(z) = (λx, λy, λ2t), the former 2N vector

fields satisfy hypotheses (H0)–(H1)–(H2). We set

T := [Xj , Yj ] = −4 ∂t .

Given ζ = (ξ, η, τ ), z = (x, y, t) ∈ R2N+1, one has

exp

(N∑

j=1

(ξjXj + ηjYj ) + τT

)(z) = (μ(1), ν(1), ρ(1)),


where

(μ, ν, ρ)(r) = (ξ, η,−4τ + 2〈ν(r), ξ 〉 − 2〈μ(r), η〉), (μ, ν, ρ)(0) = (x, y, t).

This gives (set ζ · Z :=∑Nj=1(ξjXj + ηjYj ) + τT )

exp(ζ · Z)(z) = (x + ξ, y + η, t − 4τ + 2〈y, ξ 〉 − 2〈x, η〉),whence

Exp(ζ · Z) = (ξ, η,−4τ), Log(z′) = (x′, y′,−t ′/4) · Z. (4.14)

Fixed z, z′ ∈ R2N+1, we have

z ◦ z′ = exp(Log(z′) ·Z)(z) = (x + x′, y + y′, t + t ′ + 2〈y, x′〉− 2〈x, y′〉). (4.15)

We recognize the well-known Heisenberg–Weyl group (HN, ◦) on R2N+1. Its canon-

ical sub-Laplacian ΔHN = ∑Nj=1(X

2j + Y 2

j ) is the Kohn Laplacian on HN . As we

know from Chapter 1, ◦ induces a Lie group structure ∗ (of “Campbell–Hausdorff-type”) on the Lie algebra hN of H

N in the way re-described hereafter. For (ξ, η, τ ) ∈R

N × RN × R, we agree to set

(ξ, η, τ )Z := (ξ, η, τ ) · Z =N∑

j=1

(ξjXj + ηjYj ) + τT ∈ hN.

Then the group law ∗ on hN is defined by

(ξ1, η1, τ1)Z ∗ (ξ2, η2, τ2)Z = Log(Exp(ξ1, η1, τ1)Z ◦ Exp(ξ2, η2, τ2)Z

)

= Log((ξ1, η1,−4 τ1) ◦ (ξ2, η2,−4 τ2)

)

= Log(ξ1 + ξ2, η1 + η2,−4τ1 − 4τ2 + 2〈η1, ξ2〉 − 2 〈ξ1, η2〉

)

=(

ξ1 + ξ2, η1 + η2, τ1 + τ2 − 1

2〈η1, ξ2〉 + 1

2〈ξ1, η2〉

)

Z

.

If we drop the notation (·)Z and identify hN with R2N+1 via the basis (X1, . . . , XN,

Y1, . . . , YN , T ), we have somewhat “more intrinsic” group (R2N+1, ∗) canonicallyrelated to H

N , where

(ξ1, η1, τ1) ∗ (ξ2, η2, τ2)

=(

ξ1 + ξ2, η1 + η2, τ1 + τ2 − 1

2〈η1, ξ2〉 + 1

2〈ξ1, η2〉

). (4.16)

This group is canonically related to HN in the sense that we make precise below: if

we consider the change of coordinate system on HN induced by the exponential-type

coordinates in (4.14), i.e.

(x, y, t) = Exp((ξ, η, τ )Z

) = (ξ, η,−4τ),


then, with respect to this new coordinates, the vector fields Xj and Yj are respectivelyturned into2

Xj := ∂ξj− 1

2ηj ∂τ , Yj := ∂ηj

+ 1

2ξj ∂τ .

Starting from these new vector fields Xj ’s and Yj ’s, we turn to construct a relatedCarnot group (as we did above): after simple computations we obtain the very samegroup law ∗ as in (4.16).

4.3.3 Bony-type Sub-Laplacians

Let us consider in R1+N the operator

L =(

∂

∂t

)2

+(

t∂

∂x1+ t2 ∂

∂x2+ · · · + tN

∂

∂xN

)2

, (t, x1, . . . , xN) ∈ R1+N,

quoted by J.-M. Bony in [Bon69, Rémarque 3.1] as an example of a sum of squaressatisfying Hörmander condition but nevertheless with a “very degenerate” character-istic form. Clearly, L is not a sub-Laplacian on any Carnot group since the vectorfield

∑Nj=1 tj ∂/∂xj vanishes on the hyperplane t = 0. It is however sufficient to

add a new coordinate in order to lift L to a sub-Laplacian. Indeed, consider on R2+N ,

whose points are denoted by (t, s, x), t, s ∈ R, x ∈ RN , the following operator

L := T 2 + S2,

where

T := ∂t , S := ∂s + t ∂x1 + t2

2! ∂x2 + · · · + tN

N ! ∂xN.

It is readily verified that the pair T , S satisfies hypothesis (H0) with respect to thefamily of dilations defined by

δλ(t, s, x) := (λ t, λ s, λ2x1, λ3x2, . . . , λ

N+1xN).

For every k = 1, . . . , N , we then consider the vector field

Xk := [T , [T , · · · [T︸︷︷︸k times

, S] · · ·]] = ∂xk+ t ∂xk+1 + · · · + tN−k

(N − k)! ∂xN.

With the notation of Section 4.2, we have W(1) = span{T , S} and, for k = 1, . . . , N ,W(k+1) = span{Xk}. It is easy to recognize that the hypothesis (H1) is satisfied.Finally, we have

2 Indeed, for every u ∈ C∞(HN, R), u = u(x, y, t), we set v := u ◦ Exp, i.e. v =v(ξ, η, τ ) = u(ξ, η,−4τ ), so that it holds ∂ξj

v = (∂xj u) ◦ Exp, ∂ηj v = (∂yj u) ◦ Exp,∂τ v = −4 (∂t u) ◦ Exp. Consequently,

(Xju) ◦ Exp = ∂ξjv − 1

2ηj ∂τ v, (Yj u) ◦ Exp = ∂ηj v + 1

2ξj ∂τ v.


dim(Lie{T , S}I (0)

) = 2 + N,

whence also the hypothesis (H2) holds. As a consequence, L is a sub-Laplacian ona suitable homogeneous Carnot group (G, ◦) on R

2+N with step 1 + N and with2 generators. We now turn to construct the group multiplication ◦ on G by usingDefinition 4.2.2. Let α, β ∈ R and ξ ∈ R

N be fixed. We have

α T + β S +N∑

k=1

ξk Xk =(

α, β,

(β tj /j +

j∑

k=1

ξk tj−k/(j − k)!)

j=1,...,N

).

This yields

exp[α, β, ξ ](t, s, x)

:= exp

(αT + βS +

N∑

k=1

ξk Xk

)(t, s, x) = (τ, σ, γ )(1),

where (here j runs from 1 to N ){

τ (r) = α, τ(0) = t, σ (r) = β, σ (0) = s,

γj (r) = β τj (r)/j ! +∑j

k=1ξk τ j−k(r)/(j − k)!, γj (0) = xj .

From a direct integration it follows that exp[α, β, ξ ](t, s, x) is given by

(α + t, β + s, xj + β

(α + t)j+1 − tj+1

(j + 1)α+

j∑

k=1

ξk

(α + t)j−k+1 − tj−k+1

(j − k + 1)α

).

We agree to put ((α + t)j+1 − tj+1)/α = (j + 1)tj when α = 0. We now define thefollowing matrices

F(α) :=

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 · · · 0α2! 1 0 · · · 0

α2

3!α2! 1

. . ....

......

. . .. . . 0

αN−1

N !αN−2

(N−1)! · · · α2! 1

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

,

V (α) :=

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

α2!α2

3!α3

4!...

αN

(N+1)!

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

, U(α) :=

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

αα2

2!α3

3!...

αN

N !

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

and the functions


F (α, t) := α−1((α + t)F (α + t) − t F (t)),

V (α, t) := α−1((α + t)V (α + t) − t V (t)).

It then holds

exp[α, β, ξ ](t, s, x) = (α + t, β + s, x + F (α, t) · ξ + β V (α, t)

),

Exp(α, β, ξ) = (α, β, F (α) · ξ + β V (α)

),

Log(τ, σ, y) = (τ, σ, F−1(τ ) · (y − σ V (τ)

).

Let (t, s, x) and (τ, σ, y) ∈ R2+N be given. Then we have

(t, s, x) ◦ (τ, σ, y)

= (τ + t, σ + s, x + F (τ, t) · F−1(τ ) · (y − σ V (τ)) + σ V (τ, t)).

If we now prove that the following identities hold (see also (4.13))

F (τ, t) = exp(t BN) · F(τ), V (τ, t) − exp(t BN) · V (τ) = U(t), (4.17)

then the explicit form of the multiplication ◦ turns out to be

(t, s, x) ◦ (τ, σ, y) = (τ + t, σ + s, x + exp(t BN) · y + σ U(t))

=(

τ + t, σ + s, x1 + y1 + σ t, . . . , xN +N∑

k=1

yk tN−k 1

(N − k)! + σ tN1

N !

).

The first identity in (4.17) follows by proving that, for every i, j ∈ {1, . . . , N} withi ≥ j , one has

(τ + t)i−j+1 − t i−j+1

(i − j + 1)! τ =i∑

k=j

t i−k

(i − k)! · τ k−j

(k − j + 1)! ,

which readily follows by applying Newton’s binomial formula to the left-hand side.The second identity is equivalent to

t i

i! = (τ + t)i+1 − t i+1

(i + 1)! τ −i∑

k=1

τ k

(k + 1)! · t i−k

(i − k)! , i = 1, . . . , N,

which can be proved analogously.

4.3.4 Kolmogorov-type Sub-Laplacians

We now reconsider the examples in Sections 4.1.3 and 4.1.4, and we show how toobtain the composition law of the groups of Kolmogorov type by using the results ofSection 4.2. We consider in R

1+N (whose points are denoted by (t, x(0), . . . , x(r)))


the vector fields introduced in (4.2e). Then, by the results proved in Sections 4.1.3–4.1.4, they satisfy conditions (H0)–(H1)–(H2) in Section 4.2 with respect to the fol-lowing family of dilations

δλ(t, x(0), . . . , x(r)) = (λ t, λ x(0), . . . , λr+1x(r))

(we refer the reader directly to the notation in Sections 4.1.3–4.1.4). For k = 0, . . . , r

and j = 1, . . . , pk , we set Z(k)j := ∂/∂ x

(k)j . If t, τ ∈ R and x, ξ ∈ R

N are fixed,we have

exp[τ, ξ ](t, x) := exp

(τY +

r+1∑

k=1

pk∑

j=1

ξ(k)j Z

(k)j

)(t, x) = (μ(1), γ (1)),

where

μ(r) = τ, μ(0) = t; γ (r) = ξ + τ B · γ (r), γ (0) = x.

This yields

exp[τ, ξ ](t, x) =(

τ + t, exp(τB) · x +∫ 1

0exp(τ (1 − r)B) · ξ dr

),

Exp(τ, ξ) =(

τ,

∫ 1

0exp(τ (1 − r)B) · ξ dr

),

Log(s, y) =(

s,

(∫ 1

0exp(s(1 − r)B) dr

)−1

· y

).

As a consequence,

(t, x) ◦ (s, y) = (t + s, y + exp(sB)x).

We explicitly remark that this is the same group multiplication treated in Sec-tion 4.1.3.

4.3.5 Sub-Laplacians Arising in Control Theory

We here discuss an example of homogeneous Carnot group arising from control the-ory, and we refer to [Alt99] for a description of the relevance of this example in thatcontext. In R

N , we consider the following vector fields

X1 := ∂1 + x2 ∂3 + x3 ∂4 + · · · + xN−1 ∂N , X2 := ∂2.

For every k = 3, . . . , N , we have Xk := [Xk−1, X1] = ∂k , whence it is readilyverified that X1 and X2 fulfill hypotheses (H0)–(H1)–(H2) with respect to the familyof dilations

δλ(x1, x2, x3, . . . , xN) := (λ x1, λ x2, λ2x3, . . . , λ

N−1xN).


As a consequence,L = X2

1 + X22

is a sub-Laplacian on a suitable homogeneous Carnot group (G, ◦) on RN with step

N − 1 and with 2 generators. In [Alt99] it is given a representation of G by means ofmatrices of the following form

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 x2 x3 x4 · · · xN

0 1 x1x2

12! · · · xN−2

1(N−2)!

0 0 1 x1. . .

...

0 0 0 1. . .

x21

2!...

.... . .

. . .. . . x1

0 0 · · · 0 0 1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

≡ (x1, x2, . . . , xN) ∈ G,

whereas the Lie group law is given by the matrix product. We hereafter show how toobtain the composition law following the lines described in Section 4.2. Let ξ ∈ R

N

be fixed. We have

N∑

k=1

ξk Xk = (ξ1, ξ2, ξ3 + ξ1 x2, . . . , ξN + ξ1 xN−1) = ξ + ξ1 Hx,

where H is the following N × N matrix (see also (4.13))

H :=(

0 00 BN−2

).

This gives exp[ξ ](x) := exp(∑N

k=1 ξk Xk)(x) = γ (1), where γ (r) = ξ + ξ1 Hγ (r),γ (0) = x, whence

γ (r) = exp(ξ1 r H)x +∫ r

0exp(ξ1 (r − t) H)ξ dt.

In particular,

exp[ξ ](x) = exp(ξ1 H)x +∫ 1

0exp(ξ1 (1 − t) H)ξ dt,

Exp(ξ) =∫ 1

0exp(ξ1 (1 − t) H)ξ dt.

It is straightforward to recognize that, for every ρ ∈ R, we have

exp(ρ H) =(

1 00 exp(ρ BN−1)

).

Given y = (y1, y) ∈ RN , the equation y = Exp(ξ) is equivalent to the following

system (setting ξ = (ξ1, ξ ) ∈ RN )


y1 = ξ1, y =∫ 1

0exp

(ξ1(1 − t)BN−1

)ξ dt.

As a consequence,

Log(y) =(

y1,

(∫ 1

0exp(y1(1 − t)BN−1) dt

)−1

· y

).

For any fixed x, y ∈ RN , this gives

x ◦ y = exp(Log(y))(x) = y + exp(y1 H)x = (y1 + x1, y + exp(y1 BN−1)x)

=(

y1 + x1, y2 + x2, y3 + x3 + y1 x2, . . . , yN +N∑

j=2

1

(N − j)! yN−j

1 xj

).

4.3.6 Filiform Carnot Groups

In this section, we give the definition of filiform Carnot group. To this end, we recallthe definition of filiform Lie algebra (see, e.g. [OV94, page 61]).

Definition 4.3.1 (Filiform Lie algebra). Let h be a Lie algebra of finite dimensionn ≥ 3. For every k ∈ N, we recall that the terms of the lower (or descending) centralseries for h are

h1 := h, h2 := [h, h], . . . , hk := [h, hk−1] = [h[h[· · · [h, h] · · ·]]]︸︷︷︸k times

, k ∈ N.

Then, the Lie algebra h is called filiform if

codimh(hk) = k for every k such that 3 ≤ k ≤ n.

In other words, h is filiform if and only if

dim(hk) = n − k for every k: 3 ≤ k ≤ n (where n = dim(h)). (4.18)

We explicitly remark that for the lower central series we have

h1 ⊇ h2 ⊇ h3 ⊇ · · · ⊇ hk−1 ⊇ hk ∀ k ∈ N.

Remark 4.3.2. If dim(h) = 3, (4.18) says that h is filiform iff dim(h3) = 0. Hence,simple arguments show that the only filiform Lie algebras of dimension three are:

1) Lie{X1, X2, X3} with {X1, X2, X3} linearly independent and

[X2, X1] = [X3, X1] = [X3, X2] = 0.

Any Lie group with such a Lie algebra is isomorphic to the classical (R3,+).


2) Lie{X1, X2} with {X1, X2, [X2, X1]} linearly independent and

[X2, [X2, X1]] = [X1, [X2, X1]] = 0.

Any Lie group with such a Lie algebra is isomorphic to the Heisenberg–Weylgroup H

1 on R3.

Let us now suppose that n := dim(h) ≥ 4. From (4.18) we have

dim(hn−1) = 1 and dim(hn) = 0,

whence every filiform algebra is nilpotent, and, precisely, (if n ≥ 4) an n-dimensionalfiliform algebra is nilpotent of step n−1 (i.e. any commutator of length ≥ n vanishes,and there exists at least one non-vanishing commutator of length = n − 1). A veryuseful fact is that the converse is also true, as the following proposition states.

Proposition 4.3.3 (Characterization). Let h be a Lie algebra of finite dimensionn ≥ 3.

• If n ≥ 4, then h is filiform if and only if it is nilpotent of step n − 1.• If n = 3, the same is true as in the previous case, except the case of the commu-

tative R3.

Note 4.3.4. Since the linear dimension, the step of nilpotency and being isomorphicto R

3 are all invariants of isomorphic Lie algebras, by Proposition 4.3.3, we derive:if g is a Lie algebra isomorphic to a filiform one, then g is filiform too.

Proof (of Proposition 4.3.3). The “only if” part follows from Remark 4.3.2. We turnto the “if” part. It can be proved by an inductive argument.

For example, we give the proof when n = 4. Suppose that h is nilpotent of step3. We have to prove that h fulfills (4.18), i.e.

dim(h3) = 1 and dim(h4) = 0.

The second equality follows from the step-3-nilpotence of h. By contradiction, sup-pose that dim(h3) ≥ 2. Since h2 ⊇ h3 and h2 must contain a commutator of length2 which is not of length 3, then dim(h2) ≥ 3. Finally, since there must exist at leasttwo elements in h1 \h2, then the inequality dim(h2) ≥ 3 implies dim(h1) ≥ 5, whichcontradicts the assumption dim(h1) = dim(h) = 4. This completes the proof forn = 4. The proof in the general case follows these ideas and is left to the reader. �Example 4.3.5. Let h be a Lie algebra of finite dimension n satisfying the followingcommutator identities: if {X1, . . . , Xn} is a basis (in the sense of vector spaces) forh,we have

X3 = [X1, X2], X4 = [X1, X3], X5 = [X1, X4], . . . , Xn = [X1, Xn−1],whereas all other commutators of the Xi’s vanish identically. In other words, it holds

4.3 Further Examples 209⎧⎪⎨

⎪⎩

[X1, Xi] = Xi+1 for every i = 2, . . . , n − 1,

[X1, Xn] = 0,

[Xi,Xj ] = 0 for every 1 < i < j ≤ n.

Since h is evidently nilpotent of step n−1, then, by Proposition 4.3.3, h is filiform. InSections 4.3.3 and 4.3.5 we have furnished two explicit models for a similar algebra.

Vice versa, the following remarkable fact holds.

Theorem 4.3.6 (Bratzlavsky [Bra74]). Let h be a filiform Lie algebra of finite di-mension n. Then there exists a basis {Xi}i for h (in the sense of vector spaces) suchthat Xi+1 = [X1, Xi] ( for every i = 2, . . . , n − 1) and [X1, Xn] = 0.

Hence any filiform Lie algebra has a basis of the following form:

X1, X2, [X1, X2]︸︷︷︸=:X3

, [X1, [X1, X2]]︸︷︷︸=:X4

, [X1, [X1, [X1, X2]]]︸︷︷︸=:X5

, . . . , [n − 2 times︷︸︸︷

X1, · · · [X1, X2] · · ·]︸︷︷︸=:Xn

.

In general, nothing is said about the remaining commutators [Xi,Xj ] for 1 < i <

j ≤ n.

Definition 4.3.7 (Filiform Carnot group). A Carnot group is said filiform if its Liealgebra is a filiform Lie algebra.

(Note. If G and H are isomorphic Carnot groups and G is filiform, then H is filiformtoo. This follows from Note 4.3.4.)

By Proposition 4.3.3 and Remark 4.3.2, we have the following result.

Proposition 4.3.8. A Carnot group on RN (with N ≥ 3) is filiform if and only if it is

nilpotent of step N − 1 (except the trivial case of the usual Euclidean (R3,+)).

Remark 4.3.9. For example, among the Heisenberg–Weyl groups Hn the only fil-

iform one is H1; analogously, the only filiform Carnot group of step two (hence

nilpotent of step two) has necessarily dimension 3: up to isomorphism, this is H1.

Moreover, the Carnot groups considered in Sections 4.3.3 and 4.3.5 are filiform.Again from Proposition 4.3.3 it follows that a filiform Carnot group G on R

N

(non-Euclidean) is characterized by a stratification of its algebra g of the followingtype

g = V1 ⊕ · · · ⊕ VN−1 with dim(V1) = 2, dim(Vi) = 1 ∀ i = 2, . . . , N − 1.

In particular, a non-Euclidean filiform Carnot group has necessarily two gene-rators. �


4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2)

In this section, we exhibit some examples of vector fields not satisfying one of thehypotheses (H0), (H1) or (H2) on page 191: we formally try to construct a group foreach situation and we show that what we obtain is not a Carnot group!

4.4.1 Fields not Satisfying Hypothesis (H0): The Group is not Well Posed

We consider on R2 the following two polynomial vector fields

X := (1 + x2) ∂x, Y := (1 + y2) ∂y.

It is easily seen that they satisfy hypotheses (H1) and (H2) on page 191, but not(H0). Indeed, we have

dim(W(1)

) = 2 = dim(span {(1 + x2, 0), (0, 1 + y2)}) ∀ (x, y) ∈ R

2,

and all the vector spaces W(k)’s for k ≥ 2 reduce to {0}, for X and Y commute.However, X and Y are not δλ-homogeneous with respect to any dilation on R

2 (fortheir component functions contain zero degree terms), hence (H0) is not fulfilled.

This invalidates our construction of a Carnot group canonically related to X

and Y , the construction being heavily dependent on the well-behaved propertiesof the fields and, in particular, dependent on the delicate Campbell–Hausdorff-typeLemma 4.2.4.

It is nonetheless important to remark that a Campbell–Hausdorff formula holdsin a more general setting than the one we presented here. For instance, it holds3 forvector fields satisfying the Hörmander condition (i.e. our hypothesis (H2)) and hencein the present situation too.

However, we remark that the present case is complicated by the lack of homo-geneity of X and Y , which implies that the exponential series

∑k≥0 XkI/k! may fail

to converge everywhere.We formally try to construct a group in the present situation. As done in previous

sections, we fix ζ := (ξ, η) ∈ R2 and consider the vector field ζ · Z := ξ X + η Y .

We formally let, as usual, Exp(ζ · Z) := (x(1), y(1)), where (x(s), y(s)) solves{

(x(s), y(s)) = (ζ · Z)I (x(s), y(s)) = (ξ (1 + x2(s)), η (1 + y2(s))),

(x(0), y(0)) = (0, 0).

After a simple computation, we get

Exp(ζ · Z) = (tan ξ, tan η), Log(x, y) = (arctan x, arctan y) · Z.

In particular, we explicitly remark that Exp is only defined for

3 The reader is referred to Nagel–Stein–Wainger [NSW85].

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2) 211

(ξ, η) ∈(

−π

2,+π

2

)×(

−π

2,+π

2

).

We now fix (x1, y1), (x2, y2) ∈ R2, and, again, we formally set

(x1, y1) ◦ (x2, y2) := (x(1), y(1)),

where (x(s), y(s)) solves{

(x(s), y(s)) = ( arctan x2 (1 + x2(s)), arctan y2 (1 + y2(s)))

(x(0), y(0)) = (x1, y1).

After a simple computation, we derive

(x1, y1) ◦ (x2, y2) = (tan(arctan x1 + arctan x2), tan(arctan y1 + arctan y2))

=(

x1 + x2

1 − x1 x2,

y1 + y2

1 − y1 y2

).

Where defined, this operation is commutative, associative, (0, 0) is the neutral ele-ment and has the inverse (x, y)−1 = (−x,−y). However, the operation is definedonly away from the subset of R

2 × R2

{((x1, y1), (x2, y2)

) : x1x2 = 1, or y1y2 = 1}.

Moreover, there does not exist any ε > 0 such that |x| < ε, |y| < ε implies |(x +y)/(1 − xy)| < ε. This makes it impossible to define a group from ◦.

Another formal argument is more successful in this case. Indeed, we formallycompute the Campbell–Hausdorff-type operation on the “algebra”: we identify(x, y) ∈ R

2 with x X + y Y and we set

(ξ1, η1) ∗ (ξ2, η2) ≡ Log(Exp(((ξ1, η1) · Z) ◦ (Exp(ξ2, η2) · Z))

)

= Log((tan ξ1, tan η1) ◦ (tan ξ2, tan η2)

)

= Log

(tan ξ1 + tan ξ2

1 − tan ξ1 tan ξ2,

tan η1 + tan η2

1 − tan η1 tan η2

)

= Log(

tan(ξ1 + ξ2), tan(η1 + η2))

= (ξ1 + ξ2, η1 + η2) · Z ≡ (ξ1 + ξ2, η1 + η2) ∈ R2.

Thus, the operation on the “algebra” of the formal group related to ◦ is indeed agroup law (the usual Euclidean structure on R

2! reflecting the commutative nature ofthe algebra generated by the commuting vector fields {X, Y }).

Analogously, we consider the change of coordinate system induced by the Logmap, i.e. we consider new coordinates defined by

(ξ, η) := (arctan x, arctan y).


With respect to these new coordinates, the vector fields X and Y are respectivelyturned into4

X = ∂ξ , Y = ∂η.

These fields do fulfill hypotheses (H0), (H1) and (H2) of page 191, and the relevanthomogeneous Carnot group is obviously the usual additive structure on R

2, exactlyas for the operation ∗ above.

For another example of polynomial vector fields satisfying hypotheses (H1) and(H2), but not (H0), see Ex. 18 at the end of this chapter.

4.4.2 Fields not Satisfying Hypothesis (H1): The Operation is not Associative

We consider on R3 (whose points are denoted by (x, y, t)) the vector fields

X := ∂x + y2 ∂t , Y := ∂y.

It holds

[X, Y ] = −2y ∂t , [X, [X, Y ]] = 0, [Y, [X, Y ]] = −2∂t ,

and the commutators of length > 3 vanish identically. It is immediately seen thathypothesis (H2) is fulfilled, whereas (H1) is not. Indeed, W(2) = span{−2y ∂t } isone-dimensional, whereas

{ZI (0) : Z ∈ W(2)} = {(0, 0, 0)}is zero dimensional. We remark that X and Y are δλ-homogeneous of degree 1 withrespect to the (unusual) “dilation”:

δλ : R3 −→ R

3, δλ(x, y, t) := (λx, λy, λ3t).

We try formally to consider the relevant “exponential” map: we fix ζ := (ξ, η, τ ) ∈R

3 and consider the vector field ζ · Z := ξ X + η Y + τ [X, Y ]. We formally let, asusual, Exp(ζ · Z) := (x(1), y(1), t (1)), where (x(s), y(s), t (s)) solves

{(x(s), y(s), t(s)) = (ξ, η, ξ y2(s) − 2τ y(s)

)

(x(0), y(0), t (0)) = (0, 0, 0).

4 Indeed, let v = v(ξ, η) ∈ C∞(R2, R) and set u = u(x, y) := v(arctan x, arctan y). Wealso set Log(x, y) := (arctan x, arctan y), so that u = v ◦ Log. We have

∂xu(x, y) = 1

1 + x2(∂ξ v) ◦ Log, ∂yu(x, y) = 1

1 + y2(∂ηv) ◦ Log,

i.e.

∂xu(x, y) =(

1

1 + tan2(ξ)(∂ξ v)

)◦ Log, ∂yu(x, y) =

(1

1 + tan2(η)(∂ηv)

)◦ Log.

This immediately gives Xu(x, y) = (∂ξ v) ◦ Log, Yu(x, y) = (∂ηv) ◦ Log.

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2) 213

After a simple computation, we get

Exp((ξ, η, τ ) · Z) =(

ξ, η,1

3ξ η2 − τ η

),

and this map is not globally invertible since

Exp((0, 0, 0) · Z) = Exp((0, 0, 1) · Z).

The singularity of this map reflects the fact that the system of vector fieldsX, Y, [X, Y ] is pointwise non-everywhere linearly independent! On the other hand,the everywhere well-posedness of the map is a consequence of the δλ-homogeneityof the vector fields X, Y .

Since the pointwise dependence of X, Y, [X, Y ] plays a negative rôle, a nat-ural question arises. What if we considered the pointwise everywhere linearly in-dependent vector fields X, Y , [Y, [X, Y ]]? We proceed once again formally: let(ξ, η, τ ), (x, y, t) ∈ R

3 be fixed, and let us consider

exp(ξX + ηY + τ [Y, [X, Y ]])(x, y, t) = (γ1(1), γ2(1), γ3(1)),

where {(γ1(s), γ2(s), γ3(s)) = (ξ, η, ξ γ 2

2 (s) − 2 τ),

(γ1(0), γ2(0), γ3(0)) = (x, y, t).

A simple computation now gives

(γ1(1), γ2(1), γ3(1)) =(

x + ξ, y + η,ξ η2

3+ ξ y2 + ξ η y − 2 τ + t

).

As a consequence, when (x, y, t) = (0, 0, 0), it holds (by the slight abuse of notationExp(ξ, η, τ ) := Exp(ξ X + η Y + τ [Y, [X, Y ]]))

Exp(ξ, η, τ ) =(

ξ, η,ξ η2

3− 2 τ

).

This map is now everywhere invertible, and its inverse function is (by an analogousabuse of notation)

Log(x, y, t) =(

x, y,x y2

6− t

2

).

Finally, given (x1, y1, t1), (x2, y2, t2) ∈ R3, we set (formally following the ideas in

Definition 4.2.2)

(x1, y1, t1) ◦ (x2, y2, t2)

= exp(x2X + y2Y + (x2 y2

2/6 − t2/2)[Y, [X, Y ]])(x1, y1, t1)

= (x1 + x2, y1 + y2, t1 + t2 + x2 y21 + x2 y2 y1

). (4.19)

This binary operation on R3 is not associative, for


((0, 1, 0) ◦ (0, 1, 0)) ◦ (1, 0, 0) = (1, 2, 4) �= (1, 2, 3)

= (0, 1, 0) ◦ ((0, 1, 0) ◦ (1, 0, 0)).

Moreover, X and Y are not invariant under left “translations” with respect to ◦ (aswe know, the left-invariance is closely linked to the associativity; see the proof ofTheorem 4.2.8).

We explain more closely the fact that ◦ lacks to be associative by directlystudying the Campbell–Hausdorff formula in the present situation. First, we fixx = (x1, x2, x3), (α1, α2, α3) and (β1, β2, β3) in R

3. Then, starting from x, we pro-ceed along the integral path of the vector field

A := α1X + α2Y + α3[Y, [X, Y ]],so that, at unit time, we arrive (by definition) in exp(A)(x). Then, starting fromexp(A)(x) and following the integral curve of the field

B := β1X + β2Y + β3[Y, [X, Y ]],we arrive (at unit time) in exp(B)(exp(A)(x)). Now, the Campbell–Hausdorff for-mula says that5 we get (at unit time) to this same final point even if we start from x

and proceed along the integral path of the vector field

C = A � B = A + B + 1

2[A,B] + 1

12[A, [A,B]] − 1

12[B, [A,B]]. (4.20)

A direct computation shows that it holds

C = (α1 + β1)X + (α2 + β2)Y + 1

2(α1β2 − α2β1)[X, Y ]

+(

α3 + β3 + 1

12(α2 − β2)(α1β2 − α2β1)

)[Y, [X, Y ]].

As we explained after the statement of Lemma 4.2.4 (page 194), we could cer-tainly construct the desired associative operation on R

3, if C could be expressedas a constant-coefficient linear combination of X, Y, [Y, [X, Y ]]. But here we have

C = (α1 + β1)X + (α2 + β2)Y

+(

α3 + β3 + 1

2(α1β2 − α2β1)x2 + 1

12(α2 − β2)(α1β2 − α2β1)

)[Y, [X, Y ]],

which shows that there do not exist three constants γ1, γ2, γ3 such that

C = γ1X + γ2Y + γ3[Y, [X, Y ]].This can be seen as the very motivation for the lack of associativity of ◦ as definedin (4.19).

5 We explicitly remark that the Campbell–Hausdorff formula holds in the present case too,see Chapter 15.


Remark 4.4.1. Nonetheless, we will be able to “lift” the above vector fields X, Y tosuitable vector fields (on a larger vector space) satisfying hypotheses (H0)–(H1)–(H2), only exploiting the homogeneity property of X and Y and the fact that theyfulfil Hörmander’s condition: this will be properly explained in Chapter 17 (see, inparticular, Section 17.4, page 666), thus showing that the hypotheses of the presentchapter can be somewhat weakened to produce, in case, “lifted” groups.

4.4.3 Fields not Satisfying Hypothesis (H2): The Group is Undefined

We consider on R3 the following two vector fields

X := ∂x1 , X2 := ∂x2 .

It is easily seen that they satisfy hypothesis (H0) with dilation

δλ(x1, x2, x3) = (λx1, λx2, λ2x3)

and hypothesis (H1), but they do not satisfy hypothesis (H2) on page 191.Obviously, the exponential map

(ξ1, ξ2, ξ3) �→ exp(ξ1 X1 + ξ2 X2) = (ξ1, ξ2, 0)

does not even define a bijection from R3 onto R

3. Formally, the related “composi-tion” would be

(x1, x2, x3) ◦ (y1, y2, y3) = (x1 + y1, x2 + y2, 0)

which possess no inverse map!

Bibliographical Notes. For some topics on filiform Lie algebras, we followedA.L. Onishchik and E.B. Vinberg [OV94, page 61].

Some of the topics presented in this chapter also appear in [Bon04].


Ex. 1) Consider on R4 the vector fields

X1 := ∂1 − 1

2x2 ∂3 − 1

2x3 ∂4 − 1

12x1x2 ∂4,

X2 := ∂2 + 1

2x1 ∂3 + 1

12x2

1 ∂4.


Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with thedilation δλ(x1, x2, x3, x4) := (λx1, λx2, λ

2 x3, λ3 x4). Set

Z(1)1 := X1, Z

(1)2 := X2, Z

(2)1 := [X1, X2], Z

(3)1 := [X1, [X1, X2]].

Following the notation of Section 4.2, verify that exp(ξ · Z)(x) equals⎛

⎜⎜⎝

x1 + ξ1x2 + ξ2

x3 + ξ3 + 12 (ξ2x1 − ξ1x2)

x4 + ξ4 + 12 (ξ3x1 − ξ1x3) + 1

12 (x1 − ξ1)(ξ2x1 − ξ1x2)

⎞

⎟⎟⎠ ,

whence Exp(ξ) = ξ and Log(x) = x (i.e. precisely, Exp(ξ · Z) = ξ andLog(x) = x · Z). Finally, prove that the relevant composition law is

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + 12 (y2x1 − y1x2)

x4 + y4 + 12 (y3x1 − y1x3) + 1

12 (x1 − y1)(y2x1 − y1x2)

⎞

⎟⎟⎠ .

Ex. 2) Write down the canonical sub-Laplacian ΔG of the group obtained in Ex. 1and observe that ΔG contains the first order differential term 1

6 x2 ∂4.Ex. 3) Consider on R

4 the vector fields

X1 := ∂1 + x2 ∂3 + x22 ∂4, X2 := ∂2.

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with thesame dilation as in Ex. 1. Set

Z(1)1 := X1, Z

(1)2 := X2, Z

(2)1 := [X1, X2], Z

(3)1 := [X2, [X1, X2]].

Following the notation of Section 4.2, verify that

exp(ξ · Z)(x) =

⎛

⎜⎜⎝

x1 + ξ1x2 + ξ2

x3 − ξ3 + 12 ξ1ξ2 + ξ1x2

x4 − 2ξ4 + 13 ξ1ξ

22 − ξ2ξ3 + ξ1x

22 + ξ1ξ2x2 − 2ξ3x2

⎞

⎟⎟⎠ .

Deduce that

Exp(ξ · Z) =

⎛

⎜⎜⎝

ξ1ξ2

−ξ3 + 12 ξ1ξ2

−2ξ4 + 13 ξ1ξ

22 − ξ2ξ3

⎞

⎟⎟⎠

and

Log(x) =

⎛

⎜⎜⎝

x1x2

−x3 + 12 x1x2

− 12 x4 − 1

12 x1x22 + 1

2 x2x3

⎞

⎟⎟⎠ · Z.


Finally, prove that the relevant composition law is

x ◦ y =

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + y1x2

x4 + y4 + y1x22 + 2 x2y3

⎞

⎟⎟⎠ .

Ex. 4) Show that the inverse map for the Lie group in Ex. 3 is given by

x−1 =

⎛

⎜⎜⎝

−x1−x2

−x3 + x1x2

−x4 + 2 x2x3 − x1x22

⎞

⎟⎟⎠ .

Ex. 5) a) Consider the Lie group and the notation in Ex. 3. For every ξ, η ∈ RN ,

prove that Log(Exp(ξ · Z) ◦ Exp(η · Z)) equals

⎛

⎜⎜⎝

ξ1 + η1ξ2 + η2

ξ3 + η3 + 12 (ξ1η2 − ξ2η1)

ξ4 + η4 + 12 (ξ2η3 − ξ3η2) + 1

12 (ξ2 − η2) (ξ1η2 − ξ2η1)

⎞

⎟⎟⎠ .

Denote the above composition law in R4 by ξ ∗ η. Deduce that F =

(R4, ∗) is a Carnot group isomorphic to G = (R4, ◦). Find the Lie-group isomorphism turning F into G. Considering the natural identi-fication R

4 � ξ ←→ ξ · Z ∈ g (g being the Lie-algebra of G) de-fine a composition law on g dual to ∗. Compare this operation to theCampbell–Hausdorff composition law � in (2.43), motivating your re-marks.

b) Consider on R4 the change of coordinates modeled on the Log map, i.e.

the new coordinates ξ on R4 defined by

ξ = L(x) :=

⎛

⎜⎜⎝

x1x2

−x3 + 12 x1x2

− 12 x4 − 1

12 x1x22 + 1

2 x2x3

⎞

⎟⎟⎠ .

Prove that X1 and X2 are respectively turned6 by L into the followingvector fields:

X1 := ∂1 − 1

2ξ2∂3 − 1

12ξ2

2 ∂4,

X2 := ∂2 + 1

2ξ1∂3 +

(1

12ξ1ξ2 − 1

2ξ3

)∂4.

6 I.e. if v = v(ξ) and u(x) = v(L(x)), then Xiu(x) = (Xiv)(L(x)).


c) Carry out an exercise similar to Ex. 3 for the above vector fields X1 andX2 on R

4. Verify that the relevant composition law is the same as ∗ inEx. 5-(a). Remark that the linear change of coordinates in R

4 given by

(ξ1, ξ2, ξ3, ξ4) �→ (x1, x2, x3, x4) := (ξ2, ξ1,−ξ3,−ξ4)

turns X1 and X2 into X2 and X1, respectively, of Ex. 1. (Why?)Ex. 6) Let α ∈ R be fixed. Consider on R

4 the vector fields

Z1 = ∂1 − 1

2x2 ∂3 −

(1

2x3 + 1

12x2(x1 + α x2)

)∂4,

Z2 = ∂2 + 1

2x1 ∂3 +

(−1

2α x3 + 1

12x1(x1 + α x2)

)∂4.

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with a suit-able dilation. Verify that the relevant composition law is the same as ◦ inEx. 1-(a) of Chapter 1.

Ex. 7) For (x1, y1), (x2, y2) ∈ R2 \ {(x, y) ∈ R

2 : xy = 1}, set

(x1, y1) ◦ (x2, y2) :=(

x1 + x2

1 − x1 x2,

y1 + y2

1 − y1 y2

).

Prove that ◦ is commutative, associative, has (0, 0) as neutral element andhas the inverse (x, y)−1 = (−x,−y). However, prove that there does notexist any ε > 0 such that |x| < ε, |y| < ε implies |(x + y)/(1 − xy)| < ε.This makes it impossible to define a group from ◦.


X1 := ∂1 − 1

2x2 ∂3 + 1

6x2

2 ∂4,

X2 := ∂2 + 1

2x1 ∂3 +

(x3 − 1

6x1 x2

)∂4.

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the di-lation δλ(x1, x2, x3, x4) := (λx1, λx2, λ

2 x3, λ3 x4). Verify that the relevant

composition law is

x ◦ y =

⎛

⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + 12 (x1y2 − x2y1)

x4 + y4 + (x3y2 − x2y3) + 16 (y2 − x2) (x1y2 − x2y1)

⎞

⎟⎟⎠ .

Verify that the related Jacobian basis is

X1, X2, [X1, X2], −1

2[X2, [X1, X2]].


Ex. 9) Write the following second order constant-coefficient strictly elliptic opera-tor L on R

2 as a sum of squares of vector fields

L = 10 (∂x1)2 + 10 ∂x1,x2 + 5 (∂x2)

2.

Is L a sub-Laplacian on a suitable homogeneous Carnot group? Which one?Ex. 10) Following the notation of Section 4.1.4, prove that an example of K-type

group is given by the choice p0 = p1 = p2 = · · · = pr = 1, B1 = B2 =· · · = Br = (1), whence

B =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 · · · 0 01 0 · · · 0 0

0 1. . .

......

......

. . . 0 00 0 · · · 1 0

⎞

⎟⎟⎟⎟⎟⎟⎠, N = p0 + p1 + · · · + pr = 1 + r,

so that the relevant K-type group B is R2+r (whose points are denoted by

(t, x1, x2, x3, . . . , xr+1)) equipped with the operation

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

t

x1x2x3...

xr+1

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

◦

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

s

y1y2y3...

yr+1

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

s + t

y1 + x1y2 + x2 + s x1

y3 + x3 + s x2 + s2

2 x1...

yr+1 + xr+1 + s xr + s2

2 xr−1 + · · · + sr

r! x1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

and the dilation

δλ(t, x1, x2, x3, . . . , xr+1) = (λt, λx1, λ2x2, λ

3x3, . . . , λr+1xr+1).

Ex. 11) Following the notation of Section 4.1.4, find an explicit general expressionfor the composition law on the K-type group such that r = 1 and B1 = Ip0 .


Z1 = ∂x1 + x1 ∂x4 , Z1 = ∂x2 , Z1 = ∂x3 + x2 ∂x4 .

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with thedilation δλ(x1, x2, x3, x4) := (λx1, λx2, λx3, λ

2x4). Moreover, [Z1, Z2] =0 = [Z1, Z3], [Z2, Z3] = ∂x4 . Set Z4 := [Z2, Z3]. Following the notationof Section 4.2, verify that

exp(ξ · Z)(x) =

⎛

⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3

x4 + ξ4 + x1ξ1 + 12 ξ2

1 + ξ3x2 + 12 ξ3ξ2

⎞

⎟⎟⎠ ,


whence

Exp(ξ · Z) =

⎛

⎜⎜⎝

ξ1ξ2ξ3

ξ4 + 12 ξ2

1 + 12 ξ3ξ2

⎞

⎟⎟⎠ ,

Log(y) =

⎛

⎜⎜⎝

y1y2y3

y4 − 12 y2

1 − 12 y3y2

⎞

⎟⎟⎠ · Z.

We explicitly remark that

Exp(Z1) = (1, 0, 0, 1/2) �= (1, 0, 0, 0),

which shows that the Jacobian basis of g do not necessarily correspondto the canonical basis of G ≡ R

N via the exponential map. (But see alsoProposition 2.2.22, page 139.)Finally, prove that the relevant composition law is

⎛

⎜⎜⎝

x1 + y1x2 + y2x3 + y3

x4 + y4 + y3x2 + y1x1

⎞

⎟⎟⎠ .

Instead, consider the basis {X1, . . . , X4} of g corresponding to the canonicalbasis {e1, . . . , e4} of R

4 via the exponential map, i.e. Xi := Log(ei) · Z.Verify that

X1 = Z1 − 1

2Z2, X2 = Z2, X3 = Z3, X4 = Z4.

Now, equip R4 with the following composition law: given ξ, η ∈ R

4, wedefine ξ ∗ η ∈ R

4 as the only vector such that

Log(Exp(ξ · X) ◦ Exp(η · X)) = (ξ ∗ η) · X.

Obviously, (R4, ∗) is a Lie group isomorphic to (R4, ◦). However, checkout that

ξ ∗ η =

⎛

⎜⎜⎝

ξ1 + η1ξ2 + η2ξ3 + η3

ξ4 + η4 − 12 ξ1 − 1

2 η1 − 12 ξ3η2 + 1

2 ξ2η3

⎞

⎟⎟⎠ ,

so that (R4, ∗) is not a homogeneous Carnot group, for the only “dilations”on R

4 which are homomorphisms of (R4, ∗) have the form

δλ(x1, x2, x3, x4) = (λα1x1, λα2x2, λ

α3x3, λα4x4)

with α4 = α1 = α2 + α3.


Ex. 13) According to the definition given in Section 4.1.5, page 190, write the com-position law of the sum of the Heisenberg–Weyl groups H

1 and H2. Do

the same for H1 and H

1. The groups obtained are prototype H-type groupsaccording to the definition given in Section 3.6, page 169? Why?

Ex. 14) Consider the fields on R2

X1 := ∂1, X2 := x1 ∂2.

Verify that [X1, X2] = ∂2, whereas all commutators of length > 2 vanish.It holds

W(1)I (x) = span{(1, 0), (0, x1)}, W(2)I (x) = span{(0, 1)}.Whence dim(W(1)I (x)) = 2 only when x1 �= 0, whereas

dim(W(2)I (x)

) = 1

for every x ∈ R2, whence hypothesis (H1) is not satisfied, but (H2) is.

Moreover, X1 and X2 are homogeneous of degree 1 w.r.t. the “dilation”δλ(x1, x2) := (λx1, λ

2x2). Try to construct formally the Exp, Log and ◦maps as in Section 4.4.2, page 212, and verify that

Exp(ξ1, ξ2) =(

ξ1,1

2ξ1ξ2

),

Log(x1, x2) =(

x1,2 x2

x1

), x ◦ y =

(x1 + y1, x2 + y2 + 2 x1y2

y1

).

Observe that these maps are not everywhere defined, ◦ is not associative andX1, X2 are not invariant w.r.t. ◦.

Ex. 15) Let us consider on R2 the following polynomial vector fields

X1 := ∂x1 , X2 := ∂x2 + x1 ∂x1 .

It is easily seen that they satisfy hypotheses (H1) and (H2) on page 191, butnot7 (H0).We explicitly remark that W(k) = span{X1} for every k ∈ N, k ≥ 2, whenceLie{X1, X2} is not a nilpotent algebra. Prove that the same holds for thegroup formally related to this case, according to the construction of Sec-tion 4.2. Following our usual notation, verify that the following facts hold.8

γ (t) =(

x1 etξ2 + ξ1etξ2 − 1

ξ2, x2 + tξ2

)

7 Precisely, the only “dilation” map for which X1 and X2 are homogeneous is δλ(x1, x2) =(λαx1, x2) (for every α; X1 is δλ-homogeneous of degree α and X2 of degree 0).

8 We agree to denote by (eξ − 1)/ξ the (analytic) function equal to 1 when ξ = 0 and(eξ − 1)/ξ when ξ �= 0.


solves

{γ = (ξ1X1 + ξ2X2)I (γ ),

γ (0) = x,

exp(ξ1X1 + ξ2X2)(x1, x2) =(

ξ1eξ2 − 1

ξ2, ξ2

),

Log(y1, y2) =(

y1y2

ey2 − 1

)X1 + y2X2,

x ◦ y = (x1 ey2 + y1, x2 + y2).

Moreover, verify that (R2, ◦) is a Lie group (not nilpotent) and that X1 andX2 Lie-generate the algebra of this group.

Ex. 16) Give a complete proof of Proposition 4.3.3.Ex. 17) Let n ∈ N be fixed. Let us consider the Lie algebra kn with a basis

{X, Y1, Y2, . . . , Yn} with commutator relations

[Yi, Yj ] = 0, 1 ≤ i, j ≤ n,

[X, Yj ] = Yj+1, 1 ≤ j ≤ n − 1,

[X, Yn] = 0.

Then kn is an (n + 1)-dimensional Lie-algebra nilpotent of step n; also, kn

is stratified with stratification

kn = span{X, Y1} ⊕ span{Y2} ⊕ span{Y2} ⊕ · · · ⊕ span{Yn}.Prove that a model for kn is given by the algebra of vector fields on R

1+n

(the points are denoted by (x, y) with x ∈ R, y = (y1, . . . , yn) ∈ Rn)

spanned by

X = ∂x, Yj =n∑

k=j

xk−j

(k − j)!∂yk, j = 1, . . . , n.

Show that X, Y1 fulfill hypotheses (H0)–(H1)–(H2) of page 191 with thedilation

δλ(x, y1, y2, . . . , yn) = (λx, λy1, λ2y2, . . . , λ

nyn).

Prove that the relevant composition law is

(x, y) ◦ (ξ, η)

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x + ξ

y1 + η1y2 + η2 + η1x

y3 + η3 + η2x + η1x2

2!y4 + η4 + η3x + η2

x2

2! + η1x3

3!...

yn + ηn + ηn−1x + ηn−2x2

2! + · · · + η2xn−2

(n−2)! + η1xn−1

(n−1)!

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.


Compare to the Bony-type sub-Laplacians in Section 4.3.3, page 202. An-other realization of kn as a matrix algebra can be described as follows. De-note by Kn the set of the (n + 1) × (n + 1) matrices of the form

M(x, y) :=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 x 0 · · · 0 yn

0 0 x. . . 0 yn−1

... 0. . .

......

. . . x 0 y30 x y2

0 y10 · · · 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Consider the map

ϕ : kn → Kn, ϕ

(xX +

n∑

j=1

yjYj

)= M(x, y).

Then ϕ is a Lie-algebra isomorphism. Now, consider the set

Kn := {exp(M) : M ∈ Kn},where exp denotes the exponential of a square matrix. It is remarkable toobserve that (Kn, ·) (where · denotes the usual product of matrices) is a Liegroup with the Lie algebra isomorphic to kn. In particular, Kn is nilpotentof step n and a Carnot group, for kn is stratified. Unfortunately, Kn is not ahomogeneous group (but, obviously, it is isomorphic to (kn,�), � being theCampbell–Hausdorff multiplication, which is a homogeneous Carnot groupif we identify kn to R

n+1 in the obvious way).We sketch the verification that Kn is closed under the operation ·, for thisinvolves the Campbell–Hausdorff formula. We aim to prove that if A,A′ ∈Kn, then A·A′ ∈ Kn. To this end, let M,M ′ ∈ Kn be such that A = exp(M),A′ = exp(M ′). Moreover, since ϕ is an isomorphism between kn and Kn,there exist X,X′ ∈ kn such that M = ϕ(X), M ′ = ϕ(X′). Then

A · A′ = exp(ϕ(X)) · exp(ϕ(X′))

= exp

(ϕ(X) + ϕ(X′) + 1

2[ϕ(X), ϕ(X′)] + · · ·

)

= exp

(ϕ(X + X′ + 1

2[X,X′] + · · ·)

)

= exp(ϕ(X � X′)

) ∈ Kn, since X � X′ ∈ kn.

In the second equality, we have used the Campbell–Hausdorff formula forthe exponential of matrices, which here involves a finite sum, since in Kn


we have strictly upper triangular matrices. In the third one, we used the factthat ϕ is a Lie-algebra morphism. Incidentally, we have also proved that

exp ◦ϕ : (kn,�) → (Kn, ◦)

is a Lie-group isomorphism.Ex. 18) With reference to the preceding exercise (we adopt the therein notation), we

study the Lie algebra k3. We have k3 = span{X, Y1, Y2, Y3} with

[X, Y1] = Y2, [X, Y2] = Y3, [X, Y3] = 0, [Yi, Yj ] = 0,

i, j ∈ {1, 2, 3}.Let us also set

K3 =

⎧⎪⎪⎨

⎪⎪⎩M(x, y) :=

⎛

⎜⎜⎝

0 x 0 y30 0 x y20 0 0 y10 0 0 0

⎞

⎟⎟⎠ : (x, y) = (x, y1, y2, y3) ∈ R4

⎫⎪⎪⎬

⎪⎪⎭.

Verify that K3 (equipped with the usual bracket of matrices) is a Lie algebraand that the map

ϕ : k3 → K3, ϕ

(xX +

3∑

j=1

yjYj

)= M(x, y)

is a Lie-algebra isomorphism. Now, consider the set

K3 := {exp(M) : M ∈ K3},where exp denotes the exponential of a square matrix. Verify that

exp(M(x, y)) =

⎛

⎜⎜⎝

1 x x2

2! y3 + 12!xy2 + 1

3! x2y1

0 1 x y2 + 12! xy1

0 0 1 y10 0 0 1

⎞

⎟⎟⎠ .

Verify that (K3, ·) is a Lie group (here, · is the usual product of matrices),by first checking that

exp(M(x, y)) · exp(M(ξ, η)) = exp(M((x, y) ◦ (ξ, η)

)),

where

(x, y) ◦ (ξ, η) =

⎛

⎜⎜⎝

x + ξ

y1 + η1

y2 + η2 + 12 (xη1 − ξy1)

y3 + η3 + 12 (xη2 − ξy2) + 1

12 (x − ξ)(xη1 − ξy1)

⎞

⎟⎟⎠ .

Deduce that (K3, ·) is isomorphic to the Lie group (k3,�), where � is theCampbell–Hausdorff multiplication.


Ex. 19) Analogously to what we did in the preceding two exercises, we studythe Lie algebra h1 (related to the Heisenberg group H

1). We have h1 =span{X, Y,Z} with

[X, Y ] = Z, [X,Z] = [Y,Z] = 0.

Let us also set

H1 =⎧⎨

⎩M(x, y, z) :=⎛

⎝0 x z

0 0 y

0 0 0

⎞

⎠ : (x, y, z) ∈ R3

⎫⎬

⎭ .

Verify that H1 (equipped with the usual bracket of matrices) is a Lie algebraand that the map

ϕ : h1 → H1, ϕ(xX + yY + zZ) = M(x, y, z)

is a Lie-algebra isomorphism. Now, consider the set

H1 := {exp(M) : M ∈ H1},where exp denotes the exponential of a square matrix. Verify that

exp(M(x, y, z)) =⎛

⎝1 x z + 1

2 xy

0 1 y

0 0 1

⎞

⎠ .

Verify that (H1, ·) is a Lie group (here, · is the usual product of matrices),by first checking that

exp(M(x, y, z)) · exp(M(ξ, η, ζ )) exp(M((x, y, z) ◦ (ξ, η, ζ )

)),

where

(x, y, z) ◦ (ξ, η, ζ ) =⎛

⎝x + ξ

y + η

z + ζ2 + 12 (xη − ξy)

⎞

⎠ .

Deduce that (H1, ·) is isomorphic to the Lie group (h1,�), where � is theCampbell–Hausdorff multiplication, which, in turn, is isomorphic to theusual Heisenberg–Weyl group H

1 on R3.

Ex. 20) We here consider the Lie algebra n4 of the strictly upper triangular matricesof dimension 4. We use the notation

n4 =

⎧⎪⎪⎨

⎪⎪⎩M(x) :=

⎛

⎜⎜⎝

0 x1 x4 x60 0 x2 x50 0 0 x30 0 0 0

⎞

⎟⎟⎠ : x = (x1, . . . , x6) ∈ R6

⎫⎪⎪⎬

⎪⎪⎭.

Clearly, n4 is a Lie algebra of dimension 6, nilpotent of step 3. Now, con-sider the set


N4 := {exp(M) : M ∈ N4},where exp denotes the exponential of a square matrix. Verify that

exp(M(x))

=

⎛

⎜⎜⎝

1 x1 x4 + 12x1x2 x6 + 1

2! (x1x5 + x3x4) + 13! x1x2x3

0 1 x2 x5 + 12 x2x3

0 0 1 x30 0 0 1

⎞

⎟⎟⎠ .

Verify that (N4, ·) is a Lie group (here, · is the usual product of matrices),by first checking that

exp(M(x)) · exp(M(y)) = exp(M(x ◦ y)),

where

x ◦ y =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x1 + y1x2 + y2x3 + y3

x4 + y4 + 12 (x1y2 − x2y1)

x5 + y5 + 12 (x2y3 − x3y2)

x6 + y6 + 12 (x1y5 − x5y1) + 1

2 (x4y3 − x3y4)

+ 112 (x3 − y3)(x2y1 − x1y2)

+ 112 (x1 − y1)(x2y3 − x3y2)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Deduce that (N4, ·) is isomorphic to the Lie group (n4,�), where � is theCampbell–Hausdorff multiplication.Verify that the above ◦ defines on R

6 a homogeneous Carnot group of step 3with 3 generators and dilations

δλ(x) = (λx1, λx2, λx3, λ2x4, λ

2x5, λ3x6).

The first three vector fields of the Jacobian basis are

X1 = ∂x1 − 1

2x2∂x4 +

(−1

2x5 + 1

12x2x3

)∂x6 ,

X2 = ∂x2 + 1

2x1∂x4 − 1

2x3∂x5 − 1

6x1x3∂x6 ,

X3 = ∂x3 + 1

2x2∂x5 +

(1

2x4 + 1

12x1x2

)∂x6 .

The commutator relations are

[X1, X2] = ∂x4 − 1

2x3∂x6 , [X2, X3] = ∂x5 + 1

2x1∂x6 ,

[X1, [X2, X3]] = ∂x6 = −[X3, [X1, X2]],whereas all other commutators are zero.

5

The Fundamental Solution for a Sub-Laplacian andApplications

In this chapter, we enter the core of the study of the sub-Laplacians L on the ho-mogeneous Carnot groups (and hence on the stratified Lie groups) of homogeneousdimension Q ≥ 3, by showing that they possess a fundamental solution Γ resem-blant to the fundamental solution cN |x|2−N of the usual Laplace operator Δ on R

N ,N ≥ 3. This property is one of the most striking analogies between L and the classi-cal Laplace operator. Indeed, we shall see that it holds

Γ = d2−Q,

where Q is the homogeneous dimension of G and d is a symmetric homogeneousnorm on G, smooth out of the origin (the relevant definitions will be given in Sec-tion 5.1). We shall also call d an L-gauge.

To do this, we first fix some results on homogeneous norms and the Carnot–Carathéodory distance. Then, in Section 5.3, we define the fundamental solution Γ ,whose existence follows from the hypoellipticity and the homogeneity propertiesof L. We then collect many of its remarkable properties.

As a first application, we provide mean value formulas for L, generalizing tothe sub-Laplacian setting the Gauss theorem for classical harmonic functions. Theseformulas will play a central rôle throughout the book and are proved by only usingintegration by parts and the coarea theorem (see Theorems 5.5.4 and 5.6.1).

From the mean value formulas, we derive Harnack-type inequalities for L andthe Brelot convergence property for monotone sequences of L-harmonic functions(see Theorem 5.7.10). Furthermore, as an application of the Harnack theorem, inSection 5.8 we derive several Liouville-type theorems for L. As another applicationof the properties of the fundamental solution of L, we prove the Sobolev–Stein em-bedding theorem in the stratified group setting (see Section 5.9).

To end with the applications of Γ , we provide three sections devoted to thefollowing topics: some remarks on the analytic-hypoelliptic sub-Laplacians, L-harmonic approximations, and finally an integral representation formula for the fun-damental solution by R. Beals, B. Gaveau and P. Greiner [BGG96].

Finally, three appendices close the chapter. The first is devoted to the weak andthe strong maximum principles for L. The second one provides an improved ver-

228 5 The Fundamental Solution for a Sub-Laplacian and Applications

sion of the pseudo-triangle inequality. In the third appendix, we prove in details theexistence of geodesics on Carnot groups. As a direct application of the maximumprinciples, we give a decomposition theorem for L-harmonic functions, resemblantto the decomposition of a holomorphic function on an annulus of C into the sum ofthe regular and singular parts from its Laurent expansion.

Convention. Throughout this chapter, we fix a stratified group H of step r and m

generators. Q denotes the homogeneous dimension of H.

Together with H,a stratification V = (V1, . . . , Vr) of the algebra of H will be fixed.

Moreover, L will be any sub-Laplacian on H related to the given stratification. Werecall that any stratification of the algebra of H brings along a homogeneous Carnotgroup on R

N isomorphic to H. Hence, together with the couple (H, V ), we fixG = (RN, ◦, δλ), a homogeneous Carnot group isomorphic to H, as described inProposition 2.2.22. We let Ψ : G → H be the Lie-group isomorphism, as in thecited proposition. We still denote by L the sub-Laplacian on G which is Ψ -related tothe sub-Laplacian L on H (see (2.68), page 147).

Obviously, the “homogeneous version” G of H depends upon the stratificationV but not on the sub-Laplacian L.

Thus, any definition and result given henceforward for homogeneous Carnotgroups has its counterpart (and is actually intended) for any couple (H, V ), where H

is an abstract stratified group and V is a stratification for G.

Notation. We introduce the notation for the homogeneous Carnot group G =(RN, ◦, δλ). Its dilations {δλ}λ>0 are denoted by

δλ(x) = δλ(x(1), . . . , x(r)) = (λx(1), . . . , λrx(r)), x(i) ∈ R

Ni , 1 ≤ i ≤ r.

We denote by m := N1 the number of generators of G and assume that the homoge-neous dimension

Q = N1 + 2 N2 + · · · + r Nr ≥ 3.

As we showed in Chapter 1, Section 1.4 (page 56), the sub-Laplacian L on G can bewritten as follows (see (1.90a))

L =m∑

j=1

X2j = div(A(x)∇T ),

where {X1, . . . , Xm} is a family of vector fields that form a linear basis of the firstlayer of g, the Lie algebra of G. The matrix A is given by

A(x) = (X1I (x) · · · XmI (x)) ·⎛

⎝(X1I (x))T

...

(XmI (x))T

⎞

⎠

5.1 Homogeneous Norms 229

and takes the following block form (see (1.91))

A =(

A1,1 A1,2A2,1 A2,2

),

where A1,1 is a strictly positive definite constant m×m matrix.The characteristic form of L

qL(x, ξ) := 〈A(x)ξ, ξ 〉 =N1∑

j=1

〈XjI (x), ξ 〉2, x, ξ ∈ RN, (5.1a)

is non-negative definite and, for every fixed x ∈ RN , the set

{ξ ∈ RN | qL(x, ξ) = 0}

is a linear space of dimension N −m. The vector-valued operator

∇L := (X1, . . . , Xm) (5.1b)

is called the L-gradient operator in G. Due to identity (5.1a), we have

|∇Lu|2 =m∑

j=1

|Xju|2 = 〈A∇T u,∇T u〉, u ∈ C1(RN, RN). (5.1c)

5.1 Homogeneous Norms

Definition 5.1.1. We call homogeneous norm on (the homogeneous Carnot group)G, every continuous1 function d : G→ [0,∞) such that:

1. d(δλ(x)) = λ d(x) for every λ > 0 and x ∈ G;2. d(x) > 0 iff x �= 0.

Moreover, we say that d is symmetric if3. d(x−1) = d(x) for every x ∈ G.

Example 5.1.2. Define

|x|G :=(

r∑

j=1

|x(j)| 2r!j

) 12r!

, x = (x(1), . . . , x(r)) ∈ G, (5.2)

where |x(j)| denotes the Euclidean norm on RNj . Then | · |G is a homogeneous norm

on G smooth out of the origin. It is symmetric if x−1 = −x for any x ∈ G. Ingeneral, if G is any Carnot group (with inverse x−1 not necessarily equal to −x) themap x �→ |Log (x)|G is a symmetric homogeneous norm on G smooth out of theorigin. This follows from the facts that Log (δλ(x)) = δλ(Log (x)) and Log (x−1) =−Log (x).

1 With respect to the Euclidean topology.


Example 5.1.3 (Control norm). Let d be the control distance related to a system ofgenerators of G (see Section 5.2). Define

d0(x) := d(x, 0), x ∈ G.

Then (see Theorem 5.2.8 in Section 5.2) we shall see that d0 is a symmetric homo-geneous norm on G.

From the next (elementary) proposition it will follow that the homogeneousnorms on G are all equivalent.

Proposition 5.1.4 (Equivalence of the homogeneous norms). Let d be a homoge-neous norm on G. Then there exists a constant c > 0 such that

c−1 |x|G ≤ d(x) ≤ c |x|G ∀ x ∈ G, (5.3)

where | · |G has been defined in (5.2).

Proof. Due to the δλ-homogeneity of d and | · |G, inequalities (5.3) hold taking c :=max{H, 1/h}, where

H := sup{d(x) : |x|G = 1}, h := inf{d(x) : |x|G = 1}.We explicitly remark that H <∞ and h > 0, since the set

{x : |x|G = 1}is a compact subset of G not containing the origin and d is a continuous functionstrictly positive in G \ {0}. ��Corollary 5.1.5. For every fixed (non-necessarily symmetric) homogeneous norm d

on G, there exists a constant c > 0 such that

c−1 d(x) ≤ d(x−1) ≤ c d(x) ∀ x ∈ G. (5.4)

Proof. The function x �→ d(x−1) is a homogeneous norm on G. Indeed, recall thatδλ is an automorphism of G, whence δλ(x

−1) = (δλ(x))−1. Then the assertion fol-lows from Proposition 5.1.4. ��

Any homogeneous norm turns out to be locally Hölder continuous with respectto the Euclidean metric in the following sense.

Proposition 5.1.6. Let d be a homogeneous norm on G. Then, for every compact setK ⊂ R

N , there exists a constant cK > 0 such that

d(y−1 ◦ x) ≤ cK |x − y|1/r ∀ x, y ∈ K, (5.5)

where r is the step of G.

5.1 Homogeneous Norms 231

(See also Proposition 5.15.1 (page 309) in Appendix C for an estimate frombelow of d(y−1 ◦ x).)

Proof. Let K ⊂ RN be a compact set. It is easy to see that there exists a constant

c = c(K) > 0 such that |x|G ≤ c |x|1/r for every x ∈ K , where | · |G has beendefined in (5.2). We now use Proposition 5.1.4, and we obtain that there exists aconstant c = c(K) > 0 such that d(x) ≤ c |x|1/r for every x ∈ K . Hence, (5.5) willfollow if we prove that there exists a constant c = c(K) > 0 such that |y−1 ◦ x| ≤c |x − y| for every x, y ∈ K . If we apply the mean value theorem to the functionF(x, y) := y−1 ◦ x, we obtain

|y−1 ◦ x| = |F(x, y)− F(x, x)|≤ max

t∈[0,1]‖JF (x, t x + (1− t)y)‖ |x − y| ≤ c |x − y|,

and the assertion is proved. ��Any homogeneous norm satisfies a kind of pseudo-triangle inequality.

Proposition 5.1.7 (Pseudo-triangle inequalities. I). Let d be a homogeneous normon G. Then there exists a constant c > 0 such that:

1) d(x ◦ y) ≤ c(d(x)+ d(y)),2) d(x ◦ y) ≥ 1

c d(x)− d(y−1),3) d(x ◦ y) ≥ 1

c d(x)− c d(y)

for every x, y ∈ G.

Proof. Due to the δλ-homogeneity of d , inequality 1) is equivalent to the followingone

d(x ◦ y) ≤ c if d(x)+ d(y) = 1.

This inequality holds true taking

c := max{d(x ◦ y) : d(x)+ d(y) = 1}.Obviously, 1 ≤ c <∞. From inequality 1) we now obtain

d(x) = d((x ◦ y) ◦ y−1) ≤ c(d(x ◦ y)+ d(y−1)),

whence 2). Now, 3) follows from 2) and Corollary 5.1.5 with a suitable change ofthe constant c. ��

Given a homogeneous norm d0 on G, the function

G×G � (x, y) �→ d(x, y) := d0(y−1 ◦ x)

is a pseudometric on G. Indeed, we have the following proposition.

Proposition 5.1.8 (Pseudo-triangle inequalities. II). With the above notation, thereexists a positive constant c > 0 such that:


1) d(x, y) ≤ c d(y, x) for every x, y ∈ G (here c can be taken = 1 iff d0 is alsosymmetric),

2) d(x, y) ≤ c(d(x, z) + d(z, y)) for every x, y, z ∈ G (the pseudo-triangle in-equality for d),

3) d(x, y) = 0 iff x = y.

Proof. It immediately follows from Corollary 5.1.5 and Proposition 5.1.7. ��

5.2 Control Distances or Carnot–Carathéodory Distances

We begin with some important definitions.

Definition 5.2.1 (X-subunit path). Let X = {X1, . . . , Xm} be any family of vectorfields in R

N . A piece-wise regular path γ : [0, T ] → RN is said to be X-subunit if

〈γ (t), ξ 〉2 ≤m∑

j=1

〈XjI (γ (t)), ξ 〉2 ∀ ξ ∈ RN,

almost everywhere in [0, T ]. We shall denote by S(X) the set of all X-subunit paths,and we put

l(γ ) = T

if [0, T ] is the domain of γ ∈ S(X).

We explicitly remark that every integral curve of ±Xj (j ∈ {1, . . . , m}) isX-subunit.

Convention. We assume RN is X-connected in the following sense (a proof of this

fact in the case of stratified vector fields will be given in Theorem 19.1.3 on page716):

For every x, y ∈ RN , there exists

γ ∈ S(X), γ : [0, T ] → RN such that γ (0) = x and γ (T ) = y.

Then the following definition makes sense.

Definition 5.2.2 (X-Carnot–Carathéodory distance). Suppose RN is X-connected.

Then, for every x, y ∈ RN , we set

dX(x, y) := inf{l(γ ) : γ ∈ S(X), γ (0) = x, γ (T ) = y

}. (5.6)

Under suitable hypotheses, the above inf is actually a minimum. For example,in Appendix C, we show that this occurs if {X1, . . . , Xm} are generators of the firstlayer of the stratified algebra of a homogeneous Carnot group (see also [HK00]).

Proposition 5.2.3 (dX is a metric). If RN is X-connected, then the function (x, y) �→

dX(x, y) is a metric on RN , called the X-control distance or the Carnot–Carathéodo-

ry distance related to X.

5.2 Control Distances or Carnot–Carathéodory Distances 233

In what follows, when there is no risk of confusion, we shall simply write d

instead of dX.

Proof. It is quite easy to see that d is non-negative, symmetric and satisfies the tri-angle inequality. To prove positivity, i.e.

(d(x, y) = 0

) �⇒ (x = y), (5.7)

we compare d with the Euclidean metric (which has an interest in its own). Givenx ∈ R

N and r > 0, define

M(x, r) := sup

{n∑

j=1

|XjI (z)| : |z− x| ≤ r

}, (5.8a)

where | · | denotes the Euclidean norm. We next show the following inequality

M(x, |x − y|) d(x, y) ≥ |x − y| ∀ x, y ∈ RN. (5.8b)

This will obviously imply (5.7). By contradiction, assume (5.8b) is false for some x

and y in RN . Then there exists γ ∈ S(X), γ : [0, T ] → R

N , such that γ (0) = x,γ (T ) = y and M(x, |x − y|) T < |x − y|. As a consequence, if we put

t∗ := sup{t ∈ [0, T ] : |γ (s)− x| < |x − y| for 0 ≤ s ≤ t

},

we have

|γ (t∗)− x| =∣∣∣∣∫ t∗

0γ (s) ds

∣∣∣∣ ≤m∑

j=1

∫ t∗

0|XjI (γ (s))| ds

≤ M(x, |x − y|) t∗ ≤ M(x, |x − y|) T < |x − y|.It follows that t∗ = T and |y−x| = |γ (T )−x| < |x−y|. This contradiction proves(5.8b). ��

If the vector fields X1, . . . , Xm are left invariant w.r.t. the translations on a Liegroup G = (RN, ◦), then the X = {X1, . . . , Xm}-control distance has the sameproperty. Indeed, we have the following proposition.

Proposition 5.2.4 (Control distance of a left-invariant family). Let G = (RN, ◦)be a Lie group on R

N , and let d be the control distance related to a family of leftinvariant vector fields X = {X1, . . . , Xm} on G. Then

d(x, y) = d(y−1 ◦ x, 0) ∀ x, y ∈ G, (5.9a)

andd(x−1, 0) = d(x, 0) ∀ x ∈ G. (5.9b)

The proof of this proposition will easily follow from the next lemma.


Lemma 5.2.5. In the hypotheses of Proposition 5.2.4, let γ : [0, T ] → RN be a

X-subunit curve. Then α ◦ γ is X-subunit for every α ∈ G.

Proof. If we denote by Γ the path α ◦ γ , we have

Γ (s) = Jτα (γ (s)) · γ (s) almost everywhere in [0, T ].Then, for every ξ ∈ R

N ,

〈Γ (s), ξ 〉2 = ⟨γ (s),(Jτα (γ (s))

)Tξ⟩2 (since γ is X-subunit)

≤m∑

j=1

⟨XjI (γ (s)),

(Jτα (γ (s))

)Tξ⟩2

=m∑

j=1

⟨Jτα (γ (s)) ·XjI (γ (s)), ξ

⟩2(Proposition 1.2.3, page 14)

=m∑

j=1

⟨XjI (α ◦ γ (s)), ξ

⟩2 =m∑

j=1

⟨XjI (Γ (s)), ξ

⟩2.

This proves that Γ is X-subunit. ��Proof (of Proposition 5.2.4). Let x, y ∈ G, and let γ be a X-subunit path connectingx and y. For every α ∈ G, by the previous lemma, α ◦γ is a X-subunit path connect-ing α ◦ x and α ◦ y. Then d(α ◦ x, α ◦ y) ≤ d(x, y). Since x, y, α are arbitrary, thisinequality obviously also implies d(x, y) ≤ d(α ◦ x, α ◦ y). Thus, we have proved

d(α ◦ x, α ◦ y) = d(x, y) ∀ x, y, α ∈ G. (5.10)

Choosing in (5.10) α = y−1, we obtain (5.9a).Putting x = 0 in (5.9a), we obtain d(y−1, 0) = d(0, y) = d(y, 0), which

is (5.9b). This completes the proof. ��The control distance related to homogeneous vector fields is homogeneous. More

precisely, the following assertion holds.

Proposition 5.2.6 (Control distance of a homogeneous family). Let d be the con-trol distance related to a family X = {X1, . . . , Xm} of smooth vector fields in R

N .Assume the Xj ’s are �λ-homogeneous of degree one with respect to the “dilations”

�λ : RN → RN, �λ(x1, . . . , xN) = (λσ1x1, . . . , λ

σN xN),

where σ1, . . . , σN are positive real numbers. Then

d(�λ(x), �λ(y)) = λ d(x, y) ∀ x, y ∈ RN ∀ λ > 0. (5.11)

For the proof of this proposition we need the following lemma.

5.2 Control Distances or Carnot–Carathéodory Distances 235

Lemma 5.2.7. In the hypotheses of Proposition 5.2.6, let γ : [0, T ] → RN be a

X-subunit curve. Then, for every λ > 0, the curve

Γ : [0, λ T ] → RN, Γ (s) := �λ(γ (s/λ))

is a X-subunit path.

Proof. For every ξ ∈ RN , we have (note that �λ is a linear and symmetric map for it

is represented by a diagonal matrix)

〈Γ (s), ξ 〉2 = λ−2⟨�λ(γ (s/λ)), ξ⟩2 = λ−2⟨γ (s/λ), �λ(ξ)

⟩2

(since γ is X-subunit) ≤ λ−2m∑

j=1

⟨XjI (γ (s/λ)), �λ(ξ)

⟩2

=m∑

j=1

⟨λ−1�λ(Xj I (γ (s/λ))), ξ

⟩2

(Corollary 1.3.6, page 35) =m∑

j=1

⟨XjI (�λ(γ (s/λ))), ξ

⟩2.

Since �λ(γ (s/λ)) = Γ (s), this proves the lemma. ��Proof (of Proposition 5.2.6). Let x, y ∈ R

N , and let γ : [0, T ] → RN be a X-

subunit curve connecting x and y. By the previous lemma, Γ (s) = �λ(γ (s/λ))

(0 ≤ s ≤ λ t) is a X-subunit curve, so that, since Γ connects �λ(x) and �λ(y),

d(�λ(x), �λ(y)) ≤ l(Γ ) = λ T = λ l(γ ).

Then, being γ an arbitrary X-subunit curve connecting x and y,

d(�λ(x), �λ(y)) ≤ λ d(x, y) ∀ x, y ∈ RN ∀ λ > 0. (5.12)

This inequality obviously implies (replace x and y with �1/λ(x) and �1/λ(y), respec-tively, and then λ with 1/λ; then remove “∼”)

d(x, y) ≤ 1

λd(�λ(x), �λ(y)) ∀ x, y ∈ R

N ∀ λ > 0.

Then (5.12) holds with the equality sign and the proposition is proved. ��Theorem 5.2.8 (Control distance of a homogeneous Carnot group). Let G be ahomogeneous Carnot group on R

N , and let d be the control distance related to anyfamily of generators for G. Then

G � x �→ d0(x) := d(x, 0) (5.13)

is a symmetric homogeneous norm on G.


Proof. By means of Propositions 5.2.3, 5.2.4 and 5.2.6, we are only left to prove thatd0 is continuous. For the proof of this fact, we refer to Theorem 19.1.3 on page 716.��Remark 5.2.9. The homogeneous norm (5.13), in general, is not smooth.

Corollary 5.2.10. Let G be a Carnot group. Denote by d the control distance relatedto a family of generators for G. Then, for every compact subset K of G, there existsa positive constant C(K) such that

d(x, y) ≤ C(K) |x − y|1/r ∀ x, y ∈ K,

where r denotes the step of G.

(See also Proposition 5.15.1 (page 309) in Appendix C for an estimate frombelow of d(x, y).)

Proof. It follows from Propositions 5.1.6, 5.2.4 and Theorem 5.2.8. ��

5.3 The Fundamental Solution

Throughout the sequel, we shall make use of some maximum principles for sub-Laplacians, which (for the reader’s convenience) we postpone to Appendix A of thepresent chapter (see Section 5.13).

For our purposes, it is convenient to give the definition of fundamental solutionof a sub-Laplacian L on a homogeneous Carnot group as follows.

Definition 5.3.1 (Fundamental solution). Let G be a homogeneous Carnot groupon R

N . Let L be a sub-Laplacian on G. A function Γ : RN \ {0} → R is a funda-

mental solution for L if:

(i) Γ ∈ C∞(RN \ {0});(ii) Γ ∈ L1

loc(RN) and Γ (x) −→ 0 when x tends to infinity;

(iii) LΓ = −Dirac0, being Dirac0 the Dirac measure supported at {0}. More explic-itly (recall that L∗ = L, being L∗ the formal adjoint of L),

∫

RN

Γ Lϕ dx = −ϕ(0) ∀ϕ ∈ C∞0 (RN). (5.14)

Theorem 5.3.2 (Existence of the fundamental solution). Let L be a sub-Laplacianon a homogeneous Carnot group G (whose homogeneous dimension Q is > 2). Thenthere exists a fundamental solution Γ for L.

(Note. Such a fundamental solution is indeed unique, as it will be proved inProposition 5.3.10.)

5.3 The Fundamental Solution 237

Proof. The existence of such a fundamental solution may be proved by means ofvery general arguments from the theory of distributions, based on the hypoellipticityof L and of its formal adjoint L∗ (= L), jointly with the well-behaved homogeneityproperties of L.

Indeed, from the hypoellipticity of L (see property (A0), page 63) we infer theexistence of a “local” fundamental solution satisfying LΓ = −Dirac0 on a neighbor-hood of the origin (see F. Trèves [Tre67, Theorems 52.1, 52.2]). Moreover, by usingthe homogeneity properties of L, a “local-to-global” argument can be performed. Itis out of our scopes here to give the details. The complete proof is due to G.B. Fol-land and can be found in [Fol75, Theorem 2.1] (see also L. Gallardo [Gal82] forsome further properties of Γ obtained via probabilistic techniques). An alternativeproof can be found in [BLU02, Theorem 3.9]. ��

From the integral identity (5.14) and condition (i) in the above Definition 5.3.1we immediately get the L-harmonicity of Γ out of the origin. Indeed, if we replacein (5.14) a test function ϕ with support in R

N \ {0}, by the smoothness of Γ out ofthe origin, we can integrate by parts obtaining

∫

RN

(LΓ ) ϕ dx = 0 ∀ ϕ ∈ C∞0 (RN \ {0}).

This obviously impliesLΓ = 0 in R

N \ {0}. (5.15)

A simple change of variable and the left-invariance of L w.r.t. the translations on G

give the following theorem.

Theorem 5.3.3 (Γ left-inverts L). Let L be a sub-Laplacian on a homogeneousCarnot group G. If Γ is a fundamental solution for L, then

∫

RN

Γ (y−1 ◦ x)Lϕ(x) dx = −ϕ(y) ∀ϕ ∈ C∞0 (RN) (5.16)

and every y ∈ RN .

Proof. The change of variable z = y−1 ◦ x gives∫

RN

Γ (y−1 ◦ x)Lϕ(x) dx =∫

RN

Γ (z) (Lϕ)(y ◦ z) dz. (5.17)

On the other hand, since L is left-invariant on G, then

(Lϕ)(y ◦ z) = L(ϕ(y ◦ z)).

Replacing this identity in (5.17) and using (5.14) with ϕ(·) replaced by ϕ(y ◦ ·), onegets the thesis. ��


Remark 5.3.4. The integral identity (5.16) means that

L(Γ (y−1 ◦ ·)) = −Diracy

in the weak sense of distributions. Here Diracy denotes the Dirac measure supportedat {y}.

Due to identity (5.16), we can say that Γ is a left inverse of L. We next provethat Γ is a right inverse too.

Theorem 5.3.5 (Γ right-inverts L). Let L be a sub-Laplacian on a homogeneousCarnot group G. If Γ is a fundamental solution for L, then, for every ϕ ∈ C∞0 (RN),the function

RN � y �→ u(y) :=

∫

RN

Γ (y−1 ◦ x) ϕ(x) dx (5.18a)

is smooth and satisfies the equation

Lu = −ϕ. (5.18b)

The proof of this theorem requires some prerequisites. First of all, we note that achange of variable in the integral at the right-hand side of (5.18a) gives

u(y) =∫

RN

Γ (z) ϕ(y ◦ z) dz.

Then we can differentiate under the integral sign and get the smoothness of u. More-over, if supp(ϕ) ⊆ {x : d(x) ≤ R} (here d denotes a fixed homogeneous normon G), then

|u(y)| ≤ sup{|Γ (z)| : d(y ◦ z) ≤ R}∫

RN

|ϕ(z)| dz

=: C(y)

∫

RN

|ϕ(z)| dz. (5.19)

On the other hand, by Corollary 5.1.5 and Proposition 5.1.7,

d(z) ≥ 1

cd(y)− cd(y ◦ z),

for a suitable positive constant c independent of x, y, z. As a consequence,

C(y) ≤ sup

{|Γ (z)| : d(z) ≥ 1

cd(y)− cR

},

so that, since Γ (z) vanishes as z goes to infinity, inequality (5.19) implies

limy→∞ u(y) = 0. (5.20)

We then show a crucial property of the (ε, G)-mollifiers. The relevant definition isthe following one.


Definition 5.3.6 (Mollifiers). Let G = (RN, ◦, δλ) be a homogeneous Carnot groupon R

N . Let O be a fixed non-empty open neighborhood of the origin 0. Let also begiven a function J ∈ C∞0 (RN, R), J ≥ 0, such that

supp(J ) ⊂ O and∫

RN

J = 1.

For any ε > 0, we set Jε(x) := ε−Q J(δ1/ε(x)).Let u ∈ L1

loc(RN). We set, for every x ∈ R

N ,

uε(x) := (u ∗G Jε)(x) :=∫

RN

u(y) Jε(x ◦ y−1) dy

=∫

δε(O)

u(z−1 ◦ x) Jε(z) dz. (5.21)

We call uε a mollifier of u (or (ε, G)-mollifier) related to the kernel J . Note that thismollifier depends only on G = (RN, ◦, δλ) and J .

For the use of mollifiers in a context of subelliptic PDE’s, see also [CDG97].

Example 5.3.7. Let G = (RN, ◦, δλ) be a homogeneous Carnot group on RN . Let �

be a fixed homogeneous symmetric norm on G. We set a notation which will be usedthroughout the book. For every x ∈ G and every r > 0, we set

B�(x, r) := {y ∈ G : �(x−1 ◦ y) < r}.

We say that B�(x, r) is the �-ball with center x and radius r . Also, fixed a pointx ∈ G and a set A ⊂ G, we let

�-dist(x,A) := inf{�(x−1 ◦ a) : a ∈ A

}.

We call �-dist(x,A) the �-distance of x from A. The notation dist�(x,A) will alsobe available.

Let now a function J ∈ C∞0 (RN), J ≥ 0 be given such that

supp(J ) ⊂ B�(0, 1) and∫

RN

J = 1.

For any ε > 0, we set Jε(x) := ε−Q J(δ1/ε(x)).Let u ∈ L1

loc(Ω), Ω ⊆ RN open. For the open set

Ωε := {x ∈ Ω : �-dist(x, ∂Ω) > ε},we define

uε(x) := (u ∗G Jε)(x) :=∫

B�(x, ε)u(y) Jε(x ◦ y−1) dy

=∫

B�(0,ε)

u(y−1 ◦ x) Jε(y) dy (5.22)

for every x ∈ Ωε.


We call uε a mollifier of u (or (ε, G)-mollifier) related to the homogeneousnorm �. Note that this mollifier depends only on G = (RN, ◦, δλ), J and �.

Remark 5.3.8. Let the notation in Definition 5.3.6 be fixed. Let u ∈ L1loc(R

N). Thenthe following fact holds:

(�) uε → u as ε → 0 in L1loc(R

N).

Indeed, we have (perform the change of variable y = δ1/ε(z))

uε(x) =∫

δε(O)

u(z−1 ◦ x) ε−QJ(δ1/ε(z)) dz =∫

O

u(δε(y−1) ◦ x)J (y) dy.

As a consequence (recall that∫O

J = 1),

∫

RN

|uε(x)− u(x)| dx =∫

RN

∣∣∣∣∫

O

u(δε(y−1) ◦ x) J (y) dy − u(x)

∣∣∣∣ dx

=∫

RN

∣∣∣∣∫

O

(u(δε(y

−1) ◦ x)− u(x))J (y) dy

∣∣∣∣ dx

≤∫

O

{∫

RN

∣∣ u(δε(y−1) ◦ x)− u(x)

∣∣ dx

}J (y) dy.

Given σ > 0, by well-known results, there exists ε = ε(σ, G,O) > 0 such that if0 < ε < ε, then the integral in braces is ≤ σ for every fixed y ∈ O. This provesthat

∫

RN

|uε(x)− u(x)| dx ≤ σ

∫

O

J(y) dy = σ ∀ 0 < ε < ε,

i.e. (�) holds. ��Proposition 5.3.9 (L-harmonicity of the mollifier). Let G = (RN, ◦, δλ) be a ho-mogeneous Carnot group on R

N . Let L be a sub-Laplacian on G. Let u ∈ L1loc(R

N)

be a weak solution to Lu = 0 in RN , i.e.

∫

RN

uLϕ dy = 0 ∀ϕ ∈ C∞0 (RN). (5.23)

Then, if uε denotes the mollification on G w.r.t. any kernel J (as in Definition 5.3.6),we have

Luε = 0 in G for every ε > 0. (5.24)

Proof. First, note that, being supp(Jε) ⊂ δε(O),

uε(x) =∫

RN

u(y−1 ◦ x) Jε(y) dy.

For every test function ϕ, we thus have (Fubini–Tonelli’s theorem certainly applies)


∫

RN

uε(x)Lϕ(x) dx =∫

RN

Lϕ(x)

(∫

RN

u(y−1 ◦ x) Jε(y) dy

)dx

=∫

RN

Jε(y)

(∫

RN

(Lϕ)(x)u(y−1 ◦ x) dx

)dy

=∫

RN

Jε(y)

(∫

RN

(Lϕ)(y ◦ z) u(z) dz

)dy

=∫

RN

Jε(y)

(∫

RN

L(z �→ ϕ(y ◦ z)

)u(z) dz

)dy.

The inner integral in the far right-hand side is equal to zero by the hypothesis (5.23).Then ∫

RN

uε(x)Lϕ(x) dx = 0 ∀ϕ ∈ C∞0 (RN),

so that the claimed (5.24) follows from the hypoellipticity of L and the fact thatL∗ = L. ��

With Proposition 5.3.9 at hand, it is easy to prove the uniqueness of the funda-mental solution.

Proposition 5.3.10 (Uniqueness of the fundamental solution). Let L be a sub-Laplacian on a homogeneous Carnot group G. The fundamental solution of L (whoseexistence is granted by Theorem 5.3.2) is unique.

Proof. Let Γ and Γ ′ be fundamental solutions for L. Then the function u = Γ −Γ ′has the following properties: u ∈ L1

loc(RN), u(x)→ 0 as x →∞ and

∫

RN

uLϕ dy = 0 ∀ϕ ∈ C∞0 (RN).

As a consequence, by Proposition 5.3.9, Luε = 0 in RN for every ε > 0.

Thus, since uε(x) → 0 as x → ∞ (argue as in (5.20)), the maximum principlein Section 5.13 implies uε = 0 in R

N . On the other hand, uε → u as ε → 0 inL1

loc(RN) (see Remark 5.3.8). Then u = 0 almost everywhere in R

N , so that Γ = Γ ′in R

N \ {0}. ��We now prove that Γ is “G-symmetric” with respect to the origin. More pre-

cisely, the following assertion holds.

Proposition 5.3.11 (Symmetry of Γ ). Let L be a sub-Laplacian on a homogeneousCarnot group G. Let Γ be the fundamental solution of L. Then

Γ (x−1) = Γ (x) ∀ x ∈ G \ {0}.Proof. Given ϕ ∈ C∞0 (RN), define

u(x) := −∫

RN

Γ (y−1 ◦ x)Lϕ(y) dy, x ∈ G.


The function u is smooth and vanishes at infinity (see (5.20)). Moreover, for everyψ ∈ C∞0 (RN),

∫

RN

Lu(x)ψ(x) dx =∫

RN

u(x)Lψ(x) dx

= −∫

RN

Lϕ(y)

(∫

RN

Γ (y−1 ◦ x)Lψ(x) dx

)dy

=∫

RN

Lϕ(x)ψ(x) dx (by Theorem 5.3.3).

This proves that L(u − ϕ) = 0 in G. Thus, since u − ϕ vanishes at infinity, by themaximum principle, u = ϕ in R

N . In particular,

ϕ(0) = u(0) = −∫

RN

Γ (y−1 ◦ x)Lϕ(y) dy ∀ ϕ ∈ C∞0 (RN),

so that x �→ Γ (x−1) is a fundamental solution of L (see Definition 5.3.1). Theuniqueness of the fundamental solution (Proposition 5.3.10) implies Γ (x−1) = Γ (x)

for every x ∈ G \ {0}. ��Finally, we are in the position to prove Theorem 5.3.5.

Proof (of Theorem 5.3.5). Let u be the function defined in (5.18a). Then u ∈C∞(RN) and, for any test function ψ ∈ C∞0 (RN), one has

∫

RN

(Lu)(y)ψ(y) dy =∫

RN

u(y)Lψ(y) dy

=∫

RN

Lψ(y)

(∫

RN

Γ (y−1 ◦ x) ϕ(x) dx

)dy

=∫

RN

ϕ(x)

(∫

RN

Γ (y−1 ◦ x)Lψ(y) dy

)dx

=∫

RN

ϕ(x)

(∫

RN

Γ (x−1 ◦ y)Lψ(y) dy

)dx.

Here we used Proposition 5.3.11 ensuring that Γ (y−1 ◦ x) = Γ ((x−1 ◦ y)−1) =Γ (x−1 ◦ y). Now, by identity (5.16), the inner integral at the far right-hand side isequal to −ψ(x). Then

∫

RN

(Lu)(y)ψ(y) dy = −∫

RN

ϕ(x)ψ(x) dx ∀ψ ∈ C∞0 (RN).

This gives identity (5.18b). ��From the uniqueness of Γ we easily obtain its δλ-homogeneity.

Proposition 5.3.12 (δλ-homogeneity of Γ ). Let L be a sub-Laplacian on a ho-mogeneous Carnot group G. Let Γ be the fundamental solution of L. Then Γ isδλ-homogeneous of degree 2−Q, i.e.

Γ (δλ(x)) = λ2−QΓ (x) ∀ x ∈ G \ {0} ∀ λ > 0.


Proof. For any fixed λ > 0, define

Γ ′(x) := λQ−2Γ (δλ(x)) ∀ x ∈ G \ {0}.It is quite obvious that Γ ′ ∈ C∞(RN \ {0}) ∩ L1

loc(RN) and Γ ′(x)→ 0 as x →∞.

Moreover, for every test function ϕ ∈ C∞0 (RN),

∫

RN

Γ ′(x)Lϕ(x) dx = λQ−2∫

RN

Γ (δλ(x))Lϕ(x) dx

(by using the change of variable y = δλ(x))

= λ−2∫

RN

Γ (y) (Lϕ)(δ1/λ(y)) dy

(since L is δλ-homogeneous of degree 2)

=∫

RN

Γ (y)L(ϕ(δ1/λ(y))

)dy = −ϕ(δ1/λ(0)) = −ϕ(0).

This proves that Γ ′ is a fundamental solution of L. Then, by Proposition 5.3.10,Γ = Γ ′ and the assertion is proved. ��

From the strong maximum principle in Theorem 5.13.8 we obtain another im-portant property of Γ .

Proposition 5.3.13 (Positivity of Γ ). Let L be a sub-Laplacian on a homogeneousCarnot group G. Let Γ be the fundamental solution of L. Then

Γ (x) > 0 ∀ x ∈ G \ {0}.Proof. Let ϕ ∈ C∞0 (RN), ϕ ≥ 0. Define

u(y) :=∫

RN

Γ (y−1 ◦ x)ϕ(x) dx, y ∈ G.

The function u is smooth, vanishes at infinity and satisfies the equation Lu = −ϕ

(see (5.20) and Theorem 5.3.5). Then

Lu ≤ 0 in G and limy→∞ u(y) = 0.

By the maximum principle in Corollary 5.13.6, it follows that u ≥ 0 in G. Hence∫

RN

Γ (y−1 ◦ x) ϕ(x) dx ≥ 0 ∀ϕ ∈ C∞0 (RN), ϕ ≥ 0.

Thus, Γ ≥ 0, so that, since it is L-harmonic in G\{0}, the strong maximum principlein Theorem 5.13.8 implies Γ ≡ 0 or Γ (x) > 0 for any x �= 0. The first casewould contradict identity (5.14). Then Γ > 0 at any point of G \ {0}. This ends theproof. ��


Corollary 5.3.14 (Pole of Γ ). Let L be a sub-Laplacian on a homogeneous Carnotgroup G. The fundamental solution Γ of L has a pole at 0, i.e.

limx→0

Γ (x) = ∞. (5.25)

Proof. Since Γ is smooth and strictly positive out the origin, we have

h := min{Γ (x) : d(x) = 1} > 0.

Here d denotes any fixed homogeneous norm on G. Then, by Proposition 5.3.12,

Γ (x) = d(x)2−QΓ (δ1/d(x)(x)) ≥ h d2−Q(x).

From this inequality (5.25) immediately follows. ��

5.3.1 The Fundamental Solution in the Abstract Setting

The aim of this brief section is to show how to derive a “fundamental solution” fora sub-Laplacian L on an abstract stratified group H, starting from the fundamentalsolution of the related sub-Laplacian L on a homogeneous-Carnot-group copy G

of H.Many other alternative “more intrinsic” definitions may be certainly provided, for

example, by making use of the integration on an abstract Lie group. We consideredmore “in the spirit” of our exposition to pass through the homogeneous group G.Besides, this also makes unnecessary to furnish the (lengthy) theory of integrationon manifolds.

Let H be a stratified group with an algebra h. Let L = ∑mj=1 X2

j be a sub-Laplacian on H, and let V = (V1, . . . , Vr ) be the stratification of h related to L,according to Definition 2.2.25, page 144. Let also E be a basis for h adapted to thestratification V .

By Proposition 2.2.22 on page 139 (whose notation we presently follow), thereexists a homogeneous Carnot group G = (RN,�E ) which is isomorphic to H. LetΨ : G → H be the isomorphism as in Proposition 2.2.22-(1). With the thereinnotation, we have

Ψ = Exp ◦ π−1E ,

where, for every ξ ∈ RN , π−1

E (ξ) is the vector field in h having ξ as N -tuple of thecoordinates w.r.t. the basis E .

Let X1, . . . , Xm be the vector fields in g (the algebra of G) which are Ψ -relatedto X1, . . . , Xm, respectively, i.e.

dΨ (Xj ) = Xj for every j = 1, . . . , m.

We set

L :=m∑

j=1

(Xj )2,


and (since L is a sub-Laplacian on the homogeneous Carnot group G) we let Γ

denote its (unique) fundamental solution.A possible definition of a fundamental solution for L is

Γ := Γ ◦ Ψ−1. (5.26)

Our task here is to show how Γ depends on the (arbitrary) choice of the above ba-sis E . We show that, roughly speaking “up to a multiplicative factor”, Γ in (5.26) isintrinsic (see below for the precise statement).

To this aim, let E1 and E2 be two bases of h adapted to the stratification V . Withthe above notation, let (for i = 1, 2) Gi = (RN,�Ei

) and gi denotes the algebraof Gi . Moreover, we set

Ψi := Exp ◦ π−1Ei

, i = 1, 2. (5.27)

Again for i = 1, 2, we also let

X(i)1 , . . . , X(i)

m

be the vector fields in gi which are Ψi-related to X1, . . . , Xm, respectively, i.e.

dΨi(X(i)j ) = Xj for every j = 1, . . . , m. (5.28)

We set

Li :=m∑

j=1

(X(i)j )2, i = 1, 2.

Finally, Γi denotes the fundamental solution of Li . We claim that

Γ2 = c1,2 · Γ1 ◦ (Ψ−11 ◦ Ψ2), where c1,2 :=

∣∣ det(πE1 ◦ π−1

E2

)∣∣. (5.29)

This will giveΓ2 ◦ Ψ−1

2 = c1,2 · Γ1 ◦ Ψ−11 , (5.30)

which proves the “almost” intrinsic nature of the definition (5.26) of Γ . We explicitlyremark that πE1 ◦π−1

E2is a linear automorphism of R

N , so that its determinant is well-posed.

(Note. We explicitly remark that the uniqueness of the fundamental solution ofL up to a multiplicative factor is a matter of fact. Indeed, even in the homogeneousCarnot setting, the fundamental solution depends on the measure within the integralin (5.14) of Definition 5.3.1. In order to have a unique fundamental solution, thechoice of the Lebesgue measure on R

N was quite natural (though arbitrary). Instead,in the context of an abstract Lie group H, the Haar measure is unique only up to amultiplicative factor, and there is no effective way to prefer a Haar measure insteadof another. These remarks show that the constant in (5.30) is perfectly justified.)

We now turn to the claimed (5.29). By invoking Proposition 5.3.10, set

Γ1,2 := c1,2 · Γ1 ◦ (Ψ−11 ◦ Ψ2),


then (5.29) will follow if we show that Γ1,2 satisfies (i), (ii) and (iii) of Defini-tion 5.3.1 w.r.t. the sub-Laplacian L2.

To begin with, observe that (thanks to (5.27))

Ψ−11 ◦ Ψ2 =

(Exp ◦ π−1

E1

)−1 ◦ (Exp ◦ π−1E2

) = πE1 ◦ π−1E2

, (5.31)

and this last map is a linear isomorphism of RN . As a consequence, since Γ1 ∈

C∞(RN \ {0}), we immediately infer that the same holds for Γ1,2. This proves (i).Moreover, since Γ1 ∈ L1

loc(RN) and Γ1(x) −→ 0 when x tends to infinity, the same

holds for Γ1,2, again thanks to (5.31). This proves (ii).Finally, we prove (iii). First, note that from (5.28) one gets

dΨ1(X(1)j ) = Xj = dΨ2(X

(2)j ) for every j = 1, . . . , m,

i.e. for every j = 1, . . . , m,

X(1)j = d(Ψ−1

1 ◦ Ψ2)(X(2)j ).

Set Ψ1,2 := Ψ−11 ◦ Ψ2. This gives

L1 = dΨ1,2(L2),

i.e. it holds

(L1f ) ◦ Ψ1,2 = L2(f ◦ Ψ1,2) ∀ f ∈ C∞(RN, R). (5.32)

Let now ϕ ∈ C∞0 (RN). Then we have∫

RN

Γ1,2(L2ϕ) = c1,2 ·∫

RN

Γ1(Ψ1,2(x))(L2ϕ)(x) dx

(by the linear change of variable y = Ψ1,2(x) = πE1 ◦ π−1E2

, see (5.31))

= c1,2 ·∫

RN

Γ1(y)(L2ϕ)(Ψ−1

1,2 (y)) dy

c1,2(see (5.32))

=∫

RN

Γ1(y)L1(ϕ ◦ Ψ−1

1,2

)(y) dy

(Γ1 is the fundamental solution of L1 and ϕ ◦ Ψ−11,2 ∈ C∞0 (RN))

= −(ϕ ◦ Ψ−11,2

)(0) = −ϕ(0).

This proves (iii), and the proof is complete. ��

5.4 L-gauges and L-radial Functions

On every homogeneous Carnot group, there exist distinguished smooth symmetrichomogeneous norms playing a fundamental rôle for the sub-Laplacians. We callthese norms gauges, according to the following definition.

5.4 L-gauges and L-radial Functions 247

Definition 5.4.1 (L-gauge). Let L be a sub-Laplacian on a homogeneous Carnotgroup G. We call L-gauge on G a homogeneous symmetric norm d smooth out of theorigin and satisfying

L(d2−Q) = 0 in G \ {0}. (5.33)

An L-radial function on G is a function u : G \ {0} → R such that

u(x) = f (d(x)) ∀ x ∈ G \ {0}for a suitable f : (0,∞)→ R and a given L-gauge d on G.

The L-gauges are deeply related to the fundamental solution of L.

Proposition 5.4.2. Let L be a sub-Laplacian on a homogeneous Carnot group G.Let Γ be the fundamental solution of L. Then

d(x) :={

(Γ (x))1/(2−Q) if x ∈ G \ {0},0 if x = 0

is an L-gauge on G.

Proof. The assertion follows from condition (i) in Definition 5.3.1, (5.15), Proposi-tions 5.3.11, 5.3.12, 5.3.13 and Corollary 5.3.14. ��

In the next section, we shall show the reverse part of Proposition 5.4.2 (see The-orem 5.5.6): if d is an L-gauge on G, then there exists a positive constant βd suchthat Γ = βd d2−Q is the fundamental solution of L. As a consequence, by Proposi-tion 5.3.10, the L-gauge is unique up to a multiplicative constant.

In Section 9.8, we shall also prove the following fact: if d is a homogeneous normon G, smooth out of the origin and such that

L(dα) = 0 in G \ {0}for a suitable α ∈ R, α �= 0, then α = 2 − Q and d is an L-gauge on G (seeCorollary 9.8.8, page 461, for the precise statement).

The sub-Laplacian of an L-radial function takes a noteworthy form.

Proposition 5.4.3. Let L be a sub-Laplacian on a homogeneous Carnot group G.Let f (d) be a smooth L-radial function on G \ {0}. Then

L(f (d)) = |∇Ld|2(

f ′′(d)+ Q− 1

df ′(d)

), (5.34)

where, if L =∑mj=1 X2

j , we set ∇L = (X1, . . . , Xm).


Proof. An easy computation gives (see also Ex. 6, Chapter 1)

L(f (d)) =m∑

j=1

X2j (f (d)) = f ′′(d)

m∑

j=1

(Xjd)2 + f ′(d)

m∑

j=1

X2j (d),

so thatL(f (d)) = f ′′(d) |∇Ld|2 + f ′(d)L(d). (5.35)

(Note that (5.35) holds with the only assumption that f and d are smooth functionson some open subsets of G and R, respectively, and f ◦ d is defined.)

Applying this formula to the function f (s) = s2−Q and keeping in mind (5.33),we obtain

0 = (1−Q)d−Q |∇Ld|2 + d1−Q L(d),

hence

L(d) = |∇Ld|2 Q− 1

d.

Identity (5.34) follows by replacing this last identity in (5.35). ��If � is any (sufficiently smooth) homogeneous norm on G, the integration of a

�-radial function over a “�-radially-symmetric” domain reduces to an integration ofa single variable function.

Proposition 5.4.4. Let L be a sub-Laplacian on a homogeneous Carnot group G.Let � be any homogeneous norm on G smooth on G \ {0}. Let f (�) be a functiondefined on the �-ball B�(0, r) of radius r centered at the origin,

B�(0, r) := {x ∈ G : �(x) < r}.Then, if f (�) ∈ L1(B�(0, r)), it holds

∫

B�(0,r)

f (�(x)) dx = Qω�

∫ r

0sQ−1 f (s) ds, (5.36)

where ω� denotes the Lebesgue measure of B�(0, 1),

ω� := |B�(0, 1)|.Proof. The coarea formula gives

∫

B�(0,r)

f (�(x)) dx =∫ r

0f (s)

(∫

{�=s}1

|∇� | dHN−1)

ds. (5.37)

On the other hand, by using the δλ-homogeneity of �, we have∫ r

0

(∫

{�=s}1

|∇�| dHN−1)

ds = |B�(0, r)| = ω� rQ

for every r > 0. Differentiating this last identity with respect to r , we obtain∫

{�=r}1

|∇�| dHN−1 = Qω� rQ−1. (5.38)

By using this identity in (5.37), we immediately get (5.36). ��

5.4 L-gauges and L-radial Functions 249

Corollary 5.4.5. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let� be any homogeneous norm on G. The function �α is locally integrable in R

N if andonly if α > −Q.

Proof. If � is also smooth on G \ {0}, by Proposition 5.4.4,∫

B�(0,r)

�α(x) dx = Qω�

∫ r

0sα+Q−1 ds,

and the assertion trivially follows.If � is only continuous, we argue as follows. If α ≥ 0, the assertion is trivial. If

α < 0, we have∫

B�(0,r)

�α(x) dx =N∑

k=0

∫

{r/2k+1≤�<r/2k}�α(x) dx

≤ (r/2)αN∑

k=0

1

2kα

∫

{r/2k+1≤�(x)<r/2k}dx (by x = δr/2k (y))

= (r/2)αN∑

k=0

1

2kα

( r

2k

)Q∫

{1/2≤�(y)<1}dy

= c� rQ (r/2)αN∑

k=0

2−k(α+Q),

hence, if α > −Q, then �α is integrable on B�(0, r). The reverse assertion followsby the same computation as above, by taking the obvious bound from below. ��

A couple of examples of explicit L-gauges are in order. See also Example 5.10.3and Chapter 18.

Example 5.4.6 (Δ-gauge). The classical Laplace operator in RN , N ≥ 3,

Δ :=N∑

j=1

∂2xj

is the canonical (sub-)Laplacian on the Euclidean group

E := (RN,+, δλ)

with δλx = λ x. The homogeneous dimension of E is N and a Δ-gauge function isthe Euclidean norm

x �→ |x| :=(

N∑

j=1

x2j

)1/2

, x = (x1, . . . , xN).

Indeed, | · | is smooth and strictly positive out of the origin, δλ-homogeneous ofdegree 1 and, as it is well known,

Δ(|x|2−N) = 0 ∀ x �= 0. ��


On every group of Heisenberg type, noteworthy explicit L-gauges are known.They were discovered by A. Kaplan [Kap80].

Example 5.4.7 (ΔH-gauges). Let H = (Rm+n, ◦, δλ) be a (prototype) group ofHeisenberg type (see Section 3.6, page 169). Denoting by (x, t) the points of H,x ∈ R

m, t ∈ Rn, we know that the canonical sub-Laplacian ΔH on H takes the form

(see (3.14) on page 171)

ΔH = Δx + 1

4|x|2Δt +

n∑

k=1

〈B(k)x,∇x〉 ∂tk . (5.39a)

Here, the B(k)’s are n skew-symmetric m × m orthogonal matrices satisfying therelation

B(i) B(j) = −B(j) B(i) for every i, j ∈ {1, . . . , n} with i �= j .

The dilations {δλ}λ>0 are given by

δλ(x, t) = (λ x, λ2t).

ThenQ = m+ 2 n (5.39b)

is the homogeneous dimension of H. We want to show that

d(x, t) := (|x|4 + 16 |t |2)1/4 (5.39c)

is a ΔH-gauge. Here |x| and |t | denote respectively the Euclidean norm of x ∈ Rm

and t ∈ Rn. First of all, we remark that d is strictly positive and smooth out of

the origin, it is δλ-homogeneous of degree one and symmetric, since (x, t)−1 =(−x,−t). Now, we aim to compute ΔH(d2−Q). To this end, it is convenient to fixthe following notation:

v(r, s) := r4 + 16 s2, r = |x|, s = |t |; α = (2−Q)/4.

ThenG(x, t) := (d(x, t))2−Q = (r4 + 16 s2)α = vα(r, s).

Since B(k) is skew-symmetric, we have

〈B(k)x,∇xG(x, t)〉 = 4α vα−1r2 〈B(k)x, x〉 = 0

for every x ∈ Rm, t ∈ R

n and 1 ≤ k ≤ n. Then, by using (5.39a),

ΔHG = ΔxG+ r2

4ΔtG

= αvα−1(

Δxv + r2

4Δtv

)

+ α(α − 1) vα−2(|∇xv|2 + r2

4|∇t v|2

). (5.39d)

5.5 Gauge Functions and Surface Mean Value Theorem 251

Now, thanks to the radial symmetry of v with respect to x and t , we easily obtain

Δxv = (8+ 4m)r2,r2

4Δtv = 8nr2 (5.39e)

and

|∇xv|2 = 16r6,r2

4|∇t v|2 = (16 rs)2. (5.39f)

Replacing (5.39e) and (5.39f) in (5.39d), we get

ΔHG = 4α vα−1r2(2+m+ 2n)+ 16α(α − 1) vα−2r2(r4 + 16 s2)

= 4α vα−1r2(2+Q+ 4(α − 1))

= 4α vα−1r2(Q− 2+ 4α) = 0 (being α = (2−Q)/4).

ThenΔH(d2−Q) = 0 in H \ {0},

and d in (5.39c) is a ΔH-gauge. ��Remark 5.4.8 (Cylindrically-symmetric functions on H). In the notation of the aboveExample 5.4.7, we shall call cylindrically-symmetric any function u on H such that

u(x, t) = v(|x|, t)for some function v = v(r, t), r ∈ R, r > 0, t ∈ R

n. Assume v is smooth. Then

〈B(k)x,∇xu(x, t)〉 = 〈B(k)x, x/|x|〉 ∂rv(|x|, t) = 0,

since 〈B(k)x, x〉 = 0 for every x ∈ Rm. As a consequence, keeping in mind (3.14)

and (3.16), we obtain the following form of ΔH and |∇H| for cylindrically-symmetricfunctions u(x, t) = v(|x|, t):

ΔHu = vrr + m− 1

rvr + 1

4r2 Δt,

(5.39g)

|∇Hu|2 = v2r +

1

4r2 |∇t v|2.

Here, r = |x| and vr = δrv, vrr = δrrv.

5.5 Gauge Functions and Surface Mean Value Theorem

Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. The aim of this section is to prove a mean value theorem on the boundary ofthe d-balls for the L-harmonic functions. When L is the classical Laplace operator,our result will give back the Gauss theorem for classical harmonic functions. We shallobtain the mean value theorem as a byproduct of a representation formula for general


C2 functions. These representation formulas will play a major rôle throughout thebook.

For any x ∈ G and r > 0, we recall that we have already defined the d-ball ofcenter x and radius r as follows:

Bd(x, r) := {y ∈ G : d(x−1 ◦ y) < r}. (5.40)

ThenBd(x, r) = x ◦ Bd(0, r).

By using the translation invariance of the Lebesgue measure and the δλ-homogeneityof d , one also easily recognizes that

|Bd(x, r)| = rQ |Bd(0, 1)| =: ωd rQ. (5.41)

We explicitly remark that

∂Bd(x, r) := {y ∈ G : d(x−1 ◦ y) = r}is a smooth manifold of dimension N − 1. Indeed, by Sard’s lemma, this holds truefor almost every r > 0. The assertion then follows for every r > 0, since ∂Bd(x, r)

is diffeomorphic to ∂Bd(x, 1) via the dilation δr . (Note that, so far, d may be anyhomogeneous norm smooth out of the origin.)

Definition 5.5.1 (The kernels of the mean value formulas). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. We set,for x ∈ G \ {0},

ΨL(x) := |∇Ld|2(x).

Moreover, for every x, y ∈ G with x �= y, we define the functions

ΨL(x, y) := ΨL(x−1 ◦ y) and KL(x, y) := |∇Ld|2(x−1 ◦ y)

|∇(d(x−1 ◦ ·))|(y). (5.42)

Remark 5.5.2. We explicitly remark that

ΨL is δλ-homogeneous of degree zero,

a fact which will be used repeatedly. We would like to recall that ΨL appears in the“radial” form of L, see (5.34).

We also explicitly remark that, while ΨL is translation-invariant (i.e. ΨL(α ◦x, α ◦y) = ΨL(x, y)), the function KL does not necessarily share the same property.Moreover, when L = Δ is the classical Laplace operator, then ΨL = KL = 1. Fora general sub-Laplacian L, we shall prove the following fact (see Proposition 9.8.9,page 462):

The function ΨL is constant if and only ifG is the Euclidean group.


For instance, consider the following example.

Example 5.5.3. Let H = (Rm+n, ◦, δλ) be a group of Heisenberg type. Denoting by(x, t) the points of H, x ∈ R

m, t ∈ Rn, we proved in Example 5.4.7 that the “Folland

function”d(x, t) := (|x|4 + 16 |t |2)1/4

is a ΔH-gauge. Using (5.39g), we obtain

ΨH(x, t) = |∇Hd(x, t)|2

=(

r

d

)6

+ 16

(r s

d3

)2

=(

r

d

)2(r4

d4+ 16

s2

d4

),

where r = |x| and s = |t |. Then

ΨH(x, t) = |x|2√|x|4 + 16 |t |2 , (x, t) �= (0, 0). ��

Let Ω ⊆ G be an open set and u, v ∈ C2(Ω). By using the divergence form ofthe sub-Laplacian L (see (1.90a) on page 64),

L = div(A(x) · ∇T ),

we easily get

vLu− uLv = div(v A · ∇T u)− div(uA · ∇T v). (5.43a)

Let us now assume that Ω is bounded with boundary ∂Ω of class C1 and exteriornormal ν = ν(y) at any point y ∈ ∂Ω . Then, if u, v are of class C2 in a neighborhoodof Ω , integrating (5.43a) on Ω and using the divergence theorem, we obtain theGreen identity2

∫

Ω

(vLu− uLv) dHN =∫

∂Ω

(v 〈A · ∇T u, ν〉 − u 〈A · ∇T v, ν〉) dHN−1. (5.43b)

Hereafter, dHN (respectively, dHN−1) stands for the N -dimensional (respectively,(N − 1)-dimensional) Hausdorff measure in R

N . If we choose v ≡ 1 in (5.43b), we

2 Another way to write the Green identity is the following one: if L =∑mj=1 X2

j, we have

∫

Ω(vLu− uLv) dHN =

∫

∂Ω

⎛

⎝v

m∑

j=1

Xju 〈Xj I, ν〉 − u

m∑

j=1

Xjv 〈Xj I, ν〉⎞

⎠dHN−1.

This follows from (5.43b), recalling that (see (1.90b), page 64) A is the N × N sym-metric matrix A(x) = σ(x) σ (x)T , where σ(x) is the N × m matrix whose columns areX1I (x), . . . , XmI (x).


obtain3 ∫

Ω

Lu dHN =∫

∂Ω

〈A · ∇T u, ν〉 dHN−1, (5.43c)

so that ∫

Ω

Lu dHN = 0 ∀ u ∈ C∞0 (Ω). (5.43d)

Let us now consider an arbitrary open set O ⊆ RN such that Bd(x, r) ⊂ O for a

suitable r > 0. For 0 < ε < r , we define the “d-ring”

Dε,r := Bd(x, r) \ Bd(x, ε) = {y ∈ RN : ε < d(x−1 ◦ y) < r}.

Given u ∈ C2(O), we apply the Green identity (5.43b) to the functions u and v :=d2−Q(x−1 ◦ ·) on the open set Dε,r . Since v is L-harmonic in G \ {0} (note that herewe apply, for the first time and with crucial consequences, the fact d is an L-gauge),we obtain ∫

Dε,r

vLu = Sr(u)− Sε(u)+ Tε(u)− Tr(u), (5.43e)

where we have used the following notation

Sρ(u) :=∫

∂Bd(x,ρ)

v 〈A · ∇T u, ν〉 dHN−1,

Tρ(u) :=∫

∂Bd(x,ρ)

u 〈A · ∇T v, ν〉 dHN−1.

Since v is constant on ∂Bd(x, ρ), keeping in mind (5.43c), we have

Sρ(u) = ρ2−Q

∫

∂Bd(x,ρ)

〈A · ∇T u, ν〉 dHN−1 = ρ2−Q

∫

Bd(x,ρ)

Lu dHN, (5.43f)

so that, by means of (5.41),

Sε(u) = ε2 O(|Lu|) −→ 0, as ε → 0. (5.43g)

To evaluate Tρ(u), we first remark that on ∂Bd(x, ρ) one has

ν = ∇(d(x−1 ◦ ·))|∇(d(x−1 ◦ ·))| ,

and

〈A · ∇T v, ν〉 = (2−Q)d1−Q(x−1 ◦ ·) 〈A · ∇T (d(x−1 ◦ ·)),∇T (d(x−1 ◦ ·))〉

|∇(d(x−1 ◦ ·))|(see (5.1c) and (5.42)) = (2−Q)ρ1−Q ΨL(x−1 ◦ ·)

|∇(d(x−1 ◦ ·))| .3 Or, equivalently (see the previous note),

∫

ΩLu dHN =

∫

∂Ω

m∑

j=1

Xju 〈Xj I, ν〉 dHN−1.


Therefore, keeping in mind the second definition in (5.42),

Tρ(u) = (2−Q)ρ1−Q

∫

∂Bd(x,ρ)

u(y)KL(x, y) dHN−1(y), (5.43h)

so thatTε(u) = (u(x)+ o(1)) Tε(1), as ε → 0. (5.43i)

To compute Tε(1), we observe that (5.43e) with u = 1 gives

Tε(1) = Tr(1) for 0 < ε < r <∞. (5.43j)

Hence, for every ε > 0,

Tε(1) = T1(1) = (2−Q)

∫

∂Bd(x,1)

KL(x, ·) dHN−1. (5.43k)

Finally, since v = d2−Q(x, ·) ∈ L1(Bd(x, r)) (see Corollary 5.4.5)

limε→0

∫

Dε,r

vLu dHN =∫

D0,r

vLu dHN.

Therefore, as ε → 0, identity (5.43e) together with (5.43f)-(5.43k) give∫

Bd(x,r)

vLu dHN = r2−Q

∫

Bd(x,r)

Lu dHN + T1(1) u(x)

− (2−Q) r1−Q

∫

∂Bd(x,r)

uKL(x, ·) dHN−1. (5.43l)

We now observe that

T1(1) = (2−Q)

∫

∂Bd(x,1)

KL(x, ·) dHN−1

does not depend on x, i.e.

(Q− 2)

∫

∂Bd(x,1)

KL(x, ·) dHN−1

= (Q− 2)

∫

∂Bd(0,1)

KL(0, ·) dHN−1 =: (βd)−1. (5.43m)

Indeed, from (5.43j) we have

T1(1)1

Q=∫ 1

0Tr(1) rQ−1 dr

= (2−Q)

∫ 1

0

∫

∂Bd(x,r)

KL(x, y) dHN−1(y) dr

(by the coarea formula) = (2−Q)

∫

Bd(x,1)

ΨL(x, y) dHN−1(y)

= (2−Q)

∫

Bd(0,1)

ΨL dHN.


Here we used the very definition (5.42) of the functions ΨL and KL and the leftinvariance of ΨL. Incidentally, we have also proved that

(βd)−1 = Q(Q− 2)

∫

Bd(0,1)

ΨL dHN. (5.43n)

From identity (5.43l), moving terms around, we obtain

u(x) = (Q− 2) βd

rQ−1

∫

∂Bd(x,r)

uKL(x, ·) dHN−1

− βd

∫

Bd(x,r)

(d2−Q(x−1 ◦ ·)− r2−Q)Lu dHN. (5.44)

We have thus proved the following fundamental result.

Theorem 5.5.4 (Surface mean value theorem). Let L be a sub-Laplacian on thehomogeneous Carnot group G, and let d be an L-gauge on G. Let O be an opensubset of G, and let u ∈ C2(O, R).

Then, for every x ∈ O and r > 0 such that Bd(x, r) ⊂ O, we have

u(x) =Mr (u)(x)−Nr (Lu)(x), (5.45)

where

Mr (u)(x) = (Q− 2) βd

rQ−1

∫

∂Bd(x,r)

KL(x, z) u(z) dHN−1(z),

(5.46)Nr (w)(x) = βd

∫

Bd(x,r)

(d2−Q(x−1 ◦ z)− r2−Q)w(z) dHN(z),

and βd and KL are defined, respectively, in (5.43m) and (5.42).In particular, if Lu = 0, i.e. u is L-harmonic in O, we have

u(x) =Mr (u)(x). (5.47)

Remark 5.5.5. When L = Δ is the classical Laplace operator in RN , N ≥ 3, the

kernel KL is constant (KL ≡ 1). Then, in this case,

Mr (u)(x) = 1

σN rN−1

∫

|x−y|=r

u(y) dHN−1(y) =: −−∫

|x−y|=r

u(y) dHN−1y

(see (5.43m) and (5.46)) and (5.47) gives back the Gauss theorem for classical har-monic functions. ��

From Theorem 5.5.4, we straightforwardly also obtain the following result (seealso Corollary 9.8.8 on page 461 for yet another improvement).

Theorem 5.5.6 (“Uniqueness” of the L-gauges. I). Let L be a sub-Laplacian onthe homogeneous Carnot group G. Let d be an L-gauge on G, and let βd be thepositive constant defined in (5.43m). Then

Γ = βd d2−Q (5.48)

is the fundamental solution of L.

5.6 Superposition of Average Operators 257

Proof. Let ϕ ∈ C∞0 (Ω) and choose r > 0 such that supp(ϕ) ⊂ Bd(0, r). Then, bythe mean value formula (5.45) (being u ≡ 0 on ∂Bd(0, r)),

ϕ(0) = −βd

∫

Bd(0,r)

(d2−Q(z)− r2−Q)Lϕ(z) dHN(z).

On the other hand, by identity (5.43d),∫

Bd(0,r)

r2−Q Lϕ dHN = 0.

Then, if Γ is the function defined in (5.48), Γ ∈ L1loc(R

N) thanks to Corollary 5.4.5and

−ϕ(0) =∫

RN

Γ (z)Lϕ(z) dHN(z)

for every ϕ ∈ C∞0 (Ω). Moreover, Γ is smooth in G \ {0} and Γ (z)→ 0 as z→∞,since Q− 2 > 0. Thus, by Definition 5.3.1, Γ is the fundamental solution of L. ��

From Theorem 5.5.4 we obtain the following asymptotic formula for L.

Theorem 5.5.7 (Asymptotic surface formula for L). Let L be a sub-Laplacian onthe homogeneous Carnot group G, and let d be an L-gauge.

Let Ω ⊆ G be open, and let u ∈ C2(Ω, R). Then, for every x ∈ Ω , we have

limr→0+

Mr (u)(x)− u(x)

r2= ad Lu(x), (5.49a)

where

ad := βd

∫

Bd(0,1)

(d2−Q(y)− 1) dy. (5.49b)

Proof. The change of variable z = x ◦ δr (y) in the integral defining Nr gives

Nr (1)(x) = ad r2 for every x ∈ G.

As a consequence, if u ∈ C2(Ω, R), from (5.45) we obtain

Mr (u)(x)− u(x) = Nr (Lu)(x) = Nr (Lu− Lu(x))(x)+Nr (1)(x)Lu(x)

= ad r2(Lu(x)+ o(1)), as r → 0+.

This proves (5.49a). ��

5.6 Superposition of Average Operators. Solid Mean ValueTheorems. Koebe-type Theorems

As in the previous section, L and d will respectively denote a sub-Laplacian andan L-gauge on the homogeneous Carnot group G. We shall denote by Bd(x, r) the


d-ball with center x ∈ G and radius r ≥ 0 and by Q the homogeneous dimensionof G. We shall also assume, as usual, Q ≥ 3.

Given an open set O ⊆ RN and a real number r > 0, we let

Or := {x ∈ O | d-dist(x, ∂O) > r},where

d-dist(x, ∂O) := infy∈∂O

d(y−1 ◦ x).

Since the d-balls are connected, one easily recognizes that Bd(x, ρ) ⊆ O for everyx ∈ O and 0 < ρ < d-dist(x, ∂O). It follows that

Bd(x, ρ) ⊆ O ∀ x ∈ Or and 0 < ρ ≤ r .

Let ϕ : R → R be an L1-function vanishing out of the interval ]0, 1[ and such that∫R

ϕ = 1. For r > 0, define

ϕr(t) := 1

rϕ( t

r

), t ∈ R.

Let us now consider a function u ∈ C2(O). Then, if x ∈ Or , from (5.45) we obtain

u(x) =Mρ(u)(x)−Nρ(Lu)(x) for 0 < ρ ≤ r .

We now multiply both sides of this identity times ϕr(ρ) and integrate with respectto ρ. We thus get

u(x) = Φr(u)(x)−Φ∗r (Lu)(x), x ∈ Or, (5.50a)

where

Φr(u)(x) :=∫ ∞

0ϕr(ρ)Mρ(u)(x) dρ (5.50b)

and

Φ∗r (w)(x) :=∫ ∞

0ϕr(ρ)Nρ(w)(x) dρ. (5.50c)

By using the coarea formula, the average operator Φr can be written as follows (weagree to let u = 0 out of O)

Φr(u)(x) =∫

RN

u(z) φr(x−1 ◦ z) dz (5.50d)

withφr(z) := r−Q φ

(δ1/r (z)

)

and

φ(z) := (Q− 2)βd ΨL(z)ϕ(d(z))

d(z)Q−1. (5.50e)

We note that φ vanishes out of Bd(0, 1) and


∫

RN

φ(z) dz = (Q− 2)βd

∫ ∞

0

ϕ(ρ)

ρQ−1

(∫

d(z)=ρ

|∇Ld(z)|2|∇d(z)| dHN−1(y)

)dρ

=∫ ∞

0ϕ(ρ) dρ = 1.

Here we used (5.42), (5.43h), (5.43k) and (5.43m).If the function ϕ is smooth and its support is contained in ]0, 1[, then φ ∈

C∞0 (RN), supp(φ) ⊆ Bd(0, 1) and

x �→ Φr(u)(x)

is a smooth map in Or , whenever u is just an L1loc(O)-function. We would like to

explicitly remark that in the integral (5.50d) the function

z �→ φr(x−1 ◦ z) vanishes out of Bd(x, r).

As a consequence, if x ∈ Or , that integral is performed on a compact set containedin O, since Bd(x, r) ⊆ O.

If we choose

ϕ(t) ={

QtQ−1 if 0 < t < 1,

0 otherwise,

thenΦr(u) = Mr (u) and Φ∗r (w) = Nr (w),

where

Mr (u)(x) := md

rQ

∫

Bd(x,r)

ΨL(x−1 ◦ y) u(y) dy, (5.50f)

and

Nr (w)(x) := nd

rQ

∫ r

0ρQ−1

(∫

Bd(x,ρ)

(d2−Q(x−1 ◦ y)− ρ2−Q

)w(y) dy

)dρ,

(5.50g)being

md := Q(Q− 2)βd and nd := Qβd . (5.50h)

Hence, from (5.50a), we obtain the following theorem.

Theorem 5.6.1 (Solid mean value theorem). Let L be a sub-Laplacian on the ho-mogeneous Carnot group G, and let d be an L-gauge on G. Let O be an open subsetof G, and let u ∈ C2(O, R).

Then, for every x ∈ O and r > 0 such that Bd(x, r) ⊂ O, we have

u(x) = Mr (u)(x)− Nr (Lu)(x), (5.51)

where


Mr (u)(x) = md

rQ

∫

Bd(x,r)

ΨL(x−1 ◦ y) u(y) dHN(y),

Nr (w)(x) = nd

rQ

∫ r

0ρQ−1

(∫

Bd(x,ρ)

(d2−Q(x−1 ◦ y)− ρ2−Q

)w(y) dy

)dρ,

and md , nd and ΨL are defined in (5.50h) (see also (5.43m)) and (5.42).In particular, if Lu = 0, i.e. u is L-harmonic in O, we have

u(x) = Mr (u)(x). (5.52)

Remark 5.6.2. When L = Δ is the classical Laplace operator in RN , N ≥ 3, one has

ΨL ≡ 1. Then, in this case,

Mr (u)(x) = 1

ωN rN

∫

|x−y|<r

u(y) dy =: −−∫

|x−y|<r

u(y) dHN(y)

and (5.52) gives back the “Solid” Gauss theorem for classical harmonic func-tions. ��

The mean value theorems given by identities (5.47) and (5.52) characterize the L-harmonic functions. Indeed, the following Gauss–Koebe–Levi–Tonelli type theoremholds.

Theorem 5.6.3 (Gauss–Koebe–Levi–Tonelli type theorem). Let L be a sub-Lapla-cian on the homogeneous Carnot group G, and let d be an L-gauge.

Let O be an open subset of G, and let u : O → R be a continuous function.Assume that one of the following conditions is satisfied:

(i) u(x) =Mr (u)(x) for every x ∈ O and r > 0 such that Bd(x, r) ⊂ O,(ii) u(x) = Mr (u)(x) for every x ∈ O and r > 0 such that Bd(x, r) ⊂ O.

Then u ∈ C∞(O) andLu = 0 in O.

Proof. Assume condition (i) is satisfied. Then

u(x) = Mρ(u)(x) for 0 < ρ ≤ r. (5.53)

Let ϕ ∈ C∞0 (]0, 1[, R) be such that∫

Rϕ = 1. Multiply both sides of (5.53) times

ϕr(ρ) = ϕ(ρ/r)/r . An integration with respect to ρ gives

u(x) = Φr(u)(x) ∀ x ∈ Or,

where Φr(u) is the integral operator (5.50b). From (5.50b) we get

u(x) =∫

O

u(z) φr (x−1 ◦ z) dz,

where φr(z) = r−Q φ(δ1/r (z)) and φ is the smooth function defined by (5.50e). Itfollows that u ∈ C∞(O). Condition (i) and identity (5.45) now give Nr (Lu)(x) = 0


for every x ∈ O and r > 0 such that Bd(x, r) ⊂ O. Since the kernel appearing in theintegral operator Nr is strictly positive, this implies Lu = 0 in O. Thus, the theoremis proved if condition (i) is fulfilled.

Let us now assume (ii). Since ρ �→Mρ(u)(x) is continuous on ]0, r] and

Mr (u)(x) = Q

rQ

∫ r

0ρQ−1 Mρ(u)(x) dρ,

from (ii) we get

QrQ−1u(x) = d

dr(rQu(x)) = Q

d

dr

∫ r

0ρQ−1Mρ(u)(x) dρ

= QrQ−1 Mr (u)(x).

Hence u(x) =Mr (u)(x) for every x ∈ O and r > 0 such that Bd(x, r) ⊂ O. Thenu satisfies condition (i), so that u ∈ C∞(O) and Lu = 0 in O. ��Remark 5.6.4 (Another Koebe-type result). In the hypotheses of Theorem 5.6.3, ifu : O → R is continuous and satisfies

u(x) = Φr(u)(x) ∀ r > 0 ∀ x ∈ Or,

where Φr is the integral operator in (5.50d) related to a smooth function ϕ ∈C∞0 (]0, 1[, R) (see also (5.50b)), then u ∈ C∞(O). As a consequence, by identity(5.50a), Φ∗r (Lu) = 0 in Or for every r > 0. It follows that

Lu = 0 in O. ��From Theorem 5.6.1 we obtain another asymptotic formula for the sub-Lap-

lacians.

Theorem 5.6.5 (Asymptotic solid formula for L). Let L be a sub-Laplacian on thehomogeneous Carnot group G, and let d be an L-gauge.

Let Ω ⊆ G be open, and let u ∈ C2(Ω, R). Then, for every x ∈ Ω , we have

limr→0+

Mr (u)(x)− u(x)

r2= ad Lu(x), (5.54)

where ad = Qad/(Q+ 2) (and ad is as in (5.49b)).

Proof. From (5.50g) we obtain (by recalling the definition (5.49b) of ad )

Nr (1)(x) = Qβd

rQ

∫ r

0ρQ+1

(∫

Bd(0,1)

(d2−Q(y)− 1

)dy

)dρ = r2 ad .

Then, by Theorem 5.6.1 (arguing as in the proof of Theorem 5.5.7), we get

Mr (u)(x)− u(x) = ad r2(Lu(x)+ o(1)), as r → 0+,

and (5.54) follows. ��


5.7 Harnack Inequalities for Sub-Laplacians

In this section, we shall prove some type of Harnack inequalities for sub-Laplacians.Our main tool will be the solid average operator Mr in (5.50f).

Let d be an L-gauge on G, let cd be the positive constant of the pseudo-triangleinequality (see Proposition 5.1.8)

d(x−1 ◦ y) ≤ cd

(d(x−1 ◦ z)+ d(z−1 ◦ y)

) ∀ x, y, z ∈ G, (5.55)

and let ΨL = |∇Ld|2 be the kernel appearing in the average operator Mr in (5.50f).The following lemma will play a fundamental rôle in this section: it will allow us

to compare the average on different d-balls of a given non-negative function.

Lemma 5.7.1. Let L be a sub-Laplacian on the homogeneous Carnot group G, andlet d be an L-gauge. For every r > 0, there exists a point z0 = z0(r) ∈ G satisfyingthe following conditions:

(i) d(z0) = λ r ,(ii) ΨL(x−1 ◦ y) ≥ μ and ΨL(y−1 ◦ x) ≥ μ

for every x ∈ Bd(z0, r) and every y ∈ Bd(0, 2 cd r). Here λ ≥ 1 and μ > 0 arereal constants independent of r (depending only on G, d and L).

Proof. We split the proof into three steps.(I) The set

{y ∈ G \ {0} |ΨL(y) = 0}has empty interior. Indeed, suppose by contradiction ΨL(y) = 0 for every y in aneighborhood U of a suitable y0 �= 0. Then, since ΨL = |∇Ld|2, this gives ∇LΨL ≡0 on U , so that, by Proposition 1.5.6 (page 69), d is constant in an open set containingy0. As a consequence, the function r �→ d(δr (y0)) = r d(y0) is constant near r = 1.This implies d(y0) = 0, which is a contradiction with the assumption y0 �= 0.

(II) From step (I) and the homogeneity of ΨL we infer the existence of a pointy0 ∈ G, d(y0) = 1, such that

ΨL(y0) > 0 and ΨL(y−10 ) > 0.

Then, by the continuity of ΨL out of the origin, there exist two positive constants σ

and μ (with σ ≤ 1) such that

ΨL(ξ ◦ y0 ◦ η) ≥ μ, ΨL(ξ ◦ y−10 ◦ η) ≥ μ (5.56)

for every ξ , η ∈ G satisfying the inequalities d(ξ), d(η) ≤ 2 cd σ .(III) For any fixed r > 0, we let z0 := δλ r (y0) with λ = 1/σ . Let x ∈ Bd(z0, r)

and y ∈ Bd(0, 2 cd r). From the second inequality in (5.56) we obtain

ΨL(x−1 ◦ y) = ΨL((x−1 ◦ z0) ◦ z−1

0 ◦ y)

(by the homogeneity of ΨL)

= ΨL(δ1/(λ r)(x

−1 ◦ z0) ◦ y−10 ◦ δ1/(λ r)(y)

) ≥ μ,

5.7 Harnack Inequalities 263

since d(δ1/(λ r)(x−1 ◦ z0)) ≤ σ and d(δ1/(λ r)(y)) ≤ 2 cd σ . Analogously, from the

first inequality in (5.56) we obtain

ΨL(y−1 ◦ x) = ΨL(y−1 ◦ z0 ◦ (z−1

0 ◦ x))

= ΨL(δ1/(λ r)(y

−1) ◦ y0 ◦ δ1/(λ r)(z−10 ◦ x)

) ≥ μ,

since d(δ1/(λ r)(z−10 ◦ x)) ≤ σ and d(δ1/(λ r)(y

−1)) ≤ 2 cd σ . ��To state the next theorem, it is convenient to introduce the following constants

θ0 := cd(1+ cd(1+ λ)), θ := cd(λ+ θ0). (5.57)

Theorem 5.7.2 (Non-homogeneous Harnack inequality). Let L be a sub-Lapla-cian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G

be open. Finally, let x0 ∈ Ω and r > 0 be such that Bd(x0, θ r) ⊂ Ω (here θ is asin (5.57), see also Lemma 5.7.1 and (5.55)).

Then, for any p ∈ (Q/2,∞], we have

supBd(x0,r)

u ≤ c{

infBd(x0, r)

u+ r2−Q/p ‖ Lu ‖Lp(Bd(x0, θ r))

}(5.58)

for every non-negative smooth function u : Ω → R. Here c is a positive constantdepending only on G, d , L and p and not depending on u, r , x0 and Ω .

Proof. Since L is left invariant, we may assume x0 = 0.We split the proof in several steps. Mr will be the average operator in (5.50f) and

z0 = z0(r) the point of G given by Lemma 5.7.1.(I) There exists an absolute constant4 c > 0 such that

Mr (u)(x) ≤ c Mθ0 r (u)(z0) ∀ x ∈ Bd(0, r). (5.59)

Indeed, for every x ∈ Bd(0, r), we have

Bd(x, r) ⊆ Bd(0, 2 cd r) ⊆ Bd(0, θ0 r) ⊂ Ω,

whence (being u non-negative)

Mr (u)(x) ≤(

supG\{0}

ΨL) md

rQ

∫

Bd(x,r)

u(z) dz

(by the first inequality in Lemma 5.7.1-(ii))

≤ c1

rQ

∫

Bd(x,r)

ΨL(z−10 ◦ z) u(z) dz,

where c1 = md/μ supG\{0} ΨL. We remark that c1 <∞, since ΨL is smooth out ofthe origin and δλ-homogeneous of degree zero. Then, since we also have

4 Hereafter, we call absolute constant any positive real constant independent of u, r , x0and Ω .


Bd(x, r) ⊆ Bd(z0, θ0 r) ⊆ Bd(0, θ r) ⊂ Ω,

we get (again by the non-negativity of u)

Mr (u)(x) ≤ c1

rQ

∫

Bd(z0,θ0 r)

ΨL(z−10 ◦ z) u(z) dz = c1

md

Mθ0 r (u)(z0).

This proves (5.59) with c = c1/md .(II) There exists an absolute constant c > 0 such that

Mr (u)(z0) ≤ c Mθ0 r (u)(y) ∀ y ∈ Bd(0, r). (5.60)

This inequality can be proved just by proceeding as in the previous step, by using thesecond inequality in Lemma 5.7.1-(ii) and the inclusions

Bd(z0, r) ⊆ Bd(y, θ0 r) ⊆ Bd(0, θ r) ⊂ Ω.

(III) Let finally x, y ∈ Bd(0, r). Then, by repeatedly using the solid mean valuetheorem 5.6.1, we have

u(x) = Mr (u)(x)− Nr (Lu)(x) (by (5.59))

≤ cMθ0r (u)(z0)− Nr (Lu)(x)

= c(u(z0)+ Nθ0r (Lu)(z0)

)− Nr (Lu)(x)

= c(Mr (u)(z0)− Nr (Lu)(z0)+ Nθ0 r (Lu)(z0)

)−Nr(Lu)(x).

On the other hand, from (5.60),

Mr (u)(z0) ≤ c Mθ0 r (u)(y) = c(u(y)− Nθ0 r (Lu)(y)

).

By using this last estimate in the previous one, we infer that u(x) is bounded fromabove by a suitable absolute constant c times

u(y)+ |Nθ0 r (Lu)(y)| + |Nr (Lu)(z0)| + |Nθ0 r (Lu)(z0)| + |Nr(Lu)(x)|for all x, y ∈ Bd(0, r).

An elementary computation based on Hölder’s inequality shows that

|Nr (w)(x)| ≤ cp r2−Q/p ‖ w ‖Lp′(Bd(x, r))

(5.61)

for Q/2 < p ≤ ∞, p′ = p/(p − 1) and

cp := nd

(∫

d(z)<1(d(z)2−Q − 1) dz

)1/p′

.

Thus, keeping in mind the inclusions

Bd(y, θ0 r), Bd(z0, r), Bd(z0, θ0 r), Bd(x, r) ⊆ Bd(0, θ r),

from the upper bound of u(x) and from (5.61) we obtain (5.58) when x0 = 0. Thisends the proof. ��


Theorem 5.7.2 contains the following “homogeneous” Harnack inequality.

Corollary 5.7.3 (Homogeneous invariant Harnack inequality). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω

be an open subset of G, and let u : Ω → R be a non-negative smooth solution toLu = 0. Then

supBd(x0,r)

u ≤ c infBd(x0,r)

u (5.62)

for every x0 ∈ Ω and r > 0 such that Bd(x0, θ r) ⊂ Ω . The constant c depends onlyon G, L and d and does not depend on u, r , x0 and Ω .

By using a covering argument, from the non-homogeneous Harnack inequalityof Theorem 5.7.2 one obtains the following theorem.

Theorem 5.7.4 (Non-homogeneous, non-invariant Harnack inequality). Let L bea sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. LetΩ be an open subset of G, and let K and K0 be compact and connected subsets ofΩ such that K ⊂ int(K0).

Then, for every p ∈ (Q/2,∞], there exists a positive constant c = c(K,K0,Ω,

L, d, p,Q) such that

supK

u ≤ c{

infK

u+ ‖ Lu ‖Lp(K0)

}(5.63)

for every u ∈ C∞(Ω, R), u ≥ 0.

Proof. Let {Dj | j = 1, . . . , q} be a finite family of d-balls Dj = Bd(xj , rj ) suchthat (here θ is as in (5.57)):

(i) K ⊂⋃q

j=1 Dj ,(ii) θ Dj := Bd(xj , θ rj ) ⊂ K0 for any j = 1, . . . , q,

(iii) Dj ∩Dj+1 �= ∅ for every j ∈ {1, . . . , q − 1}.We explicitly remark that such a covering exists, since K is compact, connected

and contained in the interior of K0.By Theorem 5.7.2, we have

supDj

u ≤ c{

infDj

u+ ‖ Lu ‖Lp(θ Dj )

}, j = 1, . . . , q,

for every u ∈ C∞(Ω, R), u ≥ 0. The constant c is independent of u.Then, inequality (5.63) will follow by a repeated application of the following

elementary lemma. ��Lemma 5.7.5. Let A1 and A2 be arbitrary sets such that A1 ∩ A2 �= ∅. Supposeu : A1 ∩ A2 → R is a non-negative function satisfying

supAi

u ≤ c{

infAi

u+ Li

}, i = 1, 2, (5.64)


for suitable constants c≥1 and Li ≥ 0, i = 1, 2. Then

supA1∪A2

u ≤ c2{

infA1∪A2

u+ L1 + L2

}.

Proof. We have to show that

u(x) ≤ c{u(y)+ L1 + L2} (5.65)

for every x, y ∈ A1 ∪ A2. Now, if x, y ∈ A1 or x, y ∈ A2, then inequality (5.65)directly follows from (5.64). Suppose x ∈ A1 and y ∈ A2 and choose a point z ∈A1 ∩ A2. By hypothesis (5.64),

u(x) ≤ c {u(z)+ L1} and u(z) ≤ c {u(y)+ L2}.Hence

u(x) ≤ c {c (u(y)+ L2)+ L1},and (5.65) follows. ��

From Theorem 5.7.4 one obtains the following improvement of Theorem 5.7.2.

Corollary 5.7.6 (Non-homogeneous invariant Harnack inequality). Let L be asub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. LetΩ ⊆ G be open, and let r, R and R0 be real constants such that 0 < r < R < R0.Assume

r

R,

R

R0≤ ρ and Bd(x0, R0) ⊂ Ω

for suitable ρ < 1 and x0 ∈ Ω .Then, for every p ∈ (Q/2,∞], there exists a constant c > 0 such that

supBd(x0,r)

u ≤ c{

infBd(x0, r)

u+ R2−Q/p ‖ Lu ‖Lp(Bd(x0, R))

}(5.66)

for every u ∈ C∞(Ω, R), u ≥ 0. The constant c depends only on G, L, d , p and ρ

and does not depend on u, r , R, R0, Ω and x0.

Proof. Since L is left invariant, we may assume x0 = 0. Let us put

uR(x) := u(δR(x)), x ∈ δ1/R(Ω).

Then, by applying Theorem 5.7.4 to the function uR , the compact sets K =Bd(0, ρ) and K0 = Bd(0, 1) and the open set Bd(0, 1/ρ) (which is contained inBd(0, R0/R) ⊆ δ1/R(Ω)), we obtain

supBd(0,ρ)

uR ≤ c{

infBd(0,ρ)

uR+ ‖ LuR ‖Lp(Bd(0,1))

}

= c{

infBd(0,ρ)

uR + R2−Q/p ‖ Lu ‖Lp(Bd(0,R))

},


with c > 0 depending only on the parameters in the assertion of the corollary. Notethat we have also used the δλ-homogeneity of L (of degree 2). From this inequal-ity (5.66) follows, since

supBd(0,ρ)

uR = supBd(0,R ρ)

u ≥ supBd(0,r)

u and infBd(0,ρ)

uR ≤ infBd(0,r)

u.

This ends the proof. ��Theorem 5.7.4 and the δλ-homogeneity of L easily imply the following Harnack

inequality on rings.

Corollary 5.7.7 (Harnack inequality on rings). Let L be a sub-Laplacian on thehomogeneous Carnot group G, and let d be an L-gauge. Let u be a smooth non-negative function on the ring

AR,c :={x ∈ G | cR < d(x) < R/c

},

where R > 0 and 0 < c < 1. Let 0 < a < b < c. Then, for every p ∈ (Q/2,∞],there exists a constant c > 0 such that

supAR,a

u ≤ c{

infAR,a

u+ R2−Q/p ‖ Lu ‖Lp(AR,b)

}. (5.67)

The constant c depends only on G, L, p, d , a, b and c and does not depend on u

and R.

Proof. The change of variable x �→ δR(x) reduces (5.67) to the analogous inequalitywith R = 1. This last one follows from Theorem 5.7.4. ��Remark 5.7.8 (The Harnack inequality in the abstract setting). According the con-vention in the incipit of the chapter, given an abstract stratified group H, via theisomorphism Ψ between H and a homogeneous Carnot group G, all the Harnackinequalities of the present section do possess a counterpart in H.

For example, it suffices to consider the results in Remark 2.2.28 in order to derivethe following result from Theorem 5.7.4.

Theorem 5.7.9 (A Harnack inequality in the abstract setting). Let H be an ab-stract stratified group, and let L be a sub-Laplacian on H. Let Ω be an open subsetof H, and let K be a compact and connected subset of Ω .

Then there exists a positive constant c = c(H,L,Ω,K) such that

supK

u ≤ c infK

u,

for every non-negative function u ∈ C∞(Ω, R) satisfying Lu = 0 in Ω .

We close this section by giving the following “monotone convergence” theorem.


Theorem 5.7.10 (The Brelot convergence property). Let H be an abstract strati-fied group, and let L be a sub-Laplacian on H. Let Ω ⊆ H be open and connected.Let {un}n∈N be a sequence of L-harmonic functions in Ω , i.e.

un ∈ C∞(Ω, R) and Lun = 0 in Ω for every n ∈ N.

Assume the sequence {un}n∈N is monotone increasing and

supn∈N

{un(x0)} <∞ (5.68)

at some point x0 ∈ Ω . Then there exists an L-harmonic function u in Ω such that{un}n∈N is uniformly convergent on every compact subset of Ω to u.

Proof. By the results in Remark 2.2.28, it suffices5 to consider the case when L is asub-Laplacian on a homogeneous Carnot group G.

Let K be a compact subset of Ω . Since Ω is connected, there exists a compactand connected set K∗ such that

K ⊆ K∗ ⊂ Ω and x0 ∈ K0.

Then, by Theorem 5.7.4,

supK

(un − um) ≤ supK∗

(un − um) ≤ cinfK∗(un − um)

≤ c(un(x0)− um(x0)) for every n ≥ m ≥ 1.

The constant c is independent of n and m. Then, by condition (5.68), {un}n is uni-formly convergent on K . Since K is an arbitrary compact subset of Ω , {un}n∈N islocally uniformly convergent to a continuous function u : Ω → R. On the otherhand, by the solid mean value Theorem 5.6.1, for every x ∈ Ω and r > 0 such thatBd(x, r) ⊂ Ω , we have

un(x) = Mr (un)(x) ∀ n ∈ N.

Letting n tend to infinity (by the uniform convergence un → u), we get

u(x) = Mr (u)(x) ∀ x ∈ Ω, r > 0 : Bd(x, r) ⊂ Ω,

and now the Koebe-type Theorem 5.6.3 implies

u ∈ C∞(Ω, R) and Lu = 0 in Ω . ��5 Indeed, following the notation in Remark 2.2.28 and in the assertion of the above theorem,

the following facts hold: the (abstract) sub-Laplacian L is Ψ -related to the sub-LaplacianL on G; set un := un ◦Ψ , Ω := Ψ−1(Ω), x0 := Ψ−1(x0), then Ω is open and connectedin G (recall that Ψ is a homeomorphism), the sequence {un}n∈N is monotone increasingon Ω , bounded in x0 and Lun = 0 on Ω . Finally, if K is a compact subset of H, thenK := Ψ−1(K) is a compact subset of G, and

supK

|un − u| = supK

|un − u|.

5.8 Liouville-type Theorems 269

The above Brelot convergence property implies the following strong minimumprinciple (see also Section 5.13 for a more exhaustive investigation of maximum–minimum principles).

Corollary 5.7.11 (Strong minimum principle). Let L be a sub-Laplacian on anabstract stratified group H. A non-negative solution to Lu = 0 on an open connectedset Ω ⊆ H vanishes identically iff it vanishes at a point.

Proof. Apply the result of Theorem 5.7.10 to the sequence {n · u | n ∈ N}. ��

5.8 Liouville-type Theorems

The classical Liouville theorem for entire harmonic functions also holds in the sub-Laplacian setting. Indeed, the Harnack inequality of Corollary 5.7.3 implies the fol-lowing theorem:

Theorem 5.8.1 (Liouville theorem for sub-Laplacians). Let H be an abstract strat-ified group. Let L be a sub-Laplacian on H. Let u ∈ C∞(H, R) be a function satis-fying

u ≥ 0 and Lu = 0 in H.

Then u is constant.

Proof. By the results in Remark 2.2.28, it suffices6 to consider the case when L is asub-Laplacian on a homogeneous Carnot group G. Define

m := infG

u and v := u−m.

Then v ≥ 0 and Lv = 0 in G. From Harnack inequality (5.62) we obtain

supBd(0,r)

v ≤ c infBd(0,r)

v, c independent of r .

From this inequality, letting r tend to infinity, we obtain

0 ≤ supG

v ≤ c infG

v = 0,

which implies v ≡ 0 and u ≡ m. ��Theorem 5.8.1 can be viewed as a particular case of the following stronger ver-

sion of the Liouville property for L.

6 Indeed, the (abstract) sub-Laplacian L is Ψ -related to the sub-Laplacian L on G and, setu := u ◦ Ψ , we have u ≥ 0 and Lu = 0 on G. Once we have proved that u is constant onG, the same follows for u = u ◦ Ψ−1.


Theorem 5.8.2 (Liouville-type theorem-polynomial lower bound). Let L be asub-Laplacian on the homogeneous Carnot group G. Let u be an entire solutionto Lu = 0, i.e. a function u ∈ C∞(G, R) such that Lu = 0 in G. Assume there existsa polynomial function p on G such that

u ≥ p in G.

Then7 u is a polynomial function and degG u ≤ degG p.

The inequality degG u ≤ degG p can be strict: take, for example, u ≡ 0, p =−x2

1 , so that degG u = 0 < 2 = degG p.

Remark 5.8.3. All the results in this section involving polynomial functions are ob-viously related to the fixed coordinates on a homogeneous Carnot group.

Abstract counterparts of these results are available in the obvious way. A poly-nomial function on the abstract stratified group H is a function P : H → R suchthat p := P ◦ Exp is a polynomial function on the vector space h (the Lie algebraof H), i.e. p is a polynomial function when expressed in coordinates w.r.t. any (orequivalently, w.r.t. at least one) basis for h. All the details are left to the reader. ��

Theorem 5.8.2 is an easy consequence of the following one.

Theorem 5.8.4 (Liouville-type theorem-polynomial sub-Laplacian). Let L be asub-Laplacian on the homogeneous Carnot group G. Let u be a smooth functionon G satisfying

u ≥ 0 and Lu = w in G,

where w is a polynomial function. Then u is a polynomial function and

degG u ≤ 2+ degG w.

More precisely,

degG u ={

degG w if degG u = 0,

2+ degG w if degG u ≥ 2.(5.69)

The case degG u = 1 cannot occur, since a polynomial of G-degree 1 cannot be anon-negative function.

The proof of this theorem will follow from a representation formula that we shalldeduce from identity (5.50a) in Section 5.6.

We first show how Theorem 5.8.2 can be obtained from Theorem 5.8.4.

Proof (of Theorem 5.8.2). We first recall that L is a differential operator, δλ-homo-geneous of degree two, and its coefficients are polynomial functions. Then, if u ≥ p

and Lu = 0, we haveu− p ≥ 0 and L(u− p) = w,

7 See Definition 1.3.3 on page 33 for the definition of the G-degree degG (p) of a polynomialp and the G-length |α|G of the multi-index α.


where w := −Lp is a polynomial whose G-degree does not exceed

max{0, degG p − 2}.From Theorem 5.8.4 it follows that u− p is a polynomial and that

degG (u− p) ≤ 2+ degG (w) ≤ 2+max{0, degG p − 2} = max{2, degG p}.Hence, u = (u− p)+ p is a polynomial function and its G-degree does not exceedmax{2, degG p}.

If degG p ≥ 2, this gives the assertion of Theorem 5.8.2. It remains to considerthe cases degG p = 0 and degG p = 1 (indeed, in these cases the above argumentonly proves that degG u ≤ 2, which is weaker than degG u ≤ degG p). In both cases,Lp ≡ 0, so that the hypotheses of Theorem 5.8.2 rewrites as

u− p ≥ 0, L(u− p) = 0 in G.

Hence, by Liouville Theorem 5.8.1, u− p is constant, whence

u = (u− p)+ p = p + constant,

so that u is a polynomial function of the same G-degree as p. ��We next prove a representation formula having its own interest and useful for the

proof of Theorem 5.8.4.

Proposition 5.8.5. Let L be a sub-Laplacian on the homogeneous Carnot group(G, ∗), and let d be an L-gauge on G. Let u ∈ C∞(G, R) be such that

Lu = w in G, (5.70a)

where w is a polynomial function. Then

u(x) = Φr(u)(x)−∑

|α|G≤m

C(α)Q w(α)(x) r2+|α|G (5.70b)

for any x ∈ G and r > 0. Φr denotes the integral operator (5.50b) related to asmooth function ϕ ∈ C∞0 ((0, 1), R). Moreover, m is the G-degree of w,

w(α)(x) := Dαy |y=0w(x ∗ y)

and, for any α with |α|G ≤ m, C(α)Q is a positive constant depending only on α,

Q and d . In particular, w(α) is a polynomial function with G-degree not exceedingm− |α|G.

Proof. Identity (5.50a) in Section 5.6 and hypothesis (5.70a) give

u(x) = Φr(u)(x)−Φ∗r (w)(x),


where Φ∗r (w)(x) = ∫∞0 ϕr(ρ)Nρ(w)(x) dρ, and

Nρ(w)(x) = βd

∫

Bd(x,ρ)

(d2−Q(x−1 ∗ y)− ρ2−Q)w(y) dHN(y)

= βd

∫

Bd(0,ρ)

(d2−Q(y)− ρ2−Q)w(x ∗ y) dHN(y).

We now claim that

w(x ∗ y) =∑

|α|G≤m

w(α)(x)

α! yα, (5.71)

where w(α) = Dαy |y=0w(x ∗ y) is a polynomial function of G-degree ≤ m − |α|G.

Taking this claim for granted for a moment, we have

Φ∗r (w)(x)

=∑

|α|G≤m

w(α)(x)

α! βd

∫ ∞

0ϕ

(ρ

r

)(∫

Bd(x,ρ)

(d2−Q(y)− ρ2−Q) yα dHN(y)

)dρ

r

(by using the change of variables y = δρ(z) and ρ = r σ )

=∑

|α|G≤m

C(α)Q w(α)(x) r2+|α|G ,

where

C(α)Q = βd

α!∫ ∞

0ϕ(σ)

(∫

Bd(0,1)

(d2−Q(z)− 1) zα dHN(z)

)dσ.

Then, we are left with the proof of (5.71). Since w is a polynomial function and(x, y) �→ x ∗ y has polynomial components too, one has

w(x ∗ y) =∑

|α|G+|β|G≤n

cα,β xα yβ

for a suitable positive integer n and real constants cα,β . We have to prove only thatn ≤ m. Now, since degG w ≤ m,

w(z) =∑

|γ |G≤m

cγ zγ , cγ ∈ R for any γ .

Then, since δλ is an automorphism of the group G,∑

|α|G+|β|G≤n

cα,β λ|α|G+|β|G xα yβ = w(δλ(x) ∗ δλ(y))

= w(δλ(x ∗ y)) =∑

|γ |G≤m

cγ λ|γ |G (x ∗ y)γ

for every x, y ∈ G and λ > 0. As a consequence,


∑

m<|α|G+|β|G≤n

cα,β xα yβ = 0 ∀ x, y ∈ G,

so thatw(x ∗ y) =

∑

|α|G+|β|G≤m

cα,β xα yβ,

and the claim follows. This completes the proof. ��We are now in the position to prove Theorem 5.8.4.

Proof (of Theorem 5.8.4). In the Harnack inequality (5.58) of Theorem 5.7.2, takex0 = 0, r = 2 |x| and p = ∞. We obtain

u(x) ≤ c{u(0)+ |x|m+2}, (5.72)

where c is a positive absolute constant.Let us recall the notation used in Corollary 1.5.5 (page 68). If L is the sum of

squares of the vector fields X1, . . . , XN1 and if β = (i1, . . . , ik) is a multi-index withcomponents in {1, . . . , N1}, we set

Xβ := Xi1 ◦ · · · ◦Xik and |β| = k.

Let us now use the representation formula of Proposition 5.8.5. Since w(α) in (5.70b)has G-degree ≤ m− |α|, for any non-negative multi-index β with |β| > m, we have

Xβu(x) = Xβ(Φr(u)(x)

) ∀ x ∈ G.

Then, since the Xj ’s are left-invariant on (G, ∗) and δλ-homogeneous of degree one,we have (see also (5.50d) and (5.50e))

Xβu(x) =∫

G

u(y)Xβ(φr(x

−1 ∗ y))

dy

= r−|β|∫

RN

u(y)(Xβφ

)r(y−1 ∗ x) dy,

where φ(z) := φ(z−1) and(Xβφ

)r= r−Q

(Xβφ

) ∗ δ1/r .

Hence,

Xβu(x) = r−|β|∫

Bd(0,1)

u(x ∗ δr (z−1))

(Xβφ

)(z) dz. (5.73)

Using inequality (5.72) in (5.73), we obtain

|Xβu(x)| ≤ c r−|β| (1+ |x| + r)m+2

for every x ∈ G and r > 0. The constant c depends on u(0), but it is independent ofx and r . Letting r tend to infinity, we obtain

Xβu(x) = 0 ∀ x ∈ G

and for every β with |β| > m+ 2. By the cited Corollary 1.5.5, this implies that u isa polynomial function of G-degree ≤ m+ 2. ��


Using the same argument as above, we can prove the following improvement ofTheorem 5.8.4.

Theorem 5.8.6 (Liouville: polynomial sub-Laplacian and bound). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u : G → R be a smoothfunction satisfying

u ≥ p and Lu = w in G,

where p and w are polynomial functions. Then u is a polynomial and

degG u ≤ max{degG p, 2+ degG w}.Proof. Set v = u− p. We have

v ≥ 0, Lv = w − Lp in G.

Since L has polynomial coefficients, we can apply Theorem 5.8.4 to derive

degG (u− p) ≤ 2+ degG (w − Lp) ≤ 2+max{degG w, degG Lp}≤ 2+

{max{degG w, degG p − 2} if degG p ≥ 2,

max{degG w, 0} if degG p ≤ 1

={

max{2+ degG w, degG p} if degG p ≥ 2,

2+ degG w if degG p ≤ 1= max{degG p, 2+ degG w}. ��

Summing up all the above statements, we obtain the following Liouville theorem“of polynomial type”.

Theorem 5.8.7 (Liouville-type theorem). Let L be a sub-Laplacian on the homo-geneous Carnot group G. Let u : G→ R be smooth and such that

u ≥ p and Lu = w in G,

where p and w are polynomial functions. Then u is a polynomial, and

degG u ≤{

degG p if w ≡ 0,

max{degG p, 2+ degG w} otherwise.

5.8.1 Asymptotic Liouville-type Theorems

We close this section by giving some more “asymptotic” Liouville-type theorems,easy consequences of Theorem 5.8.2.

Theorem 5.8.8 (Asymptotic Liouville. I). Let L be a sub-Laplacian on the homo-geneous Carnot group G. Let u be an entire solution to Lu = 0. Assume there existsa real number m ≥ 0 such that

u(x) = O(�m(x)), as x →∞. (5.74)


Then u is a polynomial function and

degG u ≤ [m]. (5.75)

Here � is (any fixed) homogeneous norm on G and [m] denotes the integer part ofm, i.e. [m] ∈ Z and [m] ≤ m < [m] + 1.

Proof. By condition (5.74) and Proposition 5.1.4, we get

u(x) ≥ p(x) ∀ x ∈ G,

where

p(x) = −c

(1+

r∑

j=1

|x(j)|2 r!/j)[m]+1

,

and c is a suitable positive constant. Then, by Theorem 5.8.2, u is a polynomialfunction. Let n := degG u. Assume, by contradiction, n ≥ [m] + 1. Writing

u(x) =n∑

k=0

uk(x),

where uk is δλ-homogeneous of degree k, from condition (5.74) we get

n∑

k=0

�(x)k−n uk

(δ1/�(x)(x)

) = u(x)

�(x)n−→ 0, as �(x)→∞.

Hence, un(y) = 0 for every y ∈ G such that �(y) = 1, which implies degG u ≤ n−1,a contradiction. Then n ≤ [m] and the proof is complete. ��

Theorem 5.8.8 together with Theorem 5.6.1 give the following corollary.

Corollary 5.8.9 (Asymptotic Liouville. II). Let L be a sub-Laplacian on the homo-geneous Carnot group G, and let d be an L-gauge.

Let u be an entire solution to Lu = 0. Assume there exists x0 ∈ G and a realnumber m ≥ 0 such that

−∫

Bd(x0,r)

|u(y)| dy = O(rm), as r →∞. (5.76)

Then u is a polynomial function and

degG u ≤ [m].In (5.76) we have set

−∫

D

:= 1

|D|∫

D

.


Proof. Let x ∈ G \ {0} and put r = max{d(x), d(x0)}. By the pseudo-triangleinequality (5.55), Bd(x, r) ⊆ Bd(x0, 2 cd r). Then, from the solid mean value Theo-rem 5.6.1, we obtain

|u(x)| = |Mr (u)(x)| ≤ Mr (|u|)(x)

≤ md

rQ

(sup

G\{0}ΨL) ∫

Bd(x0,2cd r)

|u(y)| dy,

so that

|u(x)| ≤ c −∫

Bd(x0,2cd r)

|u(y)| dy,

being c > 0 independent of x and r . This inequality and (5.76) give

u(x) = O(rm) = O(d(x)m), as x →∞.

Then, by Theorem 5.8.8, u is a polynomial function with the G-degree ≤ [m]. ��Remark 5.8.10. By using Proposition 5.1.4, one easily recognizes that condition(5.76) in the previous corollary is equivalent to the following one

∫

Br

|u(y)| dy = O(rm+Q), as r →∞, (5.77)

where Br := {y ∈ G : �(y) < r} is the ball of radius r centered at the origin, withrespect to a homogeneous norm � on G.

Corollary 5.8.11 (Asymptotic Liouville. III). Let L be a sub-Laplacian on the ho-mogeneous Carnot group G. Let u be an entire solution to Lu = 0, and 1 ≤ p ≤ ∞.Assume (with the notation of Remark 5.8.10)

‖ u ‖Lp(Br )= O(rm+Q/p), as r →∞. (5.78)

Then u is a polynomial function of the G-degree not exceeding [m].Proof. The Hölder inequality gives

∫

Br

|u(y)| dy ≤ c rQ(1−1/p) ‖ u ‖Lp(Br ) .

Then (5.78) implies (5.77), and the assertion follows from Remark 5.8.10 and Corol-lary 5.8.9. ��

5.9 Some Results on GGG-fractional Integrals and the Sobolev–SteinEmbedding Inequality

Let L be a sub-Laplacian on the homogeneous Carnot group G of homogeneousdimension Q. Let 0 < α < Q. Given a function f : G→ R, we formally define

5.9 Sobolev–Stein Embedding Inequality 277

Iα(f )(x) :=∫

G

f (y)

(d(x, y))Q−αdy,

where d(x, y) stands for d(y−1 ◦ x) and z �→ d(z) is an L-gauge function on G.By analogy with the Euclidean setting, we shall call Iα the G-fractional integral oforder α.

The following theorem, when G is the usual Euclidean group (RN,+), givesback a celebrated theorem by Hardy, Littlewood and Sobolev.

Theorem 5.9.1 (Hardy–Littlewood–Sobolev for sub-Laplacians). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Suppose1 < α < Q and 1 < p <

Qα

. Let q > p be defined by

1/q = 1/p − α/Q.

Then there exists a positive constant C = C(α, p, G,L, d) such that

‖Iα(f )‖q ≤ C ‖f ‖p for every f ∈ Lp(G).

Here, we use the notation ‖ · ‖r to denote the Lr norm in G ≡ RN with respect to

the Lebesgue measure.

In the classical Euclidean setting, several proofs of this theorem are known. Thesimplest proof seems to be due to L.I. Hedberg [Hed72] and it makes use of themaximal function theorem by Hardy–Littlewood–Wiener. This last theorem holds ingeneral metric spaces equipped with a doubling measure.8 In particular, it holds inour context.

Fixed a sub-Laplacian L and an L-gauge d , we define the L-maximal functionML(f ) of a function f ∈ Lp(G, C), 1 < p <∞, as follows

ML(f )(x) := supr>0

−∫

Bd(x,r)

|f (y)| dy, x ∈ G,

where we used the notation

−∫

D

= 1

|D|∫

D

.

The function x �→ ML(f )(x) is lower semicontinuous. Indeed, if ML(f )(x) > α,there exists r > 0 such that −∫

Bd(x,r)|f (y)|dy > α. Then, since

x �→ −∫

Bd(x,r)

|f (y)| dy

is a continuous function (see Ex. 5 at the end of this chapter), there exists δ > 0 suchthat

8 A Radon measure μ on a quasi-metric space (X, d) is doubling if there exists a positiveconstant Cd such that

0 < μ(B(x, 2r)) ≤ Cd μ(B(x, r))

for every d-ball with center at x and radius r .


ML(f )(z) ≥ −∫

Bd(z,r)

|f (y)| dy > α if d(x−1 ◦ z) < δ.

The L-maximal function theorem is the following one.

L-Maximal Function Theorem. Let 1 < p < ∞. With the above notation, thereexists a positive constant C = C(p, G,L, d) such that

‖ML(f )‖p ≤ C ‖f ‖p for every f ∈ Lp(G, C).

A proof of this theorem in general doubling metric spaces (which is out of ourscopes here) can be found in the monograph [Ste81, Chapter 2], by E.M. Stein.

Starting from this result, we now prove Theorem 5.9.1 by using the idea inL.I. Hedberg’s paper [Hed72].

Proof (of Theorem 5.9.1). For every fixed t > 0 and x ∈ G, we have

Iα(f )(x) =(∫

d(x,y)≤t

+∫

d(x,y)≥t

)|f (y)| (d(x, y))α−Q dy

=: I (t)α (f )(x)+ E(t)

α (f )(x).

The Hölder inequality gives

E(t)α (f )(x) ≤ ‖f ‖p

(∫

d(x,y)≤t

(d(x, y))(α−Q)p/(p−1) dy

)1−1/p

= (see (5.36)) C ‖f ‖p(∫ ∞

t

τQ−1+(α−Q)p/(p−1) dτ

)1−1/p

= C′ ‖f ‖p tα−Q/p.

On the other hand, one has

I (t)α (f )(x) =

∞∑

k=0

∫

{2−k−1<d(x,y)/t≤2−k}|f (y)| (d(x, y))α−Q dy

≤ C

∞∑

k=0

(t/2k

)α−Q∫

{d(x,y)≤t2−k}|f (y)| dy

≤ C′∞∑

k=0

(t/2k

)α−Q (2−k t)Q

ML(f )(x)

= C′ tα ML(f )(x)

∞∑

k=0

2−k α.

Then, summing up the above estimates, we derive

Iα(f )(x) ≤ C(tα−Q/p ‖f ‖p + tα ML(f )(x)

), x ∈ G,

for every t > 0. We now choose

5.9 Sobolev–Stein Embedding Inequality 279

t =( ‖f ‖p

ML(f )(x)

)p/Q

,

so thatIα(f )(x) ≤ C ‖f ‖(p α)/Q

p (ML(f )(x))1−(p α)/Q.

Hence, being q (1− pα/Q) = p, we have

‖Iα(f )‖qq ≤ C‖f ‖(pqα)/Qp

∫

G

(ML(f )(x))p dx

(by the L-maximal function theorem)

≤ C‖f ‖(p q α)/Qp ‖f ‖pp = C‖f ‖qp,

since (p q α)/Q+ p = q. The theorem is proved. ��From this theorem and the representation formula (5.16), one easily obtains an

inequality extending the classical Sobolev embedding theorem to the homogeneousCarnot groups.

Theorem 5.9.2 (Sobolev–Stein embedding). Let L be a sub-Laplacian on the ho-mogeneous Carnot group G of homogeneous dimension Q.

Suppose 1 < p < Q. Then there exists a positive constant C = C(p, G,L) suchthat

‖u‖q ≤ C ‖∇Lu‖p for every u ∈ C∞0 (RN, R),

where

1/q = 1/p − 1/Q

(i.e. q = Qp

Q− p

).

Proof. Let u ∈ C∞0 (RN, R). Using the representation formula (5.16), we have

u(x) = −∫

G

Γ (x−1 ◦ y)Lu(y) dy.

Keeping in mind that L = ∑mj=1 X2

j and X∗j = −Xj , by integrating by parts at theright-hand side, we obtain

u(x) =∫

RN

(∇LΓ )(x−1 ◦ y)∇Lu(y) dy. (5.79)

On the other hand, out of the origin, we have

∇LΓ = βd ∇L(d2−Q) = (2−Q)βd d1−Q ∇Ld,

so that, since ∇Ld is smooth in G \ {0} and δλ-homogeneous of degree zero,

|∇LΓ | ≤ C d1−Q,

for a suitable constant C > 0 depending only on L. Using this inequality in (5.79),we get


|u(x)| ≤ C

∫

G

|∇Lu(y)| d(x, y)1−Q dy = C I1(|∇Lu|)(x).

Then, by the Hardy–Littlewood–Sobolev Theorem 5.9.1,

‖u‖q ≤ C ‖I1(|∇Lu|)‖q ≤ C ‖∇Lu‖p,

where

1/q = 1/p − 1/Q

(⇔ q = Qp

Q− p

).

This ends the proof. ��

5.10 Some Remarks on the Analytic Hypoellipticity ofSub-Laplacians

In this section, we collect some results on the hypoellipticity (especially in the an-alytic sense) of sub-Laplacians. It is far from our scopes here to give proofs of theresults of this section, the interested reader will be properly referred to the existingliterature.

First of all, we recall the relevant definition.

Definition 5.10.1 ((Analytic) hypoellipticity). We say that a differential operator L

defined on an open set Ω ⊆ RN is hypoelliptic (respectively, analytic hypoelliptic)

in Ω if, for every open set Ω ′ ⊆ Ω and every f ∈ C∞(Ω ′, R) (respectively, f realanalytic in Ω ′), any solution u to the equation

Lu = f on Ω ′ (in the weak sense of distributions)

is of class C∞(Ω ′, R) (respectively, is real analytic on Ω ′).

In the sequel, we shall write u ∈ Cω(Ω) to mean that u is real analytic on Ω .Moreover, we may also use the notation C∞-hypoelliptic and Cω-hypoelliptic tomean, respectively, hypoelliptic and analytic hypoelliptic.

In the very special case of a homogeneous differential operator L with constantcoefficients in R

N , the problem of hypoellipticity is completely solved by the follow-ing result (see, e.g. [Hor69]).

Let L be a homogeneous differential operator with constant coefficients in RN .

Then the following statements are equivalent:

1) L is C∞-hypoelliptic in RN ,

2) L is Cω-hypoelliptic in RN ,

3) L is elliptic in RN .

Moreover, if L has constant coefficients (but it is not necessarily homogeneous),then the equivalence of (2) and (3) still holds true.

As a consequence of the above result, all sub-Laplacians on the Euclidean groupE = (RN,+) (see Section 4.1.1) are C∞ and Cω-hypoelliptic.

5.10 Analytic Hypoellipticity of Sub-Laplacians 281

We next focus our attention to the C∞-hypoellipticity. Let {X1, . . . , Xm} be C∞-vector fields on R

N . We recall the well-known rank (or bracket) condition (alsoreferred to as Hörmander’s hypoellipticity condition):

dim(Lie{X1, . . . , Xm}I (x)

) = N ∀ x ∈ RN,

i.e. for every x ∈ RN , there exists a set of N iterated brackets of the Xi’s which

are linearly independent at x. The following celebrated result holds (see Hörmander[Hor67]).

If {X1, . . . , Xm} are C∞-vector fields on RN , then the rank condition is suffi-

cient for the C∞-hypoellipticity of the operator L = ∑nj=1 X2

j . Moreover, if thecoefficients of the Xj ’s are analytic, then the rank condition is also necessary for theC∞-hypoellipticity of L.

(See also Derridj [Der71], Helffer–Nourrigat [HN79], Kohn [Koh73], Oleınik–Radkevic [OR73], Rothschild–Stein [RS76].) See also Bony [Bon69,Bon70] for apartial converse of the rank condition, namely

If the sum of squares L of smooth vector fields is C∞-hypoelliptic, then the rankcondition holds on an open set, dense in R

N .(See also Fediı [Fed70], Kusuoka–Stroock [KuSt85], Bell–Mohammed [BM95]

for examples of sums of squares which are C∞-hypoelliptic but do not satisfy therank condition everywhere.)

The problem of C∞-hypoellipticity is thus completely solved for any sub-Laplacian L on any homogeneous Carnot group, since L is a sum of squares ofpolynomial vector fields satisfying the rank condition. It is also interesting to remarkthat, for any homogeneous left-invariant differential operator L on a stratified Liegroup (hence in particular for our sub-Laplacians), the C∞-hypoellipticity of L isequivalent to a Liouville-type property for L, namely the property stating that theonly bounded functions u on G such that Lu = 0 are the constant functions. (See[Rot83]; see also [HN79,Gel83].)

We finally turn our attention to the Cω-hypoellipticity. The problem of analytichypoellipticity is more involved and only partial results are known. To begin with,we consider the rank condition, which played a central rôle for C∞-hypoellipticity.Unfortunately, if L is a sum of squares of analytic vector fields, then the rank-condition is not sufficient for analytic hypoellipticity. (See, for instance, Trèves[Tre78], Tartakoff [Tar80], Grigis–Sjöstrand [GS85]; see also explicit counterex-amples in [BG72,Hel82,PR80,HH91,Chr91].) In the sequel of the section, we col-lect some of these results (in particular, several explicit negative ones) for our sub-Laplacians on Carnot groups. The first one is encouraging:

The canonical sub-Laplacian on the Heisenberg–Weyl group Hn is analytic hy-

poelliptic.It is not difficult to prove this result making use of the real analyticity (out of

the origin) of the fundamental solution Γ for the canonical sub-Laplacian on Hn, for

instance (Q = 2n+ 2 denotes the homogeneous dimension of Hn)

Γ (x, t) = cQ

1

(|x|4 + |t |2)(Q−2)/4. (5.80)


Since (see Kaplan [Kap80]; see also Example 5.4.7, page 250, and Chapter 18) thefundamental solution Γ for the canonical sub-Laplacian on every H-type group hasexactly the same form as in (5.80), then the canonical sub-Laplacian on any H-typegroup is analytic-hypoelliptic.

This is only a partial result of what is true on (a class of operators containingthe) Heisenberg-type groups, as we shall see below; but, unfortunately (as we shallalso see in a moment), it must be soon realized that Cω-hypoellipticity rarely occurswithin the non-Euclidean setting of Carnot groups.

To this end, we cite a first “negative” result (see Helffer [Hel82]):If G is a Carnot group of step two, and L is a sub-Laplacian on G, then a nec-

essary condition for L to be Cω-hypoelliptic is that G is a HM-group. (See Defini-tion 3.7.3, page 174, for the definition of HM-group.)

Hence, at least within the setting of step-two Carnot groups, in order to finda Cω-hypoelliptic sub-Laplacian, we must restrict our attention to the sub-classof HM-groups. Fortunately, a complete answer on the Cω-hypoellipticity for sub-Laplacians is available on HM-groups. This is given by the following result (seeMétivier [Met81]).

If G is a HM-group and L ∈ Um(g) (i.e. L is a homogeneous operator of degreem in the relevant enveloping algebra), then L is Cω-hypoelliptic if and only if L isC∞-hypoelliptic.

Consequently, since any sub-Laplacian L on G belongs to U2(g) and L is C∞-hypoelliptic by the rank condition, then L is also Cω-hypoelliptic.

This result covers quite large classes of remarkable cases: for example, sincethe Heisenberg–Weyl groups, the Iwasawa-groups, and (more generally) the H-typegroups all belong to the HM-group class, we now know that all their sub-Laplaciansare Cω-hypoelliptic. On the converse, it is easy to exhibit a step-two group where nosub-Laplacian is real analytic: take any Carnot group where the first layer has odddimension (for, in that case, it cannot be a HM-group; see Example 5.10.2 below).

The problem finally rest on the investigation of Carnot groups of step strictlygreater than two. Unfortunately, a complete answer to the analytic-hypoellipticity inthat case is still an open problem. Quoting Rothschild [Rot84], it is reasonable toconjecture that if G is not a HM-group, then there is no L ∈ Um(g) (hence, no sub-Laplacian) which is analytic-hypoelliptic. For example, from a result by M. Christ[Chr93, Theorem 1.5] we infer that if G is a filiform Carnot group of dimension≥ 4,then no sub-Laplacian on G is analytic-hypoelliptic. In Examples 5.10.4, 5.10.5,5.10.6 below, we exhibit some other explicit negative results.

To begin, we give two examples: the first (respectively, the second) is an exam-ple of a sub-Laplacian on a homogeneous Carnot group of step two which is not(respectively, which is) analytic-hypoelliptic; in the latter case, we explicitly writethe (analytic) fundamental solution.

Example 5.10.2. Let R4 (the points are denoted by (x, y, z, t) with x, y, z, t ∈ R)

be equipped with the dilation δλ(x, y, z, t) = (λx, λy, λz, λ2t) and the compositionlaw


⎛

⎜⎝

x

y

z

t

⎞

⎟⎠ ◦⎛

⎜⎝

ξ

η

ζ

τ

⎞

⎟⎠ =⎛

⎜⎝

x + ξ

y + η

z+ ζ

t + τ + x η

⎞

⎟⎠ .

Then G = (R4, ◦, δλ) is a homogeneous Carnot group of step two, and

ΔG = (∂x)2 + (∂y + x ∂t )

2 + (∂z)2

is its canonical sub-Laplacian. It is easily seen that G is isomorphic to the sum (inthe sense of Section 4.1.5) of the Heisenberg-Weyl group H

1 on R3 and the usual

Euclidean group (R,+) (note that the canonical sub-Laplacians on both these groupsare analytic hypoelliptic!). It can be proved that ΔG is not analytic hypoelliptic!Indeed, by the cited result of Helffer [Hel82], this follows from the fact that G is nota HM-group, since the first layer of the stratification has odd dimension.

This example is a particular case of the family of not analytic hypoelliptic sub-Laplacians (see Rothschild [Rot84])

L =n∑

j=1

((∂xj

)2 + (∂yj+ xj ∂t )

2)+ (∂z)2, (5.81)

which, in turn, are inspired by a famous counterexample by Baouendi–Goulaouic[BG72]. Indeed, in [BG72] it is proved that the operator

L :=n∑

j=1

((∂xj

)2 + (xj ∂t )2)+ (∂z)

2

on Rn+2 (the points are denoted by (x, z, t), x = (x1, . . . , xn) ∈ R

n, z ∈ R, t ∈ R)is not Cω-hypoelliptic on R

n+2. Now, we notice that the operator L in (5.81) is a“lifted” version of L, so that it is easy to prove that L is not Cω-hypoelliptic onR

2n+2 if L is not Cω-hypoelliptic on Rn+2.

Indeed, suppose to the contrary that L is Cω-hypoelliptic on R2n+2. Then take a

function f = f (x, z, t) real analytic on an open set Ω ⊆ Rn+2 and a solution u =

u(x, z, t) to Lu = f on Ω . If we set f (x, y, z, t) := f (x, z, t) and u(x, y, z, t) :=u(x, z, t), then we notice that

Lu(x, y, z, t) = Lu(x, z, t) = f (x, z, t) = f (x, y, z, t) (5.82)

on the open set Ω := {(x, y, z, t) : (x, z, t) ∈ Ω, y ∈ R}. Since f is clearlyreal analytic on Ω , then (5.82) and the supposed Cω-hypoellipticity of L imply u ∈Cω(Ω). This obviously means that u is real analytic on Ω . Thus we have shown thatL is Cω-hypoelliptic on R

n+2, contrarily to what is proved in [BG72]. ��Example 5.10.3. This example is taken from Balogh–Tyson [BT02]. Let us con-sider the group G on R

5 (the points are denoted by (x1, x2, x3, x4, t) ∈ G, (x1, x2,

x3, x4) ∈ R4 corresponds to the first layer of the stratification, t ∈ R to the second


one) with dilation δλ(x1, x2, x3, x4, t) = (λx1, λx2, λx3, λx4, λ2t) and the composi-

tion law⎛

⎜⎜⎜⎝

x1x2x3x4t

⎞

⎟⎟⎟⎠ ◦

⎛

⎜⎜⎜⎝

ξ1ξ2ξ3ξ4τ

⎞

⎟⎟⎟⎠ =

⎛

⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3x4 + ξ4

t + τ + 12 (x2ξ1 − x1ξ2 + 2x4ξ3 − 2x3ξ4)

⎞

⎟⎟⎟⎠ .

Following our conventional notation, G is the homogeneous Carnot group of steptwo (with m = 4 generators and n = 1) with relevant matrix

B(1) =⎛

⎜⎝

0 1 0 0−1 0 0 00 0 0 20 0 −2 0

⎞

⎟⎠ .

Then G is obviously a HM-group, for B is a non-singular skew-symmetric matrix.In particular, by the cited result of Métivier [Met81], the canonical sub-Laplacian Lis Cω-hypoelliptic. We can see this directly, for the relevant fundamental solution Γ

has been explicitly written by Balogh–Tyson in [BT02] (making use of a remarkableformula by Beals–Gaveau–Greiner, see [BGG96]; we describe this formula closelyin Section 5.12, page 291): it is apparent that Γ is analytic out of the origin! Indeed,it holds

Γ (x1, x2, x3, x4, t) = c d2−Q(x1, x2, x3, x4, t),

where c is a suitable positive constant, Q = 6 is the homogeneous dimension of G,and d is the homogeneous norm defined by

d(x1, x2, x3, x4, t)

=((

1

2x2

1 +1

2x2

2 + x23 + x2

4

)2

+ t2)1/8

·(

1

2x2

1 +1

2x2

2 +√(

1

2x2

1 +1

2x2

2 + x23 + x2

4

)2

+ t2

)3/8

·(

1

2x2

1 +1

2x2

2 + x23 + x2

4 +√(

1

2x2

1 +1

2x2

2 + x23 + x2

4

)2

+ t2

)−1/8

. (5.83)

This formula gives the remarkable example of an explicit fundamental solution of agroup which is not a H-type group!

We then turn our attention to groups of step greater than two. Following the ideain the argument at the end of Example 5.10.2, we can give infinite examples of groupsof arbitrarily high step with a sub-Laplacian which is not Cω-hypoelliptic: it sufficesto take the sum (in the sense of Section 4.1.5) of the group G of Example 5.10.2 withanother Carnot group. Let us now quote some other examples taken or inspired bythe existing literature.


Example 5.10.4. This example is taken from a paper by Christ [Chr95] (see also[Hel82,PR80]). Consider the Carnot group on R

4 with the composition 9 low

⎛

⎜⎝

x1x2x3x4

⎞

⎟⎠ ◦⎛

⎜⎝

ξ1ξ2ξ3ξ4

⎞

⎟⎠ =⎛

⎜⎝

x1 + ξ1x2 + ξ2

x3 + ξ3 − ξ2x1x4 + ξ4 + 2ξ3x1 − ξ2x

21

⎞

⎟⎠ .

It is easily seen that G is a filiform homogeneous Carnot group of step three and twogenerators, with dilations

δλ(x1, x2, x3, x4) = (λx1, λx2, λ2x3, λ

3x4),

and such that the first two vector fields of the relevant Jacobian basis are

X1 = ∂x1 , X2 = ∂x2 − x1∂x3 − x21∂x4 .

Then, it can be proved that the canonical sub-Laplacian on G

ΔG = X21 +X2

2 = (∂x1)2 + (∂x2 − x1∂x3 − x2

1∂x4

)2

is not Cω-hypoelliptic. (Compare the group in this example to the Bony-type sub-Laplacian with N = 2 in Section 4.3.3, page 202.) ��Example 5.10.5. It is known (see [Chr91,Chr93,HH91,PR80]) that in R

3 (with coor-dinates (t, s, x)) the operator

L = (∂t )2 + (∂s − tm∂x)

2 (5.84)

is not Cω-hypoelliptic for any m ∈ N, m ≥ 2. In this example, fixed m as above,we give a suitable sub-Laplacian L “lifting” L: as a consequence (arguing as at theend of Example 5.10.2), L cannot be Cω-hypoelliptic, since L does not possess thisproperty.

Take N ∈ N, N ≥ m ≥ 2, and consider the following Bony-type sub-Laplacian(see Section 4.3.3, page 202): we equip R

2+N (whose points are denoted by (t, s, x),t, s ∈ R, x ∈ R

N ) by the composition law

(t, s, x1, x2, x3, . . . , xN) ◦ (τ, σ, ξ1, ξ2, ξ3, . . . , ξN )

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

t + τ

s + σ

x1 + ξ1 + σ t

x2 + ξ2 + ξ1 t + σ t2

2!x3 + ξ3 + ξ2 t + ξ1

t2

2! + σ t3

3!...

xN + ξN + ξN−1 t + ξN−2t2

2! + · · · + ξ1tN−1

(N−1)! + σ tN

N !

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

9 Compare to Ex. 3, Chapter 4, page 216.


and the group of dilations defined by

δλ(t, s, x1, x2, x3, . . . , xN) = (λt, λs, λ2x1, λ3x2, λ

4x3, . . . , λN+1xN).

Then, G = (R2+N, ◦, δλ) is a filiform homogeneous Carnot group of step N + 1(note that, since N ≥ m ≥ 2, then the step N + 1 is ≥ 3) and two generators, and

ΔG = (∂t )2 +

(∂s + t ∂x1 +

t2

2! ∂x2 + · · · +tN

N ! ∂xN

)2

is its canonical sub-Laplacian.We now prove that ΔG is not Cω-hypoelliptic (starting from the fact that L in

(5.84) is not). Indeed, since L is not Cω-hypoelliptic, there exists an open set Ω ⊆ R3

and a function u = u(t, s, x) on Ω such that Lu ∈ Cω(Ω) but u /∈ Cω(Ω). Let usnow consider the function (recall that m ≤ N )

u = u(t, s, x1, . . . , xN) := u(t, s,−m!xm)

defined on the open subset of R2+N

Ω := {(t, s, x1, . . . , xN) | (t, s,−m! xm) ∈ Ω}.

As a consequence, it holds

ΔG

(u(t, s, x1, . . . , xN)

) = (Lu)(t, s,−m! xm) on Ω,

whence ΔGu ∈ Cω(Ω) (for Lu ∈ Cω(Ω)) and u /∈ Cω(Ω) (for u /∈ Cω(Ω)). Thisproves that ΔG is not analytic-hypoelliptic. ��Example 5.10.6. Let m, k ∈ N be such that 0 ≤ m ≤ k. Consider the operator on R

3

(whose points are denoted by (x1, x2, x3))

L = (∂x1)2 + (xm

1 ∂x2)2 + (xk

1 ∂x3)2. (5.85)

O.A. Oleınik and E.V. Radkevic [OR72] (see also [Him98]) proved that L is Cω-hypoelliptic if and only if m = k. Our aim in this example is to “lift” (in a suitablesense) the vector fields on R

3 appearing in (5.85)

∂x1 , xm1 ∂x2 , xk

1 ∂x3 (5.86)

to three vector fields (in a larger space, namely R3+m+k) generating a homogeneous

Carnot group. It will easily follow (arguing as in the last paragraph of Example5.10.5) that the relevant sub-Laplacian is not Cω-hypoelliptic if L is not.

When m = k = 0, then L = ΔR3 is the ordinary Laplace operator on R3; when

m = 0, k = 1, we obtain a non-analytic-hypoelliptic operator already consideredin Example 5.10.2 (see also Baouendi–Goulaouic [BG72]). Let us now suppose thatk > m ≥ 1. We equip R

3+m+k with the following coordinates (the semicolon willdenote different layers in a suitable homogeneous Carnot group structure)

5.11 Harmonic Approximation 287

P = (x1, y1, z1; y2, z2; y3, z3; . . . , ym, zm; x2, zm+1; zm+2; zm+3; . . . ; zk; x3).

We define a group of dilations by setting (recall that k ≥ m+ 1)

δλ(P ) = (λx1, λy1, λz1;

λ2y2, λ2z2; λ3y3, λ

3z3; . . . ; λmym, λmzm;λm+1x2, λ

m+1zm+1;λm+2zm+2; λm+3zm+3; . . . ; λkzk;

λk+1x3).

Let us also consider on R3+m+k the following vector fields

X = ∂x1 ,

Y = ∂y1 + x1∂y2 + x21∂y3 + · · · + xm−1

1 ∂ym + xm1 ∂x2 ,

Z = ∂z1 + x1∂z2 + x21∂z3 + · · · + xk−1

1 ∂zk+ xk

1∂x3 .

It is then not difficult to see that X, Y,Z are δλ-homogeneous of degree one and theyfulfill hypotheses (H0)–(H1)–(H2) of page 191. Hence, by the results of Section 4.2(page 191), we can define a suitable homogeneous Carnot group structure on R

3+m+k

such thatΔG = X2 + Y 2 + Z2

is the relevant canonical sub-Laplacian. Now, since

ΔG(u(x1, x2, x3)) = (Lu)(x1, x2, x3)

for any smooth function u on R3+m+k depending only on x1, x2, x3, we can argue

as in the last paragraph of Example 5.10.5 to infer that ΔG is not Cω-hypoelliptic(since L is not). ��

5.11 Harmonic Approximation

Let L be a sub-Laplacian on the homogeneous Carnot group G. In this section, wegive some conditions ensuring that a L-harmonic function defined in a neighborhoodof a compact set K contained in an open set Ω can be uniformly approximated on K

by a sequence of L-harmonic functions in Ω .In some of the results of this section, we assume that L is analytic-hypoelliptic

(see Section 5.10). Γ = d2−Q, will denote its fundamental solution. To begin with,we prove the following lemma.

Lemma 5.11.1. Let L be a sub-Laplacian on the homogeneous Carnot group G =(RN, ◦, δλ), and let Γ = d2−Q be its fundamental solution. Suppose also that L isanalytic-hypoelliptic.


Let K ⊆ G be compact. Then there exists R = R(K, G,L) > 0 such that, forevery z ∈ G with d(z) > R, it holds

Γ (z−1 ◦ y) =∑

α∈(N∪{0})NCα(z) · yα, Cα(z) = 1

α! Dα

∣∣∣∣y=0

(Γ (z−1 ◦ y)

),

the series converging uniformly on the d-disc {y ∈ G : d(y) < d(K)}, whered(K) := supz∈K d(z).

Proof. Let ζ ∈ G be such that d(ζ ) = 1. Since η �→ Γ (ζ−1 ◦ η) is analytic closeto η = 0 and ∂Bd(0, 1) = {ζ ∈ G : d(ζ ) = 1} is compact, there exists ρ > 0,independent of ζ , such that

Γ (ζ−1 ◦ η) =∑

α∈(N∪{0})Naα(ζ ) · ηα, uniformly on Bd(0, ρ),

where

aα(ζ ) = 1

α! Dα

∣∣∣∣η=0

(Γ (ζ−1 ◦ η)

).

Let us now choose R > 0 such that R > d(K)/ρ. If z ∈ G, d(z) > R, then

Γ (z−1 ◦ y) = d2−Q(z) Γ((

δd−1(z)z−1) ◦ δd−1(z)(y)

)

= d2−Q(z)∑

α∈(N∪{0})Naα

(δd−1(z)z

−1) · (δd−1(z)(y))α

=:∑

α∈(N∪{0})NCα(z) · yα.

The series converges uniformly for y ∈ G such that

d(δd−1(z)(y)

) = d(y)

d(z)< ρ.

In particular, this holds for

y ∈ Bd(0, R ρ) ⊇ Bd(0, d(K)). ��The next lemma does not require the analytic-hypoellipticity of L.

Lemma 5.11.2. Let L be a sub-Laplacian on the homogeneous Carnot group G =(RN, ◦, δλ), and let Γ = d2−Q be its fundamental solution. Let u be an L-harmonicfunction on the ball Bd(0, r). Suppose

u(x) =∞∑

k=1

uk(x), x ∈ Bd(0, r),

where uk is a continuous δλ-homogeneous function in the whole G of δλ-degree mk ,k ∈ N. If the series is uniformly convergent on the compact subsets of Bd(0, r) andmk �= mh for every k �= h, then every uk is L-harmonic in G.

5.11 Harmonic Approximation 289

Proof. Let ϕ ∈ C∞0 (RN, R) with supp ϕ ⊆ Bd(0, 1). Since u is L-harmonic inBd(0, r), we have

0 =∫

Bd(0,r)

u(x)L(ϕ(δλ−1(x))

)dx ∀ λ < r.

The homogeneity of L and the change of variable y = δλ−1(x) give

0 = λQ−2∫

Bd(0,r/λ)

u(δλ(y))Lϕ(y) dy

= λQ−2∞∑

k=1

∫

Bd(0,r/λ)

uk(δλ(y))Lϕ(y) dy

= λQ−2∞∑

k=1

λmk

∫

Bd(0,r/λ)

uk(y)Lϕ(y) dy

= λQ−2∞∑

k=1

λmk

∫

Bd(0,1)

uk(y)Lϕ(y) dy.

Note that, in the last equality, we were able to replace Bd(0, r/λ) by Bd(0, 1), sinceλ < r and ϕ is supported in a compact set in Bd(0, 1). Then

∫

Bd(0,1)

uk(y)Lϕ(y) dy = 0 ∀ ϕ ∈ C∞0 (Bd(0, 1)) ∀ k ∈ N.

This means that Luk = 0 in Bd(0, 1) in the weak sense of distributions. Since L ishypoelliptic, this implies the L-harmonicity of uk in Bd(0, 1), and so in G, due tothe δλ-homogeneity of uk , k ∈ N. ��

We are now ready to prove the announced approximation theorem.

Theorem 5.11.3 (An approximation theorem). Let L be a sub-Laplacian on thehomogeneous Carnot group G. Suppose that L is analytic-hypoelliptic.

Let Ω ⊆ G be an open set such that ∂Ω = ∂Ω . Let K ⊆ Ω be compact andsatisfy the following condition:

if ω is a bounded connected component of Ω \K , then ∂ω ⊆ ∂Ω . (5.87)

Then, for every function h which is L-harmonic in a neighborhood of K , there existsa sequence (hn)n∈N of L-harmonic functions in Ω such that

limn→∞ hn = h, uniformly on K.

Proof. By general results from functional analysis, it is well known that it sufficesto prove the following statement.10

10 This is also known as “Caccioppoli’s completeness method”.


Let μ be a signed Radon measure supported on K and satisfying∫

K

h dμ = 0 (5.88)

for every L-harmonic function h in Ω . Then (5.88) holds for every function h whichis L-harmonic just in a neighborhood of K .

The crucial part of the proof is the following assertion.Claim. If μ satisfies the previous hypotheses, then

u(x) :=∫

K

Γ (y−1 ◦ x) dμ(y) =∫

K

Γ (x−1 ◦ y) dμ(y) (5.89)

is identically zero in Ω \K .We first show how to complete the proof of the theorem by using this claim.

Let h be an L-harmonic function in an open set Ω0 ⊇ K , Ω0 ⊆ Ω . Choose afunction φ ∈ C∞0 (Ω0) such that φ = 1 in an open set Ω1 ⊇ K , Ω1 ⊆ Ω0. Thenφ h ∈ C∞0 (Ω0) and φ h = h in Ω1. By the representation formula (5.16), we have

(φ h)(y) = −∫

Ω0

Γ (x−1 ◦ y)L(φ h)(x) dx

= −∫

Ω0\Ω1

Γ (x−1 ◦ y)L(φ h)(x) dx, y ∈ Ω0.

It follows that∫

K

h(y) dμ(y) =∫

K

(φ h)(y) dμ(y)

= −∫

K

(∫

Ω0\Ω1

Γ (x−1 ◦ y)L(φ h)(x) dx

)dμ(y) =: (�).

Thus, by interchanging the integrals and keeping in mind (5.89), we infer

(�) = −∫

Ω0\Ω1

L(φ h)(x)u(x) dx = 0,

since u = 0 in Ω \K ⊇ Ω0 \Ω1.Thus, we are left with the proof of the Claim. Let ω be a connected component

of Ω \K . We have to prove that u ≡ 0 in ω.We first suppose that ω is bounded. Then ∅ �= ∂ω ⊆ ∂Ω . In particular, this

implies that ∂Ω �= ∅, hence Ω �= G for ∂Ω = ∂Ω . For every x0 ∈ G \ Ω , thefunction y �→ Γ (x−1

0 ◦ y) is L-harmonic in Ω . Therefore, by the assumption (5.88),

u(x0) =∫

K

Γ (y−1 ◦ x0) dμ(y) = 0.

This proves that u ≡ 0 in G \Ω . As a consequence,

5.12 An Integral Representation Formula for Γ 291

Dαu(x) = 0 for every multi-index α (5.90)

and for every x ∈ G \ Ω . Since u ∈ C∞(G \ K), it follows that (5.90) also holdsat any point x ∈ ∂Ω = ∂Ω . In particular, since ∂ω ⊆ ∂Ω , (5.90) holds at somepoint x ∈ ∂ω. Then ω ≡ 0 in a neighborhood of x, since u is L-harmonic, hence realanalytic, close to x. The connectedness of ω and again the analyticity of u imply thatu ≡ 0 in ω.

Let us now assume that ω is unbounded. By Lemma 5.11.1, for every x ∈ ω withd(x) sufficiently large, we have

Γ (x−1 ◦ y) =∑

α∈(N∪{0})NCα(x) · yα, uniformly in Bd(0, d(K)).

Then

u(x) =∫

K

Γ (x−1 ◦ y) dμ(y) =∞∑

m=0

∫

K

um(x, y) dμ(y),

whereum(x, y) =

∑

|α|G=k

Cα(x) · yα.

The function um is δλ-homogeneous of degree m and the series

∞∑

m=0

um(x, ·)

is uniformly convergent on Bd(0, d(K)) to the L-harmonic function y �→Γ (x−1 ◦ y). Then, by Lemma 5.11.2, um(x, ·) is L-harmonic in G, so that, by theassumption (5.88),

∫

K

um(x, y) dμ(y) = 0 ∀ m ≥ 0.

Thus we have proved that u(x) = 0 for every x ∈ ω, with d(x) sufficiently large.Since ω is connected and u is analytic in ω, this implies u ≡ 0 in ω and completesthe proof of the theorem. ��

5.12 An Integral Representation Formula for the FundamentalSolution on Step-two Carnot Groups

The aim of this section is to state a remarkable result in the paper [BGG96] byR. Beals, B. Gaveau and P. Greiner. This result provides a somewhat explicit integralrepresentation formula of the fundamental solution of the canonical sub-Laplacianon a general Carnot group of step two.

We shall see in Section 16.3 (page 637 in Part III) that, given a Carnot group G1and an arbitrary sub-Laplacian L on G1, there exists a Carnot group G2 isomorphic


to G1 such that L corresponds (via the related isomorphism in the relevant Lie alge-bras) to the canonical sub-Laplacian ΔG2 on G2. Moreover, if G2 (whence G1) hasstep two, we saw in Proposition 3.5.1 (page 168) that we can perform another Lie-group isomorphism sending G2 into the homogeneous Carnot group G3 such that thecomposition law on G3 is given by (we follow our usual notation)

(x, t)◦ (ξ, τ ) =(

x+ξ, t1+τ1+ 1

2〈B(1)x, ξ 〉, . . . , tn+τn+ 1

2〈B(n)x, ξ 〉

)(5.91)

and the matrices B(i)’s are skew-symmetric. This last isomorphism also sendsthe canonical sub-Laplacian of G2 into the canonical sub-Laplacian of G3. Now,the cited result in [BGG96] furnishes an integral formula for the canonical sub-Laplacian on a homogeneous Carnot group of step two whose composition law ◦has precisely the above form and the matrices B(i)’s are skew-symmetric. Our ar-gument above shows that (and how) we can obtain a representation formula for thefundamental solution of any (not necessarily canonical) sub-Laplacian on any homo-geneous Carnot group of step two.

We now state the remarkable result in [BGG96]. We explicitly remark that in[BGG96] the formalism of complex Hamiltonian mechanics is followed: we slightlychange the therein notation.

Theorem 5.12.1 (Beals–Gaveau–Greiner, [BGG96]). Let G = Rm+n (whose points

are denoted by (x, t), x ∈ Rm, t ∈ R

n) be equipped with a homogeneous Carnotgroup structure by the dilation δλ(x, t) = (λx, λ2t) and the composition lawin (5.91), where the B(k)’s are n skew-symmetric linearly independent matrices oforder m×m. Consider the canonical sub-Laplacian ΔG =∑n

i=1X2i , where

Xi = ∂/∂xi + 1

2

n∑

k=1

(m∑

l=1

b(k)i,l xl

)∂/∂tk, i = 1, . . . , m

(here b(k)i,l denotes the entry of position (i, l) of B(k)). Then, for every (x, t) with

x �= 0, the fundamental solution Γ of ΔG is given by

Γ (x, t) = cQ

∫

Rn

√det(V(B(τ)))

( 12 〈W(B(τ)) · x, x〉 − ι 〈t, τ 〉)Q/2−1

dτ, (5.92)

where ι is the imaginary unit of C and cQ is the dimensional constant

cQ = �(Q2 − 1)

2 (2 π)Q/2. (5.93)

Here we used the following notation: Q = m + 2 n is the homogeneous dimensionof G, � in (5.93) is Euler’s Gamma function, τ = (τ1, . . . , τn) ∈ R

n,

B(τ) = 1

2(τ1 B(1) + · · · + τn B(n)),

5.13 Appendix A. Maximum Principles 293

V and W are the real-analytic functions prolonging z/ sin(z) and z/ tan(z), respec-tively, at z = 0, i.e.

V(z) =∞∑

j=0

(−1)j

(2j + 1)! z2j , W(z) =

∞∑

j=0

(−1)j 22j B2j

(2j)! z2j

(here the B2j ’s are the Bernoulli numbers). Moreover, for every t ∈ Rn \{0}, we have

Γ (0, t) = lim0�=x→0

Γ (x, t).

We remark that a general integral formula for Γ is provided in [BGG96] com-prising the case x = 0 too, by shifting the contour R

n into the complex domain Cn

(see [BGG96, Theorem 3, page 315]).The δλ-homogeneity of Γ in (5.92) (of degree 2−Q) should be noted.As we saw in Example 5.10.3 (page 283), formula (5.92) can, in some cases,

give explicit fundamental solutions. This can be done by using the fact that, ifλ1(τ ), . . . , λm(τ) and v1(τ ), . . . , vm(τ) denote the eigenvalues and correspondingeigenvectors of the matrix B(τ) (over the complex field), normalized in such a waythat |vj (τ )| = 1 for j = 1, . . . , m, we have

det(V(B(τ))

) =m∏

j=1

λj (τ )

sin(λj (τ )),

⟨W(B(τ)) · x, x

⟩ =m∑

j=1

λj (τ )

tan(λj (τ ))

∣∣〈x, vj (τ )〉∣∣2.

(In the last formula, the inner product is, of course, that of Cm.)

5.13 Appendix A. Maximum Principles

In this section, we shall prove some weak and strong maximum principles for L, anarbitrary sub-Laplacian on a homogeneous Carnot group G.

To begin with, we prove some elementary lemmas.

Lemma 5.13.1. Let Ω ⊂ RN be a bounded open set and let u : Ω → R be an

arbitrary function. Then there exists a point x0 ∈ Ω such that

lim supx→x0

u(x) = supΩ

u. (5.94)

Proof. We argue by contradiction and assume that (5.94) is false. Then, for everyx ∈ Ω , there exists an open neighborhood Vx of x such that

supΩ∩Vx

u < supΩ

u. (5.95)


The family {Vx : x ∈ Ω} is an open covering of Ω , so that, since Ω is compact, wehave

Ω ⊆p⋃

j=1

Vxj, p ∈ N,

for suitable x1, . . . , xp ∈ Ω . Then

supΩ

u = max{

supΩ∩Vxj

u : j = 1, . . . , p}. (5.96)

On the other hand, by (5.95), the right-hand side of (5.96) is strictly less than supΩ u.This contradiction proves the lemma. ��Lemma 5.13.2. Let A and B be N×N symmetric matrices with constant real entries.Assume A ≥ 0 and B ≤ 0. Then trace(A · B) ≤ 0.

Proof. Let R := A1/2 be a symmetric square root of A. Then trace(A·B) = trace(R ·R · B) = trace(R · B · R) = trace(RT · B · R) ≤ 0, since B ≤ 0. ��Lemma 5.13.3. Let L be a sub-Laplacian on the homogeneous Carnot group G. LetΩ ⊆ G be an arbitrary open set, and let u : Ω → R be a C2 real function. Assumethat u has a local maximum at x0 ∈ Ω . Then

Lu(x0) ≤ 0. (5.97)

Proof. We know that L = div(A · ∇T ), where A is a N ×N symmetric matrix withpolynomial entries and A(x) ≥ 0 at any point x ∈ R

N . Then

L = trace(A ·D2u)+ 〈b,∇u〉, (5.98)

where D2u = (∂xi xj)i,j≤N is the Hessian matrix of u and b is the vector-valued

function whose j -th component is given by

bj =N∑

i=1

∂xiai,j . (5.99)

Since u has a local maximum at x0, we have ∇u(x0) = 0 and D2u(x0) ≤ 0. Then,by Lemma 5.13.2,

Lu(x0) = trace(A(x0) ·D2u(x0)) ≤ 0.

This ends the proof. ��We are now able to give a simple proof of the following weak maximum princi-

ple.


Theorem 5.13.4 (Weak maximum principle). Let L be a sub-Laplacian on the ho-mogeneous Carnot group G. Let Ω be a bounded open subset of G. Let u : Ω → R

be a C2 function such that{Lu ≥ 0 in Ω ,

lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω .(5.100)

Then u ≤ 0 in Ω .

Proof. We know that the matrix A in (5.98) has the following block form (see also(1.91), page 64)

A =(

A1,1 A1,2A2,1 A2,2

),

where A1,1 = (ai,j )i,j≤m is a constant m × m symmetric matrix strictly positivedefinite. Then a1,1 > 0. Let b1 be given by (5.99) with j = 1. Define

λ := 2 supx∈Ω

b1(x)

a1,1, M := sup

(x1,...,xN )∈Ω

exp(λ x1),

andh(x) = h(x1, . . . , xN) := M − exp(λ x1).

A trivial computation shows that

h(x) ≥ 0 and Lh(x) < 0 for every x ∈ Ω . (5.101)

For an arbitrary ε > 0, let us now consider the function uε := u − ε h. Due toinequalities (5.101) and condition (5.100), we have

Luε > 0 in Ω and lim supx→y

uε(x) ≤ 0 for every y ∈ ∂Ω . (5.102)

By Lemma 5.13.1, there exists a point x0 ∈ Ω such that

lim supx→x0

uε(x) = supΩ

uε. (5.103)

We want to show that x0 ∈ ∂Ω . Arguing by contradiction, we assume x0 ∈ Ω . Then,by the continuity of u in Ω ,

uε(x0) = lim supx→x0

uε(x),

so that, by (5.103), uε(x0) = maxΩ uε. As a consequence, by Lemma 5.13.3,Luε(x0) ≤ 0. This contradicts the first inequality in (5.102). Thus x0 ∈ ∂Ω . Then,by (5.103) and the second condition in (5.102),

supΩ

uε = lim supx→x0

uε(x) ≤ 0.

Hence, u − ε h = uε ≤ 0 in Ω for every ε > 0. Letting ε tend to zero, we obtainu ≤ 0 in Ω . The theorem is thus completely proved. ��


Note 5.13.5. The previous proof can be applied also to continuous functions satisfy-ing the inequality Lu ≥ 0 in the asymptotic sense of Exercises 8 and 9 at the end ofthe chapter (see also Ex. 10).

Corollary 5.13.6. Let L be a sub-Laplacian on the homogeneous Carnot group G.Let Ω be an unbounded open subset of G. Let u : Ω → R be a C2 function suchthat ⎧

⎪⎨

⎪⎩

Lu ≥ 0 in Ω ,

lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω ,

lim sup|x|→∞ u(x) ≤ 0.

(5.104)


Proof. Let ε > 0 be arbitrary but fixed. The third condition in (5.104) implies theexistence of a real positive constant R such that

u(x)− ε < 0 in Ω \ΩR , (5.105)

where ΩR := {x ∈ Ω : |x| < R}. It follows that

{L(u− ε) = Lu ≥ 0 in ΩR ,

lim supx→y u(x) ≤ 0 for every y ∈ ∂ΩR .

Then, by Theorem 5.13.4, u − ε ≤ 0 in ΩR . This inequality, together with (5.105),gives u ≤ ε in Ω for every ε > 0. Hence u ≤ 0 in Ω . ��

A particular case of Corollary 5.13.6 is the following one.

Corollary 5.13.7. If L is as in Corollary 5.13.6, the only entire L-harmonic functionvanishing at infinity is the null function.

Proof. Let u : G → R be an entire L-harmonic function vanishing at infinity, i.e.u ∈ C∞(G, R) satisfies {Lu = 0 in G,

lim|x|→∞ u(x) = 0.

Then, by applying Corollary 5.13.6 both to u and −u, we get u ≡ 0. ��The rest of this section is devoted to the proof of the following strong maximum

principle.

Theorem 5.13.8 (Strong maximum principle). Let L be a sub-Laplacian on thehomogeneous Carnot group G. Let Ω be a connected open subset of G. Let u :Ω → R be a C2 function such that

u ≤ 0 and Lu ≥ 0 in Ω . (5.106)

Suppose there exists a point x0 ∈ Ω such that u(x0) = 0. Then u(x) = 0 for everyx ∈ Ω .


The proof of this theorem requires several preliminary results. In what follows,we shall denote by | · | the standard Euclidean norm and by D(z, r) the ball

D(z, r) := {x ∈ RN : |x − z| < r}.

Definition 5.13.9. Let F be a relatively closed subset of Ω . We say that a vectorν ∈ R

N \ {0} is orthogonal to F at a point y ∈ Ω ∩ ∂F if

D(y + ν, |ν|) ⊆ (Ω \ F) ∪ {y}. (5.107)

If this inclusion holds, we shall write ν⊥F at y. We also put

F ∗ := {y ∈ Ω ∩ ∂F | there exists ν: ν⊥F at y}.

With the above notation, we explicitly remark that F ∗ �= ∅ if F ⊂ Ω , F �= Ω .Indeed, since Ω is connected, Ω ∩ ∂F is not empty. Take a point z ∈ Ω ∩ ∂F , a ballD(z,R) ⊆ Ω and a point x0 ∈ D(z,R/2). Let y ∈ Ω ∩ ∂F be such that

r := |x0 − y| = dist(x0, ∂F ).

Then y ∈ F ∗ and ν := r2 (x0 − y)⊥F at y.

The following Hopf-type lemma will be crucial for the proof of the strong maxi-mum principle in Theorem 5.13.8.

Lemma 5.13.10 (A Hopf-type lemma for sub-Laplacians). Let L be a sub-Lapla-cian on the homogeneous Carnot group G. Let Ω ⊆ G be open, and let u : Ω → R

be a C2 function satisfying the inequalities in (5.106). Let

F := {x ∈ Ω : u(x) = 0}. (5.108)

Assume ∅ �= F �= Ω . Then, for every y ∈ F ∗ and ν⊥F at y, we have

qL(y, ν) = 0, (5.109)

where qL(x, ξ) := 〈A(x) · ξ, ξ 〉 is the characteristic form of L defined in (5.1a).

Proof. Let y ∈ F ∗ and ν⊥F at y. Then

D(y + ν, |ν|) ⊆ (Ω \ F) ∪ {y}.Since F ∗ ⊆ F ∩Ω , y is a maximum point for u (see (5.108)). Then ∇u(y) = 0. Wenow argue by contradiction assuming that (5.109) is false. Hence

qL(y, ν) > 0.

Let us now consider the function

h(x) := exp(−λ |x − z|2)− exp(−λ r2),


where z = y + ν and r = |ν|. The positive constant λ will be fixed later on. A directand easy computation shows that

Lh(y) = 4λ2 exp(−λ r2) · (qL(y, ν)+O(1/λ)),

as λ → ∞. Then, we can choose and fix λ > 0 in such a way that Lh > 0 ina suitable neighborhood V of y. Obviously, we may assume V ⊂ Ω . Let us nowconsider the bounded open set

U := V ∩D(z, r).

Note that ∂U = Γ1 ∪ Γ2, where

Γ1 = V ∩ ∂D(z, r) and Γ2 = D(z, r) ∩ ∂V .

Since Γ2 is a compact subset of Ω\F and u < 0 in Ω\F , there exists ε > 0 such thatu+ ε h < 0 in Γ2. On the other hand, being h = 0 on ∂D(z, r) and u ≤ 0 in Ω , wehave u+ε h ≤ 0 on Γ1. Then, since L(u+ε h) ≥ εLh ≥ 0 in U , from the maximumprinciple of Theorem 5.13.4, we obtain u + ε h ≤ 0 in U . Since u(y) = h(y) = 0,

this implies

u(y + t ν)− u(y)

t≤ −ε

h(y + t ν)− h(y)

tfor 0 < t < 1. (5.110)

Letting t tend to zero in this inequality, we get

〈∇u(y), ν〉 ≤ −ε 〈∇h(y), ν〉 = −2ε exp(−λ r2) r2.

This contradicts the condition ∇u(y) = 0 and completes the proof. ��Corollary 5.13.11. Let the hypotheses and notation of the previous lemma hold. Letalso L =∑m

j=1 X2j . Then we have

〈XjI (y), ν〉 = 0 ∀ y ∈ F ∗ ∀ ν⊥F at y

and for every j = 1, . . . , m.

Proof. It follows from the previous lemma, by just noticing that

qL(x, ξ) =m∑

j=1

〈XjI (y), ξ 〉2. ��

Another crucial definition is the following one.

Definition 5.13.12 ((Positively) invariant set w.r.t. a vector field). Let X ∈ T (RN)

be a smooth vector field in RN , and let F be a relatively closed subset of Ω . We say

that F is positively X-invariant if, for any integral curve γ of X, γ : [0, T ] → Ω

such that γ (0) ∈ F , we have γ (t) ∈ F for every t ∈ [0, T ]. We say that F isX-invariant if it is positively X-invariant with respect to both X and −X.


It is easy to verify that the condition

〈XI (y), ν〉 ≤ 0 ∀ y ∈ F ∗ ∀ ν⊥F at y (5.111)

is necessary for the positive X-invariance of F . Indeed, let y ∈ F ∗, ν⊥F at y andγ : [0, T ] → Ω be an integral curve of X such that γ (0) = y. Let F be positivelyX-invariant. Since

D(y + ν, |ν|) ⊆ (Ω \ F) ∪ {y},we have

|γ (t)− (y + ν)|2 ≥ |ν|2 and |γ (0)− (y + ν)|2 = |ν|2for every t ∈ [0, T ]. This means that the real function t �→ |γ (t) − (y + ν)|2 has aminimum at t = 0. As a consequence,

0 ≤ d

d t

∣∣∣t=0|γ (t)− (y + ν)|2 = 〈γ (0), γ (0)− (y + ν)〉 = 〈XI (y),−ν〉.

Hence (5.111) holds. We will show that this condition is also sufficient for F to bepositively X-invariant. To prove this statement, we need the following elementarylemma.

Lemma 5.13.13. Let g : [0, T ] → R be a continuous function such that

lim suph→0−

g(t + h)− g(t)

h≤ M ∀ t ∈ (0, T ], (5.112)

for a suitable M ∈ R. Then

g(t) ≤ g(0)+M t ∀ t ∈ [0, T ].Proof. Let ε > 0 be fixed. Condition (5.112) implies that the real function

t �→ g(t)− g(0)− (M + ε)t

has a maximum at t = 0. Indeed, suppose to the contrary that there exist ε0 > 0 andt0 ∈ (0, T ] such that

g(t)− g(0)− (M + ε) t ≤ g(t0)− g(0)− (M + ε0) t0 ∀ t ∈ [0, T ].In particular, for t = t0 + h and h < 0 small enough, this gives

g(t0 + h)− g(t0)

h≥ (M + ε0),

which contradicts the hypothesis. Then, g(t) − g(0) − (M + ε) t ≤ 0 for everyt ∈ [0, T ]. Letting ε tend to zero, we obtain the assertion. ��Proposition 5.13.14 (Nagumo–Bony). Let X ∈ T (RN) be a smooth vector fieldin R

N , and let F be a relatively closed subset of Ω . Then F is positively X-invariantif and only if

〈XI (y), ν〉 ≤ 0 ∀ y ∈ F ∗ ∀ ν⊥F at y. (5.113)


Proof. We only need to show the “if” part. Let γ : [0, T ] → Ω be an integral curveof X such that x0 := γ (0) ∈ F . Define

δ(t) := dist(γ (t), F ), 0 ≤ t ≤ T .

We have to prove that δ(t) = 0 for every t ∈ [0, T ].Let V be a bounded neighborhood of x0 containing γ ([0, T ]), and let

L := supx,z∈V, x �=z

|XI (x)−XI (z)||x − z| (5.114)

be the Lipschitz constant of X on V . We may suppose LT < 1/2 and V = D(x0, r)

with D(x0, 2r) ⊆ Ω . We claim that

L(t) := lim suph→0−

δ(t + h)− δ(t)

h≤ Lδ(t) ∀ t ∈ (0, T ]. (5.115)

If δ(t) = 0, inequality (5.115) is trivial, since h < 0 and δ(t + h) ≥ 0. Supposeδ(t) > 0 and choose a sequence hn ↑ 0 such that

L(t) = limn→∞

δ(t + hn)− δ(t)

hn

.

Let us now denote x := γ (t) and xn := γ (t + hn). Since γ ([0, T ]) ⊂ D(x0, r) andD(x0, 2r) ⊆ Ω , for every n ∈ N there exists a point zn ∈ F ∩Ω such that

|xn − zn| = dist(xn, F ) = δ(t + hn).

Obviously, we may suppose that zn → z ∈ F ∩D(x0, r), so that, since xn → x,

|x − z| = limn→∞ |xn − z| = lim

n→∞ dist(xn, F )

= dist(x, F ) = δ(t).

Moreover,

ν := 1

2(x − z)⊥F at z. (5.116)

Then

δ(t + hn)− δ(t) = |xn − zn| − |x − z| ≥ |xn − zn| − |x − zn|≥ |xn − x| ≥ −〈xn − x, zn − x〉

|x − zn| .

Hence

L(t) ≤ limn→∞

⟨x − zn

|x − zn| ,xn − x

hn

⟩=⟨γ (t),

x − z

|x − z|⟩

= 2

|x − z| 〈XI (x), ν〉

= 2

|x − z|(〈XI (z)−XI (x), ν〉 + 〈XI (z), ν〉).


From (5.116) and (5.113), together with (5.114), we finally get

L(t) ≤ L |x − z| = Lδ(t).

This completes the proof of (5.115). This inequality, combined with Lemma 5.13.13,gives

δ(t) ≤ δ(t)− δ(0) ≤ LT sup[0,T ]

δ,

so that sup[0,T ] δ ≤ 1/2 · sup[0,T ] δ. Hence δ ≡ 0, and the proof is complete. ��Corollary 5.13.15. The closed set F is X-invariant if and only if

〈XI (y), ν〉 = 0, ∀ y ∈ F ∗ ∀ ν⊥F at y.

Proof. It straightforwardly follows from Proposition 5.13.14 and Definiti-on 5.13.12. ��

This corollary, together with Corollary 5.13.11, immediately gives the followingresult.

Corollary 5.13.16. Let L = ∑mj=1 X2

j be a sub-Laplacian on the homogeneous

Carnot group G. Let u : Ω → R be a C2 function satisfying the inequalities (5.106).Let F := {x ∈ Ω : u(x) = 0}. Assume ∅ �= F �= Ω . Then F is invariant withrespect to X1, . . . , Xm.

Proposition 5.13.17. Let F be a relatively closed subset of the open set Ω ⊆ RN .

Assume ∅ �= F �= Ω . Then

a := {X ∈ T (RN) : F is X-invariant}is a Lie algebra of vector fields.

Proof. Let X, Y ∈ a, and let λ,μ be real constants. By Corollary 5.13.15, we have〈XI (y), ν〉 = 〈YI (y), ν〉 = 0 for every y ∈ F ∗ and for every ν⊥F at y. Then

〈λ XI (y)+ μYI (y), ν〉 = 0 ∀ y ∈ F ∗ ∀ ν⊥F at y.

Hence a is a linear space. The next lemma will complete the proof of the proposi-tion. ��Lemma 5.13.18. In the notation of Proposition 5.13.17, if X, Y ∈ a, then[X, Y ] ∈ a.

Proof. Let y ∈ F ∗ and ν⊥F at y. For every t > 0 define

Γ (t) := ( exp(−√tY ) ◦ exp(−√tX) ◦ exp(√

tY ) ◦ exp(−√tX))(y).


Here ◦ denotes the composition of maps, whereas “exp” denotes the exponentialof a vector field as introduced in Definition 1.1.2, page 8.

Let T > 0 be such that Γ (t) ∈ Ω for 0 ≤ t ≤ T . By using the Taylor expan-sion (1.7) of exp (on page 7) with n = 2, a direct computation gives

Γ (t) = y + t(JYI (y) ·XI (y)− JXI (y) · YI (y)

)+ o(t)

= y + t[X, Y ]I (y)+ o(t), as t ↓ 0.

Then

limt↓0

Γ (t)− y

t= [X, Y ]I (y). (5.117)

On the other hand, since F is X and Y invariant, Γ (t) ∈ F for every t ∈ [0, T ]. Asa consequence, since D(y + ν, |ν|) ⊆ (Ω \ F) ∪ {y} and Γ (0) = y,

|Γ (t)− (y + ν)|2 ≥ |ν|2 = |Γ (0)− (y + ν)|2.Then, by using also (5.117), we have

0 ≥ d

d t

∣∣∣∣t=0|Γ (t)− (y + ν)|2 = 2〈[X, Y ]I (y), ν〉.

Hence〈[X, Y ]I (y), ν〉 ≤ 0 ∀ y ∈ F ∗ ∀ ν⊥F at y.

By swapping X with Y , we also get 〈[Y,X]I (y), ν〉 ≤ 0, so that

〈[X, Y ]I (y), ν〉 = 0 ∀ y ∈ F ∗ ∀ ν⊥F at y.

Then, by Corollary 5.13.15, [X, Y ] ∈ a. ��Finally, we are in the position to give the proof of Theorem 5.13.8.

Proof (of Theorem 5.13.8). Let us define

F := {x ∈ Ω : u(x) = 0 }.By hypothesis, x0 ∈ F . Then F is a non-empty relatively closed subset of Ω . Wehave to prove that F = Ω . By contradiction, assume F �= Ω . Then, since Ω isconnected, F ∗ �= ∅. Let y ∈ F ∗, and let ν⊥F at y. By Proposition 5.13.17,

〈ZI (y), ν〉 = 0 ∀Z ∈ g.

Since dim(g) = N , this obviously implies ν = 0. On the other hand, by the verydefinition of a vector orthogonal to F , we have ν ∈ R

N \ {0}. This contradictioncompletes the proof. ��


5.13.1 A Decomposition Theorem for L-harmonic Functions

In this section, we give a decomposition theorem for L-harmonic functions, resem-blant to the decomposition of a holomorphic function on an annulus of C into thesum of the regular and singular parts from its Laurent expansion (for the classicalcase of the Laplace operator, see also S. Axler, P. Bourdon, W. Ramey [ABR92]).Our main tool is the maximum principle from the previous section (precisely, we useCorollary 5.13.7, page 296).

In the sequel, we assume Q ≥ 3. Moreover, in the proof of Theorem 5.13.20, weadopt the following notation: G = (RN, ◦, δλ) is a homogeneous Carnot group, L isa sub-Laplacian on G, Γ = d2−Q is the fundamental solution for L (see Proposition5.4.2). If d is the above L-gauge, A is any subset of G and λ > 0, we agree to set

Aλ := {x ∈ G | d-dist(x,A) < λ},where

d-dist(x,A) := inf{d(x−1 ◦ a) | a ∈ A}.Moreover, we use the following simple lemma.

Lemma 5.13.19. Let K be a compact subset of G, and let f be bounded on K . Thenthe function

F : G→ R, F (x) :=∫

K

Γ (y−1 ◦ x) f (y) dy

is L-harmonic on G \K and vanishes at infinity. Moreover, if μ is a Radon measureon R

N with compact support K , the same is true for G(x) := ∫RN Γ (y−1◦x) dμ(y).

Proof. It easily follows by differentiation under the integral sign (recall also that Γ

is locally integrable and vanishes at infinity). ��We are now ready to state and prove the following assertion.

Theorem 5.13.20 (The decomposition theorem). Let the hypotheses in the incipitof this section hold. Let K ⊂ Ω ⊆ G, with K compact and Ω open. If u is L-harmonic in Ω \K , then u has a decomposition of the form

u = r + s,

where r is L-harmonic in Ω and s is L-harmonic in G \ K . Furthermore, it canbe assumed that s vanishes at infinity, and in this case the above decomposition isunique.

Proof. Suppose the theorem holds true whenever Ω is bounded. We show that itholds true even for an unbounded Ω . Indeed, let u ∈ H(Ω \K), where K is compactand Ω is an (unbounded) open set. Let R > 0 be such that K ⊂ Bd(0, R). SetΩ := Ω ∩Bd(0, R). Then K ⊂ Ω and, by our assumption, u can be decomposed asu = r + s, where r ∈ H(Ω), s ∈ H(G \K) and s → 0 at infinity. We consider thefunction r := u−s on Ω . Then r is L-harmonic in Ω\K and extends L-harmonically


across K , since r coincides with r on Ω (which is an open neighborhood of K). Thisends the proof, since u = r+s, with r ∈ H(Ω), s ∈ H(G\K) and s → 0 at infinity.

We can then suppose that Ω is bounded. We fix the following notation (see alsoFig. 5.1):

B(λ) := closure of (∂Ω)λ, C(λ) := closure of Kλ, A(λ) := Ω \ (B(λ) ∪ C(λ)).

Fig. 5.1. Figure for the proof of Theorem 5.13.20

Since K is compact and ∂Ω is closed, we can choose λ > 0 small enough so that

B(λ) ∩ C(λ) = ∅.Since Ω is bounded, we can choose a cut-off function ψλ ∈ C∞0 (RN) such that

supp(ψλ) ⊂ Ω \K, ψλ ≡ 1 on A(λ).

We consider the function uψλ, and we agree to consider this function triviallyprolonged on R

N to be zero. Hence this trivial prolongation belongs to C∞0 (RN).By (5.16) in Theorem 5.3.3 (page 237) (being ψλ ≡ 1 on A(λ)),

u(x) = (uψλ)(x) = −∫

RN

Γ (x−1 ◦ y)L(uψλ)(y) dy

= −(∫

A(λ)

+∫

B(λ)

+∫

C(λ)

+∫

RN\(A(λ)∪B(λ)∪C(λ))

)

= −(∫

B(λ)

+∫

C(λ)

)Γ (y−1 ◦ x)L(uψλ)(y) dy ∀ x ∈ A(λ). (5.118)

In the last equality we used the fact that ψλ ≡ 1 on A(λ) jointly with Lu = 0 on Ω ,and the fact that ψλ is supported in Ω . We now set

rλ(x) := −∫

B(λ)

Γ (y−1 ◦ x)L(uψλ)(y) dy for x ∈ Ω \ B(λ),

sλ(x) := −∫

C(λ)

Γ (y−1 ◦ x)L(uψλ)(y) dy for x ∈ G \ C(λ).


Hence, (5.118) gives the decomposition

u(x) = rλ(x)+ sλ(x) ∀ x ∈ A(λ). (5.119)

From Lemma 5.13.19, we infer that rλ and sλ are L-harmonic on the respective setsof definition and that sλ vanishes at infinity.

Let now 0 < μ < λ. Then obviously A(λ) ⊂ A(μ). From the decompositionin (5.119), we get an analogous decomposition

u(x) = rμ(x)+ sμ(x) ∀ x ∈ A(μ). (5.120)

We claim that the decompositions (5.119) and (5.120) are compatible, i.e.

rλ(x) = rμ(x) ∀ x ∈ Ω \ B(λ),

sλ(x) = sμ(x) ∀ x ∈ G \ C(λ).(5.121)

To prove the claim, we first remark that from (5.119) and (5.120) we obtain

rλ(x)− rμ(x) = sμ(x)− sλ(x) ∀ x ∈ A(λ). (5.122)

Now, let us consider the following function

S : G→ R, S(x) :={

sμ(x)− sλ(x) for every x ∈ G \ C(λ),

rλ(x)− rμ(x) for every x ∈ Ω \ B(λ).

We claim that S has the following properties:

i) S is well-posed: indeed, thanks to (5.122), sμ− sλ coincides with rλ− rμ on theset

{G \ C(λ)} ∩ {Ω \ B(λ)} = A(λ);ii) S vanishes at infinity: indeed, this is true for sμ and sλ;

iii) S is L-harmonic on G: indeed, this is true for sμ − sλ on the open set G \ C(λ),and this is true for rλ − rμ on the open set Ω \ B(λ) (recall that C(μ) ⊂ C(λ)

and B(μ) ⊂ B(λ)).

Now, by the maximum principle (precisely, see Corollary 5.13.7, page 296) weinfer that S ≡ 0, which is equivalent to the claimed (5.121). Now, let us fix a de-creasing sequence of positive λn’s such that λn → 0 as n→∞. We set

r(x) := rλn(x) ∀ x ∈ Ω (where n ∈ N is such that x ∈ Ω \ B(λn)),

s(x) := sλn(x) ∀ x ∈ G \K (where n ∈ N is such that x ∈ G \ C(λn)).

Thanks to (5.121), the definition of r(x) and s(x) are unambiguous.11 Now, the de-composition

u(x) = r(x)+ s(x) ∀ x ∈ Ω \K

11 We are also using the trivial fact that Ω \B(λn) ↑ Ω and RN \C(λn) ↓ R

N \K as n→∞.


follows from the analogous decomposition (5.119) (and the fact that A(λ) ↑ Ω \K

as n→∞). Moreover, it is easily seen that r ∈ H(Ω), s ∈ H(G\K) and s vanishesat infinity. This gives the desired decomposition of u as in the assertion.

Finally, the uniqueness of the decomposition in the assertion (under the assump-tion that s vanishes at infinity) is another easy consequence of the same maximumprinciple quoted above. Indeed, suppose we are given two decompositions

r2 + s2 = u = r1 + s1 on Ω \K,

where

ri ∈ H(Ω), si ∈ H(G \K), limx→∞ si(x) = 0, i = 1, 2.

Then setting

S : G→ R, S(x) :={

s1(x)− s2(x) for every x ∈ G \K ,

r2(x)− r1(x) for every x ∈ Ω ,

we see that S ∈ H(G), S vanishes at infinity, and we end the proof following thesame arguments as in the previous paragraph. ��

5.14 Appendix B. The Improved Pseudo-triangle Inequality

Let G = (RN, δλ, ◦) be a homogeneous Carnot group, and let d be a symmetrichomogeneous norm on G, smooth out of the origin.12 For example, d could be anL-gauge on G for some sub-Laplacian L on G.

We know from Proposition 5.1.7 (page 231) that (even without assumptions onsmoothness or symmetry of d) d satisfies the pseudo-triangle inequality

d(a ◦ b) ≤ c (d(a)+ d(b)) for every a, b ∈ G.

Here c≥1 is a constant depending on d and G.The aim of this brief appendix is to prove the following improvement of the

pseudo-triangle inequality. For the following result, see also [DFGL05, (2.6)].

Proposition 5.14.1 (The improved pseudo-triangle inequality). Let d be a sym-metric homogeneous norm on the homogeneous Carnot group G. Furthermore, sup-pose d is smooth out of the origin. Then there exists a constant β ≥ 1 dependingonly on d and G such that

d(y ◦ x) ≤ β d(y)+ d(x) for every x, y ∈ G. (5.123)

12 In other words (see the definition at the beginning of Section 5.1, page 229), the present d

has the following properties:d ∈ C(G, [0,∞)) ∩ C∞(G \ {0}); d(δλ(x)) = λ d(x) for every λ > 0 and x ∈ G;

d(x) = 0 iff x = 0; d(x−1) = d(x) for every x ∈ G.

5.14 Appendix B. The Improved Pseudo-triangle Inequality 307

Proof. Since (5.123) holds when y = 0, we can suppose y �= 0. Since d is δλ-homogeneous of degree one, (5.123) is equivalent to

d(δ1/d(y)(y) ◦ δ1/d(y)(x)) ≤ β + d(δ1/d(y)(x)). (5.124a)

By using the symmetry of d and setting

ξ−1 = δ1/d(y)(y), η−1 = δ1/d(y)(x),

(5.124a) is equivalent to (note that d(δ1/d(y)(y)) = 1)

d(η ◦ ξ)− d(η) ≤ β for every ξ, η ∈ G: d(ξ) = 1. (5.124b)

By the usual13 pseudo-triangle inequality for d , (5.124b) holds when η ∈ Bd(0,M),i.e. d(η) ≤ M (where M = M(d, G) % 1 will be chosen in the sequel). Indeed, ifη ∈ Bd(0,M), we have

d(η ◦ ξ)− d(η) ≤ c (d(η)+ d(ξ))− d(η) = (c− 1)d(η)+ c≤(c− 1)M + c =: β.

We can hence suppose η /∈ Bd(0,M). Roughly speaking, we will show that wecan drop η from (5.124b), by an argument of left-translation along curves which aresupported away from zero, when M is large enough. Then, (5.124b) will follow fromthe classical mean value theorem.

We now make this precise. Set Z := Log (ξ) ∈ g, where Log is the logarithmicmap related to G and g is the Lie algebra of G. Consider the integral curve γ of Z

starting from η, i.e. with our usual notation

γ (t) = exp(tZ)(η) = η ◦ exp(tZ)(0) = η ◦ Exp (tZ).

Here we used Corollary 1.2.24 (page 24) and the definition of exponential map(see page 24). Obviously, we have γ (0) = η and γ (1) = η ◦ Exp (Z) = η ◦Exp (Log (ξ)) = η ◦ ξ . If we show that we can choose M % 1 such that

d(γ (t)) ≥ 1 for every t ∈ [0, 1] (5.124c)

(recall that γ depends on η, besides ξ ), then [0, 1] � t �→ u(t) := d(γ (t)) is smooth(for d is smooth out of the origin by hypothesis) so that the classical Lagrange meanvalue theorem applies and gives

d(η ◦ ξ)− d(η) = u(1)− u(0) ≤ supt∈[0,1]

|u(t)| = supt∈[0,1]

∣∣∣⟨(∇d)(γ (t)), γ (t)

⟩∣∣∣

= supt∈[0,1]

∣∣∣⟨(∇d)(γ (t)), (ZI)(γ (t))

⟩∣∣∣

= supt∈[0,1]

∣∣(Zd)(γ (t))∣∣. (5.124d)

Let X1, . . . , XN be the Jacobian basis for the Lie algebra g of G. Hence,

13 See Proposition 5.1.7-1, page 231.


Z = Log (ξ) =N∑

j=1

pj (ξ) Zj , (5.124e)

where the pj ’s are polynomials, so that there exists a constant C such that

supd(ξ)=1

|pj (ξ)| ≤ C1 for all j = 1, . . . , N , (5.124f)

since {ξ ∈ G : d(ξ) = 1} is a compact set (see, e.g. Proposition 5.1.4, page 230).Moreover, Zj is δλ-homogeneous of degree σj ≥ 1 (see Corollary 1.3.19, page 42),so that Zjd is δλ-homogeneous of degree 1 − σj ≤ 0. Consequently, it is boundedon G \ Bd(0, 1), say

supd(ζ )≥1

|(Zjd)(ζ )| ≤ C2 for all j = 1, . . . , N . (5.124g)

We now use again the claimed (5.124c) and, by collecting together (5.124f) to(5.124g), we derive that (5.124d) yields

d(η ◦ ξ)− d(η) ≤ supd(ζ )≥1

∣∣(Zd)(ζ )∣∣

= supd(ζ )≥1

∣∣∣∣∣

N∑

j=1

pj (ξ) (Zjd)(ζ )

∣∣∣∣∣ ≤ NC1C2.

This proves (5.124b). We are then left to prove (5.124c). From the pseudo-triangleinequality for d (see Proposition 5.1.7-2, page 231) we have

d(γ (t)) = d(η ◦ Exp (tZ)) ≥ 1

cd(η)− d(Exp (tZ))

≥ 1

cM − sup

t∈[0,1], d(ξ)=1d(Exp (t Log (ξ))) =: 1

cM −m(d,G),

whence (5.124c) follows by choosing M = c(1 + m(d, G)). The finiteness ofm(d, G) follows from

d(Exp (t Log (ξ))) = d

(Exp

(N∑

j=1

tpj (ξ) Zj

))

≤ sup|ζj |≤C1

d

(Exp

(N∑

j=1

ζj Zj

))<∞,

uniformly for t ∈ [0, 1], d(ξ) = 1. Here we used (5.124f) and the fact that

q(ζ ) := Exp

(N∑

j=1

ζj Zj

)

has polynomial coefficient functions (see (1.75a), page 50). This completes theproof. ��

5.15 Appendix C. Existence of Geodesics 309

Note that, β being the constant in Proposition 5.14.1, if we replace x, y in (5.123)by, respectively, y−1 ◦ z and z−1 ◦ x, we get

d(y−1 ◦ x) ≤ β d(y−1 ◦ z)+ d(z−1 ◦ x) for every x, y, z ∈ G. (5.125a)

Moreover, by interchanging y and z in the above inequality (and using the symmetryof d) one gets

|d(y−1 ◦ x)− d(z−1 ◦ x)| ≤ β d(y−1 ◦ z) for every x, y, z ∈ G. (5.125b)

5.15 Appendix C. Existence of Geodesics

Let G = (RN, δλ, ◦) be a homogeneous Carnot group. Let

g = W(1) ⊕W(2) ⊕ · · · ⊕W(r)

be a stratification of the Lie algebra of G, as in Remark 1.4.8 (page 59). Let X ={X1, . . . , Xm} be any basis of W(1). We consider the related Carnot–Carathéodorydistance dX. The aim of this section is to prove that the “inf” defining dX in (5.6) onpage 232 is actually a minimum.

In other words, fixed any x, y ∈ G, we show the existence of a X-subunit curveγ : [0, T ] → G connecting x and y (i.e. γ (0) = x, γ (T ) = y) such that T =dX(x, y). We shall call any such curve a X-geodesic (for x and y).

The existence of geodesics can be proved in many general cases (see [Bus55]and [HK00]). Our argument here will make crucial use of the δλ-homogeneity andleft-invariant properties of the system X. The resulting proof will be quite simple(for a more general proof, see the note after Theorem 5.15.5).

First, we recall that, by Propositions 5.2.4, 5.2.6 and Theorem 5.2.8,

dX(x, y) = d0(y−1 ◦ x) for every x, y ∈ G, (5.126)

whered0(z) := dX(z, 0), z ∈ G, (5.127)

and d0 is a homogeneous (symmetric) norm on G. As usual, we denote the dilationof G by

δλ(x) = δλ(x1, . . . , xN) = (λσ1x1, . . . , λσN xN), λ > 0, x ∈ G,

where 1 = σ1 ≤ · · · ≤ σN = r are consecutive integers and r is the step of nilpo-tency of G. We are ready to prove the following result.

Proposition 5.15.1. Let G be a homogeneous Carnot group, and let d be any homo-geneous norm on G.

For every compact set K ⊂ G, there exists a constant cK > 0 such that

(cK)−1|x − y| ≤ d(y−1 ◦ x) ≤ cK |x − y|1/r ∀ x, y ∈ K, (5.128)

where r is the step of G and | · | is the Euclidean norm on G ≡ RN .


In particular, (5.128) holds true if d(y−1 ◦ x) is replaced by dX(x, y), wheredX is the control distance related to any basis of generators (of the first layer of thestratification of the Lie algebra) of G.

(Note. More generally, the estimates in (5.128) also hold when d = dX (with asuitable r), where X is a system of smooth vector fields satisfying the Hörmandercondition, see [Lan83,NSW85,VSC92,Gro96]. The first equality in (5.128) holds fora general Carnot–Carathéodory distance d = dX, see [HK00] and Ex. 25 at the endof the chapter.)

Proof. Once (5.128) has been proved, the last assertion of the proposition followsfrom (5.126), by taking d = d0 (d0 as in (5.127)).

We then turn to prove (5.128). If | · | denotes the absolute value on R, set

� : G→ [0,∞), �(x) =N∑

j=1

|xj |1/σj . (5.129)

Obviously, � is a homogeneous (symmetric) norm on G. By the equivalence ofall homogeneous norms on G (see Proposition 5.1.4), there exists a constant c =c(�, d, G) ≥ 1 such that

c−1 �(x) ≤ d(x) ≤ c �(x) ∀ x ∈ G. (5.130)

Thus (5.128) will follow if we show that, given a compact set K ⊂ G,

(cK)−1|x − y| ≤ �(y−1 ◦ x) ≤ cK |x − y|1/r ∀ x, y ∈ K, (5.131)

for a suitable constant cK > 0.Now, we recall the result in Corollary 1.3.18 (page 41): for every j ∈ {1, . . . , N}

and every x, y ∈ G, we have

(y−1 ◦ x)j = xj − yj +∑

k: σk<σj

P(j)k (x, y) (xk − yk),

where P(j)k (x, y) is a polynomial function. Thus, the very definition of � gives

�(y−1 ◦ x) =N∑

j=1

∣∣∣∣ xj − yj +∑

k: σk<σj

P(j)k (x, y) (xk − yk)

∣∣∣∣1/σj

. (5.132)

Since any P(j)k is a continuous function, we immediately get from (5.132) the esti-

mate from above: for every x, y ∈ K

�(y−1 ◦ x) ≤N∑

j=1

(|xj − yj | + c

∑k: σk<σj

|xk − yk|)1/σj

≤ c′N∑

j=1

( ∑

k: σk≤σj

|xk − yk|)1/σj

≤ c′N∑

j=1

∑

k: σk≤σj

|xk − yk|1/σj


≤ c′′N∑

j=1

|xj − yj |1/σj ≤ c′′′N∑

j=1

|xj − yj |1/σN

≤ c′′′N(

N∑

j=1

|xj − yj |)1/σN

≤ c′′′N1+1/σN |x − y|1/σN .

This gives the estimate from above in (5.131), since σN = r . (Yet another proof ofthe estimate from above can be obtained by the Lagrange mean value theorem.14)

The estimate from below is just a little more delicate. We fix the notation: forevery j = 2, . . . , N , let nj be the cardinality of the set {k : σk < σj }. Let α be avector in R

n with n = n2 + · · · + nN , and let us denote α in the following way

α = (α(2), . . . , α(N))

with α(j) = (α(j)

1 , . . . , α(j)nj

), j = 2, . . . , N.

With the notation in (5.132), if K is a compact subset of G, there exists a constantM ≥ 1 such that (recall that the P

(j)k ’s are polynomial functions)

supx,y∈K

|P (j)k (x, y)| ≤ M ∀ j = 2, . . . , N ∀ k : σk < σj .

As a consequence, (5.132) gives (here M = n M)

�(y−1 ◦ x) =N∑

j=1

∣∣∣∣xj − yj +∑

k: σk<σj

P(j)k (x, y) (xk − yk)

∣∣∣∣1/σj

14 Indeed, fix any y ∈ G and set

fy : RN → R, fy(x) := (y−1 ◦ x)j .

Given any x, x0 ∈ G, by the Lagrange mean value theorem, we have

(�) |fy(x)− fy(x0)| ≤ sup|ξ−x0|≤|x−x0|

|∇fy(ξ)| |x − x0|.

Now, take x0 = y and observe that fy(x0) = fy(y) = 0. Moreover, if x, y belong to acompact set K , and if RK % 1 is such that K ⊆ B(0, RK), then

sup|ξ−y|≤|x−y|

|∇fy(ξ)| ≤ sup|ξ |≤RK+diam(K)

|∇fy(ξ)| =: Mj <∞,

for fy(ξ) is a polynomial function in ξ and y. Thus, (�) gives (set M = maxj≤N Mj )

|(y−1 ◦ x)j | ≤ M |x − y| ∀ x, y ∈ K ∀ j ≤ N.

Then, for every x, y ∈ K , we have

�(y−1 ◦ x) =N∑

j=1

|(y−1 ◦ x)j |1/σj ≤ c∑N

j=1|x − y|1/σj ≤ c′|x − y|1/r .


≥ inf|α|≤M

{N∑

j=1

∣∣∣∣xj − yj +∑

k: σk<σj

α(j)k (xk − yk)

∣∣∣∣1/σj

}=: (�).

Being x, y ∈ K , we obviously have (whenever α ≤ M)∣∣∣∣xj − yj +

∑

k: σk<σj

α(j)k (xk − yk)

∣∣∣∣ ≤ c(K, M) <∞,

hence (notice that 1/σj ≤ 1)

(�) ≥ c(K, M)1/r

c(K, M)inf|α|≤M

{N∑

j=1

∣∣∣∣xj − yj +∑

k: σk<σj

α(j)k (xk − yk)

∣∣∣∣

}

= c′(K) inf|α|≤M

{fα(x − y)

} =: (�′),

where

fα : RN → R, fα(z) =N∑

j=1

∣∣∣∣zj +∑

k: σk<σj

α(j)k zk

∣∣∣∣.

We explicitly remark that fα is homogeneous of degree 1 with respect to the Euclid-ean dilations on R

N . As a consequence (if x �= y, otherwise there is nothing toprove),

fα(x − y) = |x − y| fα

(x − y

|x − y|)≥ |x − y| inf|ξ |=1

fα(ξ).

This gives

(�′) ≥ |x − y| c′(K) inf|α|≤M, |ξ |=1fα(ξ) ≥ |x − y| c′′(K).

Indeed, since fα(ξ) is a continuous function in ξ and α, and (for every α) fα(·) ispositive outside the origin (as a simple inductive argument shows), there exist α0 andξ0 (with |α0| ≤ M and |ξ0| = 1) such that

inf|α|≤M, |ξ |=1fα(ξ) = fα0(ξ0) > 0.

This gives the lower estimate in (5.131) and completes the proof. ��From Proposition 5.15.1 we obtain useful corollaries, as we show below. In the

sequel, we write dE to denote the usual Euclidean metric on G ≡ RN .

Corollary 5.15.2. Let G be a homogeneous Carnot group. Let dX be the controldistance related to any basis X of generators (of the first layer of the stratification ofthe Lie algebra) of G.


Then A ⊆ G is bounded in the metric space (G, dX) if and only if it is boundedin the Euclidean metric space (G, dE). More precisely, there exists a constant c≥1such that (for every R > 0)

BdX(0, R) ⊆ BdE

(0, R1(R)) and BdE(0, R) ⊆ BdX

(0, R2(R)), (5.133)

whereR1(R) = cmax{R,Rr}, R2(R) = cmax{R,R1/r}.

(Note. The fact that a bounded set in (RN, dE) is also bounded in (RN, dX)

holds for a dX related to a general system of vector fields satisfying the Hörmandercondition, see the note after Proposition 5.15.1. The reverse assertion may be falseeven in the Hörmander case, see Ex. 27.)

Proof. By (5.130), we know that there exists a constant c = c(�, dX, G) ≥ 1 suchthat

c−1 �(x) ≤ dX(x, 0) ≤ c �(x) ∀ x ∈ G, (5.134)

where � is as in (5.129). Now, it is obvious that a set A ⊆ G is bounded in (G, dE) iffthere exists R > 0 such that A ⊆ B�(0, R). This remark jointly with (5.134) provesthe first assertion of the corollary.

Precisely (see the definition of �), it holds (for every R > 0)

BdE(0, R) ⊆ B�

(0, N max{R,R1/r}), B�(0, R) ⊆ BdE

(0, N max{R,Rr}).

These inclusions, together with (5.134), prove (5.133). ��Obviously, (5.133) also gives the inclusions

BdX(0, R3(R)) ⊆ BdE

(0, R) and BdE(0, R4(R)) ⊆ BdX

(0, R), (5.135)

whereR3(R) = c−1 min{R,R1/r}, R1(R) = c−r min{R,Rr}.

Corollary 5.15.3. Let the hypothesis of Corollary 5.15.2 hold.Then the map

id : (G, dX)→ (G, dE), id(x) = x

is a homeomorphism. As a consequence, the topologies of the metric spaces (G, dX),(G, dE) coincide.

(Note. The continuity of id : (G, dX) → (G, dE) holds for a general Carnot–Carathéodory distance dX, see Ex. 25 at the end of the chapter. For a general dX, thereverse continuity may be false, see Ex. 26 (see also [BR96]). Instead, this reversecontinuity holds for all smooth vector fields satisfying the Hörmander condition, seethe note after Proposition 5.15.1 and Ex. 25.)


Proof. Let {xn}n∈N be a sequence in G. Suppose xn → x in (G, dE). Then thereexists a compact set K ⊂ R

N such that x, xn ∈ K for every n ∈ N. Hence, by thesecond inequality in (5.128), we derive

dX(x, xn) ≤ cK |x − xn| ∀ n ∈ N,

so that, letting n→∞, xn → x in (G, dX) too.Vice versa, suppose xn → x in (G, dX). Then the sequence {xn}n is bounded in

(G, dX). Hence, by Corollary 5.15.2, it is also bounded in (G, dE). Consequently,there exists a compact subset K of R

N such that x, xn ∈ K for every n ∈ N. Hence,by the first inequality in (5.128), we derive

|x − xn| ≤ cK dX(x, xn) ∀ n ∈ N,

so that, letting n→∞, xn → x in (G, dE) too.We have thus proved that

xndX−→ x ⇒ id(xn)

dE−→ x, xndE−→ x ⇒ id−1(xn)

dX−→ x.

As a consequence, by the well-known characterization of continuity (in a metricspace) as sequential-continuity, we infer that id : (G, dX) → (G, dE) is a home-omorphism. This implies that id and id−1 are open maps, i.e. (G, dX) and (G, dE)

have the same open sets. ��By collecting together the above lemmas, we obtain the following result.

Proposition 5.15.4. Let G be a homogeneous Carnot group. Let dX be the controldistance related to any basis X of generators (of the first layer of the stratification ofthe Lie algebra) of G.

Then, a subset of G is, respectively, open, closed, bounded or compact in themetric space (G, dX) if and only if the same holds in the Euclidean metric spaceG ≡ R

N .In particular, the compact subsets of (G, dX) are precisely the closed and

bounded subsets of RN or, equivalently, the closed and bounded subsets of (G, dX).

Proof. Let A ⊆ G. Then, by Corollary 5.15.3, A is, respectively, open, closed orcompact in the metric space (G, dX) if and only if the same holds in (G, dE).

By Corollary 5.15.2, A is bounded in (G, dX) if and only if the same holdsin (G, dE). This also gives the last assertion of the proposition and completes theproof. ��

We are in a position to prove the following result.

Theorem 5.15.5 (Existence of X-geodesics). Let G be a homogeneous Carnotgroup. Let dX be the control distance related to any basis X of generators (of thefirst layer of the stratification of the Lie algebra) of G.

Then, for every x, y ∈ G, there exists a X-geodesic connecting x and y, i.e.there exists a X-subunit path γ : [0, T ] → G such that γ (0) = x, γ (T ) = y andT = dX(x, y).


(Note. Since we proved that the compact subsets of (G, dX) are precisely theclosed and bounded subsets of (G, dX) (see Proposition 5.15.4) this result also fol-lows from the general results in [Bus55, page 25].)

Proof. Fix x, y ∈ G. By definition of dX, there exists a sequence {γn}n∈N of X-subunit paths

γn : [0, Tn] → G, γn(0) = x, γn(Tn) = y, Tn ↘ dX(x, y).

By definition of X-subunit path, γn is an absolutely continuous curve such that, forevery n ∈ N, there exists En ⊆ [0, Tn] such that Fn := [0, Tn] \ En has vanishingLebesgue measure and

〈γn(t), ξ 〉2 ≤m∑

j=1

〈XjI (γn(t)), ξ 〉2 ∀ ξ ∈ RN ∀ t ∈ En. (5.136)

Since Tn ≤ Tn+1 for all n, it is not restrictive to suppose that every γn is defined on[0, T1], by prolonging γn to be y on (Tn, T1]. We still denote this prolongation by γn.Observe that the prolongation is still a X-subunit path (connecting x to y) because,for t ∈ (Tn, T1], γn is constant, whence the far left-hand side in (5.136) is 0. Theresulting prolongation is also trivially absolutely continuous.

We claim that the family of functions F = {γn}n on [0, T1] is uniformly boundedand equicontinuous.• F is equicontinuous: Suppose we have proved that F is uniformly bounded,

i.e.∃M > 0 : sup

t∈[0,T1]|γn(t)| ≤ M ∀ n ∈ N. (5.137)

Then the equicontinuity of F follows. Indeed, for every t, t ′ ∈ [0, T1], it holds

|γn(t)− γn(t′)| =

∣∣∣∣∫ t

t ′γn(s) ds

∣∣∣∣ ≤∫ t

t ′|γn(s)| ds

(see the note15) =∫ t

t ′sup

ξ∈RN : |ξ |=1

∣∣⟨γn(s), ξ⟩∣∣ ds

(by (5.136)) ≤∫ t

t ′sup

ξ∈RN : |ξ |=1

(m∑

j=1

〈XjI (γn(s)), ξ 〉2)1/2

ds

(Cauchy–Schwartz) ≤∫ t

t ′sup

ξ∈RN : |ξ |=1

(m∑

j=1

|XjI (γn(s))|2 |ξ |2)1/2

ds

15 Here, we use a well-known fact: if v ∈ RN , then

|v| = supξ∈RN : |ξ |=1

|〈v, ξ〉|.


=∫ t

t ′

∣∣(X1I, . . . , XmI)(γn(s))∣∣ ds

(by (5.137)) ≤ |t − t ′| sup|η|≤M

∣∣(X1I, . . . , XmI)(η)∣∣ ≤ M ′ |t − t ′|.

In the last inequality, we used the smoothness of the Xj ’s. Note that this proves morethan the equicontinuity, namely, F is a uniform Lipschitz-continuous family.• F is uniformly bounded: We are then left to prove (5.137). By the very defini-

tion of dX, for every t ∈ [0, T1], we have

dX(x, γn(t)) ≤ t ≤ T1.

As a consequence, by the triangle-inequality for the distance dX, we get

dX(0, γn(t)) ≤ dX(0, x)+ dX(x, γn(t)) ≤ dX(0, x)+ T1 <∞.

Hence, the set {γn(t) : n ∈ N, t ∈ [0, T1]} is bounded in (G, dX), so that, due toCorollary 5.15.2, the set is bounded in the Euclidean metric. This is precisely (5.137).

We are then entitled to apply the Arzelà–Ascoli theorem, ensuring that thereexists a subsequence of {γn}n∈N which converges uniformly on [0, T1], say to γ :[0, T1] → G. For the sake of brevity, we still denote this uniformly convergentsubsequence by {γn}n∈N.

Set T := dX(x, y). We aim to prove that the curve

γ ∗ : [0, T ] → G, γ ∗ := γ |[0,T ]

is X-subunit and γ connects x to y, which will end the proof since γ is defined on[0, T ].

First, γn(0) = x for every n ∈ N yields γ (0) = x, by pointwise convergence.Second, since γn converges uniformly to γ on [0, T1], and {Tn}n is a sequence in[0, T1] converging to T , we infer

γn(Tn)→ γ (T ),

so thatγ ∗(T ) = γ (T ) = lim

n→∞ γn(Tn) = y,

because γn(Tn) = y for every n ∈ N. This proves that γ ∗ connects x to y.Finally, we prove that γ is X-subunit. To begin with, we remark that γ is ab-

solutely continuous. Indeed, in proving the equicontinuity of F , we showed that thereexists a constant M ′ such that

|γn(t)− γn(t′)| ≤ M ′ |t − t ′| for every n ∈ N and every t, t ′ ∈ [0, T1].

Letting n → ∞, this shows that γ is Lipschitz continuous, whence it is absolutelycontinuous. Obviously, γ is X-subunit iff, for every fixed ξ ∈ R

N , the functions


f±(t) :=(

m∑

j=1

〈XjI (γ (t)), ξ 〉2)1/2

± 〈γ (t), ξ 〉

are non-negative a.e. on [0, T1]. To this end, it suffices to prove that (notice that f± ∈L1(0, T1)) ∫ T1

0f±(t) ϕ(t) dt ≥ 0 ∀ ϕ ∈ C∞0 (0, T1), ϕ ≥ 0. (5.138)

First, being γn X-subunit, it holds

0 ≤∫ T1

0

{(m∑

j=1

〈XjI (γn(t)), ξ 〉2)1/2

± 〈γn(t), ξ 〉}

ϕ(t) dt

=∫ T1

0

(m∑

j=1

〈XjI (γn), ξ 〉2)1/2

ϕ ±∫ T1

0〈γn, ξ 〉ϕ =: An ± Bn

for every n ∈ N and every ϕ ∈ C∞0 (0, T1), ϕ ≥ 0 (exploit (5.136)), whence

0 ≤ An ± Bn for every n ∈ N. (5.139)

Now, by dominated convergence (indeed, recall that Xj is smooth and {γn}n∈N isuniformly bounded), the limit of An is easily obtained:

An =∫ T1

0

(m∑

j=1

〈XjI (γn), ξ 〉2)1/2

ϕn→∞−→

∫ T1

0

(m∑

j=1

〈XjI (γ ), ξ 〉2)1/2

ϕ. (5.140)

Furthermore, the limit of Bn can be obtained by recalling that absolutely continuousfunctions support integration by parts:

Bn =∫ T1

0〈γn, ξ 〉ϕ =

N∑

i=1

ξi

∫ T1

0(γn)i ϕ = −

N∑

i=1

ξi

∫ T1

0(γn)i ϕ

n→∞−→ −N∑

i=1

ξi

∫ T1

0γi ϕ =

N∑

i=1

ξi

∫ T1

0γi ϕ =

∫ T1

0〈γ , ξ 〉ϕ. (5.141)

(Passing to the limit, we used another dominated convergence argument, recallingthat γn → γ uniformly.) As a consequence, letting n→∞ in (5.139), from (5.140)and (5.141) we infer

0 ≤∫ T1

0

(m∑

j=1

〈XjI (γ ), ξ 〉2)1/2

ϕ ±∫ T1

0〈γ , ξ 〉ϕ,

which is exactly (5.138) (by the very definition of f±). This completes theproof. ��


As a consequence of Theorem 5.15.5, we have the following segment propertyfor the Carnot–Carathéodory distance on a homogeneous Carnot group. (See alsoB. Franchi and E. Lanconelli [FL83] for the segment property in a sub-elliptic con-text.)

Corollary 5.15.6 (The segment property). Let G be a homogeneous Carnot group.Let dX be the control distance related to any basis X of generators (of the first layerof the stratification of the Lie algebra) of G.

Then the metric space (G, dX) has the segment property, i.e. for every x, y ∈ G,there exists a continuous curve γ : [0, T ] → G with γ (0) = x, γ (T ) = y, and suchthat

dX(x, y) = dX(x, γ (t))+ dX(γ (t), y) for every t ∈ [0, T ].For instance, γ can be any X-geodesic connecting x and y, whose existence isgranted by Theorem 5.15.5.

Proof. Let x, y ∈ G be fixed. Let γ be a X-geodesic connecting x and y: the exis-tence of such a X-geodesic is granted by Theorem 5.15.5. We claim that

dX(x, y) = dX(x, γ (t))+ dX(γ (t), y) for every t ∈ [0, T ].Fix t ∈ [0, T ]. First, by the very definition of dX (being γ a X-subunit curve), wehave

(�) dX(x, γ (t)) ≤ t.

(Actually, the equality will hold.) We next prove that

(��) dX(γ (t), y) ≤ T − t.

(Actually, the equality will hold.) Indeed, consider the curve

γ : [0, T − t] → G, γ (s) := γ (s + t).

Obviously, γ is X-subunit and γ (0) = γ (t), γ (T − t) = γ (t) = y. Hence, again bythe definition of dX, one has

dX(γ (t), y) = dX(γ (0), γ (T − t)) ≤ T − t,

i.e. (��) holds. Consequently, by (�) and (��) together with the triangle-inequality,we get

T = dX(x, y) ≤ dX(x, γ (t))+ dX(γ (t), y) ≤ t + (T − t) = T .

Hence, these inequalities are in fact equalities, and the proof is complete. (Inciden-tally, this also proves that the equality holds in (�) and (��).) ��Bibliographical Notes. Some of the topics presented in this chapter also appear in[BL01].


For other equivalent definitions of the Carnot–Carathéodory distance, see, e.g.[JSC87,NSW85]. See [HK00] and the references therein for applications of Carnot–Carathéodory distances in PDE’s problems. In particular, see the collection of papers[BR96] for an introduction to the geometry of C-C spaces. For explicit estimates ofthe Carnot–Carathéodory distance for “diagonal” vector fields, see [FL82]; for thesegment property, see [FL83]. See also [GN98].

The gauge functions on the Heisenberg group and on H-type groups were dis-covered by G.B. Folland [Fol75] and A. Kaplan [Kap80], respectively.

For mean value formulas for the Hörmander sum of squares, see [Lan90,CGL93];see [Gav77] for mean value formulas for the Heisenberg group; for a survey on meanvalue formulas in the classical setting of Laplace’s operator, see [NV94] (see also thelist of references therein for related results in a non-Riemannian setting).

The bibliography on Harnack-type inequalities for sub-elliptic operators is ex-tremely vast: see, e.g. [FL83] for the first result on these topics. See also [BM95,Bon69,FGW94,FL82,GL03,LM97a,LK00,SaC90,SCS91,Varo87].

For Liouville-type theorems for homogeneous operators, see [Gel83,KoSt85,LM97a,Luo97]. For results on the maximum principles and propagation, see [Ama79,Bon69,Hil70,Hop52,PW67,Red71,Spe81,Tai88].


Ex. 1) Give a detailed proof of Lemma 5.13.19, page 303.Ex. 2) The following operator

L :=n∑

j=1

((∂xj

)2 + (∂yj+ xj ∂t )

2)+ (∂z)2

in R2n+2 (the points are denoted by (x, y, z, t) with x = (x1, . . . , xn) ∈ R

n,y = (y1, . . . , yn) ∈ R

n, z ∈ R, t ∈ R) is not analytic-hypoelliptic (see[Rot84]). Find an explicit homogeneous Carnot group G such that L is thecanonical sub-Laplacian of G.

Ex. 3) Construct explicitly the Carnot group referred to in Example 5.10.6, page 286.(Hint: Model the needed composition law on the composition law in Exam-ple 5.10.5, page 285.)

Ex. 4) Consider the Kolmogorov-type group in Ex. 7 of Chapter 3, page 179. Bymeans of formula (5.92), verify that the fundamental solution of its canoni-cal sub-Laplacian is given by (when x �= 0)


Γ (x, t)

= c

∫

R2dτ1dτ2

|τ |sinh |τ |

×( {(τ2x2 − τ1x3)

2 + 2 |τ |tanh |τ | (|τ |2 x2

1 + (τ1x2 + τ2x3)2)}

{. . . the above braces . . .}2 + (t1τ1 + t2τ2)2

) 52

where c is the dimensional constant

c = 3

27 π3.

Ex. 5) Prove that if f ∈ L1loc(R

N), then

x �→ −∫

Bd(x,r)

|f (y)| dy

is a continuous function.Ex. 6) Provide a detailed proof of (5.61) at page 264.Ex. 7) Let u ∈ C2(Bd(0, R)) ∩ H

(Bd(0, R) \ Bd(0, r)

), 0 < r < R. Prove that

there exist constants C0, C1 such that

Mρ(u)(0) = C0 + C1

ρQ−2for every ρ ∈ (r, R).

Ex. 8) (The surface asymptotic sub-Laplacian). Let Ω ⊆ G be open, and letu ∈ C(Ω, R). Given x ∈ Ω , we say that the surface asymptotic sub-Laplacian of u at x is non-negative, written ALu(x) ≥ 0, if

lim infr→0+

Mr (u)(x)− u(x)

r2≥ 0.

If AL(−u)(x) ≥ 0, we write ALu(x) ≤ 0. Prove the following statements:(i) If ALu ≥ 0 and ALv ≥ 0, then AL(λ u+μ v) ≥ 0 for every λ,μ ≥ 0.

(ii) If ALu ≥ 0 and λ ≤ 0, then AL(λ u) ≤ 0.(iii) If u ∈ C2(Ω, R) and Lu ≥ 0, then ALu(x) ≥ 0 for every x ∈ Ω .(iv) If x0 ∈ Ω is a local maximum point for u, then ALu(x0) ≤ 0.

Ex. 9) (The solid asymptotic sub-Laplacian). Let Ω ⊆ G be open, and let u ∈C(Ω, R). Given x ∈ Ω , we say that the solid asymptotic sub-Laplacian ofu at x is non-negative, written ALu(x) ≥ 0, if

lim infr→0+

Mr (u)(x)− u(x)

r2≥ 0.

If AL(−u)(x) ≥ 0, we write ALu(x) ≤ 0. Prove the statements of theprevious exercise with AL replaced by AL.


Ex. 10) (Maximum principle for AL and AL). Let Ω ⊆ G be open and bounded.Let u ∈ C(Ω, R) be such that{ALu ≥ 0 in Ω ,

lim supy→x

u(y) ≤ 0 ∀ x ∈ ∂Ω , or

{ALu ≥ 0 in Ω ,lim sup

y→xu(y) ≤ 0 ∀ x ∈ ∂Ω .

Prove that u ≤ 0 in Ω . (Hint: Proceed as in the proof of the weak maximumprinciple, Theorem 5.13.4.)

Ex. 11) Provide the detailed proof of the result in Remark 5.6.4.Ex. 12) Let L be a sub-Laplacian on a homogeneous Carnot group G on R

N . Sup-pose that L has no first order differential terms, i.e. L(xk) ≡ 0 for everyk = 1, . . . , N . Let ψ ∈ C2(G, G) be such that

L(u ◦ ψ) = (L u) ◦ ψ ∀ u ∈ C2(G). (5.142)

Following the usual notation for the “stratification” of the variables of G,we set

ψ = (ψ(1), . . . , ψ(r))

(here ψ(i) ∈ C2(G, RNi ), N1 + · · · + Nr = N and Ni is the dimension of

the i-th layer in the stratification of the algebra of G). Prove that, for every1 ≤ j ≤ N1, it holds {

0 = Lψ(1)j

1 = ∣∣∇L ψ(1)j

∣∣2 .(5.143)

(Hint: Use (1.112) in Ex. 12, Chapter 1, page 83.)Let now u

(1)j := (ψ

(1)j )2. Then u

(1)j ≥ 0, and from (5.143) we have Lu

(1)j =

2. Using Liouville Theorem 5.8.4, derive that u(1)j is a polynomial of G-

degree ≤ 2 and therefore u(1)j = p1(x

(1)) + p2(x(2)), where p1 and p2 are

polynomials of ordinary degrees 2 and 1, respectively. Deduce that p2 ≡ 0.

Since |ψ(1)j | =

√u

(1)j , using, for example, Theorem 5.8.8, derive that ψ

(1)j

is a polynomial of G-degree ≤ 1, depending at most on x(1).Now suppose by induction that ψ(1), . . . , ψ(n) have polynomial componentfunctions of G-degree respectively 1, . . . , n at most. Write L = ∑p

k=1 X2k

with Xk = ∑Ni=1 σ

(k)i ∂i . Prove that (using again (1.112)) if i = n + 1 and

1 ≤ j ≤ Ni+1, it holds{

0 = Lψ(n+1)j∑p

k=1

(σ

(k)n+1, j

)2(ψ) = ∣∣∇L ψ

(n+1)j

∣∣2 .(5.144)

Let u(n+1)j := (ψ

(n+1)j )2; then u

(n+1)j ≥ 0 and, from (5.144), derive

L u(n+1)j = 2

p∑

k=1

(σ

(k)n+1, j

)2(ψ).


On the other hand, σ(k)n+1, j is a polynomial of G-degree n thus depending

only on x(1), . . . , x(n). Consequently, by induction hypothesis,

(σ

(k)n+1, j

)2(ψ) = (σ (k)

n+1, j (ψ(1), . . . , ψ(n))

)2

has G-degree at most 2 n. Therefore (why?), u(n+1)j is a polynomial of G-

degree at most 2+ 2n, consequently |ψ(n+1)j | =

√u

(n+1)j ≤ C(1+ |x|)n+1.

This last estimate, together with Lψ(n+1)j = 0 and Theorem 5.8.2, proves

that ψ(n+1)j is a polynomial of G-degree at most n+ 1. We have proved the

following assertion.

Proposition 5.16.1. Let ψ be a C2(G, G) map such that

L ◦ ψ = ψ ◦ L,

where L is a sub-Laplacian without the first order differential terms. Follow-ing the usual notation of the coordinates on G, we let ψ = (ψ(1), . . . , ψ(r)),where ψ(j) ∈ C2(G, R

Nj ) for every j = 1, . . . , r . Then each component ofψ(j) is a polynomial function of G-degree less or equal to j .

Ex. 13) (Maps commuting with Laplace operator Δ). Consider the Laplace oper-ator Δ =∑N

i=1 ∂2i,i on R

N . Let ψ ∈ C2(RN, RN) be such that

Δ(u ◦ ψ) = (Δu) ◦ ψ ∀ u ∈ C2(RN). (5.145)

With the aid of Ex. 12, Chapter 1, page 83, prove that (5.145) holds if andonly if ⎧

⎪⎨

⎪⎩

Δψi = 0, i = 1, . . . , N,

|∇ψi |2 = 1, i = 1, . . . , N,

〈∇ψi,∇ψj 〉 = 0, 1 ≤ i, j ≤ N, i �= j.

Derive that there exists an N×N orthogonal matrix M and a vector C ∈ RN

such thatψ(x) = Mx + C.

This means that the set of the maps commuting with the Laplace operatorcoincides with the group of the isometries of R

N .Ex. 14) (Maps commuting with Δ

HHH1 ). Let us consider the Kohn Laplacian ΔH1

on the Heisenberg-Weyl group H1 ≡ R

3. We characterize the maps ψ ∈C2(R3, R

3) such that

ΔH1(u ◦ ψ) = (ΔH1 u) ◦ ψ ∀ u ∈ C2(H1). (5.146)

Set ψ(x, y, t) = (ξ(x, y, t), η(x, y, t), τ (x, y, t)). With the aid of Ex. 12,Chapter 1, page 83, prove that (5.146) is equivalent to

5.16 Exercises of Chapter 5 323⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(i)

{1 = |∇H1ξ |2,0 = ΔH1ξ,

(ii)

{1 = |∇H1η|2,0 = ΔH1η,

(iii)

{4 (ξ2 + η2) = |∇H1τ |2,0 = ΔH1τ,

(iv) 2 η = 〈∇H1ξ,∇H1τ 〉,(v) − 2 ξ = 〈∇H1η,∇H1τ 〉,(vi) 0 = 〈∇H1ξ,∇H1η〉.

Using Liouville Theorem 5.8.4, derive that ξ and η are polynomials of H1-

degree ≤ 1, τ is a polynomial of H1-degree ≤ 2. Hence, there exist con-

stants such that

ξ(x, y, t) = c0 + c1 x + c2 y,

η(x, y, t) = d0 + d1 x + d2 y,

τ (x, y, t) = e0 + e1 x + e2 y + e3 t.

From the first equation in (i) and the first equation in (ii) and from (vi) itfollows that there exists θ ∈ [0, 2 π[ such that

(c1, c2) = (cos θ, sin θ), (d1, d2) = ±(− sin θ, cos θ).

From (iv) and (v) one gets(

e1 + 2 y e3e2 − 2 x e3

)=(

2d0c1 − 2d1c0 ± 2 y

2d0c2 − 2d2c0 ∓ 2 x

),

i.e.e3 = ±1, e1 = 2d0c1 − 2d1c0, e2 = 2d0c2 − 2d2c0.

Recalling that on H1 it holds

(α, β, γ ) ◦ (x, y, t) = (x + α, y + β, t + γ + 2xβ − 2yα),

we have proved that the only maps commuting with ΔH1 have the form

ψ+(x, y, t) =⎛

⎝c0d0e0

⎞

⎠ ◦⎛

⎝M+ ·(

x

y

)

t

⎞

⎠ ,

ψ−(x, y, t) =⎛

⎝c0d0e0

⎞

⎠ ◦⎛

⎝M− ·(

x

y

)

−t

⎞

⎠ ,

where M+ and M− are (respectively) isometries with the determinant equalto +1 (respectively, −1) and (c0, d0, e0) is a fixed point in H

1.


Ex. 15) (Translation-formula in surface integrals over a d-sphere). Let f : G→R be non-negative (or with suitable summability properties). Let d be asmooth, homogeneous and symmetric norm on G, and let S(x0, r) be thed-sphere with center x0 and radius r > 0. Finally, let HN−1 denote theHausdorff (N − 1)-dimensional measure on R

N . Prove that∫

S(x0,r)

f (x) dHN−1(x) =∫

S(0,r)

f (x0 ◦ y)Kd(x0, y) dHN−1(y),

where

Kd(x0, y) := |∇(d(x−10 ◦ ·))|(x0 ◦ y)

|∇d(y)| .

Prove also that

Kd(x0, y) =|∇d(y) · Jτ

x−10

(x0 ◦ y)||∇d(y)| =

∣∣∣∣∇d(y)

|∇d(y)| · Jτx−1

0(x0 ◦ y)

∣∣∣∣.

(Hint: Consider the identity (why does it hold?)∫

S(x0,r)

f (x) dHN(x) =∫

S(0,r)

f (x0 ◦ y) dHN(y)

rewritten (by the coarea formula) as

∫ r

0

(∫

S(x0,λ)

f (x)dHN−1(x)

|∇d(x−10 ◦ ·)|(x)

)dr

=∫ r

0

(∫

S(0,λ)

f (x0 ◦ y)dHN−1(y)

|∇d(y)|)

dr

and differentiate w.r.t. r .)Ex. 16) In the sequel, d is a symmetric homogeneous norm on G. If x ∈ G and

A ⊆ G is any set, we call the d-distance of x to A, the following real non-negative number

distd(x,A) := infa∈A

d(x−1 ◦ a).

Prove the following result.

Lemma 5.16.2. Let A ⊂ G be any set. For every x ∈ G, there exists ax ∈ A

(the closure of A) such that

distd(x,A) = d(x−1 ◦ ax).

(Hint: By definition, there exists {aj }j in A such that d(x−1 ◦ aj ) →distd(x,A). Obviously, {aj }j is bounded, otherwise d(x−1◦ aj ) ≥ 1

c d(aj )−d(x) → ∞. Then, extract a subsequence ajn → ax ∈ A, as n → ∞, anduse the continuity of d .)Then prove the following result.


Proposition 5.16.3. Let A ⊆ G be any set. The following assertions hold:(a) For every x ∈ G, we have distd(x,A) = distd(x,A);(b) The d-distance from A, i.e. the function

distd(·, A) : G→ [0,∞), x �→ distd(x,A),

is a continuous function.

Hint: (a). Let α ∈ A be such that distd(x,A) = d(x−1 ◦ α). Let aj ∈ A besuch that aj → α as j →∞. Then consider the following inequalities:

distd(x,A) ≥ distd(x,A) = d(x−1 ◦ α)

= limj→∞ d(x−1 ◦ aj ) ≥ distd(x,A).

The last inequality follows from d(x−1 ◦aj ) ≥ distd(x,A) for every j ∈ A,since aj ∈ A, and by the very definition of distd(x,A).(b). Provide the details of the following arguments: Let x0 ∈ G be fixed. Itsuffices to show that from every sequence {xj }j∈N in G converging to x0 wecan extract a subsequence {xjn}n∈N such that

limn→∞ distd(xjn, A) = distd(x0, A). (5.147)

It is not restrictive to suppose that A is closed. For every j ∈ N, there existsaj ∈ A such that

distd(xj , A) = d(x−1j ◦ aj ). (5.148)

It is easy to see that {aj }j∈N is bounded. Hence, we can extract a convergingsubsequence from {aj }j∈N, say ajn → a ∈ A as n→∞. From (5.148) weinfer

limn→∞ distd(xjn, A) = lim

n→∞ d(x−1jn◦ ajn) = d(x−1

0 ◦ a). (5.149)

Thus, (5.147) will follow if we show that d(x−10 ◦ a) = distd(x0, A). Sup-

pose to the contrary that we have d(x−10 ◦ a) �= distd(x0, A). This may

occur iff

(i) : d(x−10 ◦ a) < distd(x0, A) or (ii) : d(x−1

0 ◦ a) > distd(x0, A).

Case (i) is impossible by the very definition of distd(x0, A) (since a ∈ A).Case (ii) is absurd too, as we show below. Let a0 ∈ A be such thatdistd(x0, A) = d(a−1

0 ◦ x0). Then, by (5.149), we derive

d(x−10 ◦ a) = lim

n→∞ distd(xjn, A) ≤ limn→∞ d(a−1

0 ◦ xjn)

= d(a−10 ◦ x0) = distd(x0, A).

This contradicts (ii).


Ex. 17) Prove the following improvement of Proposition 5.16.3 above, when theextra hypothesis d is smooth holds.

Proposition 5.16.4. Let d be a smooth homogeneous norm as in Proposi-tion 5.14.1, and let A ⊆ G. Prove that the function distd(x,A) is Lipschitzcontinuous with respect to d , i.e.

|distd(ξ, A)− distd(η,A)| ≤ β d(ξ, η) ∀ ξ, η ∈ G,

where β is the same constant as in Proposition 5.14.1.

(Hint: Let a ∈ A, and write (5.125a) with x = a, y = ξ , z = η,

d(ξ−1 ◦ a) ≤ d(η−1 ◦ a)+ β d(ξ−1 ◦ η).

Then take the infimum over a ∈ A, and derive distd(ξ, A) ≤ distd(η,A) +β d(ξ, η). Then interchange ξ and η.)

Ex. 18) (T : Another mean integral operator). Let L be a sub-Laplacian on thehomogeneous Carnot group G. Let Ω ⊆ G be a bounded open set. Forevery x ∈ Ω , let us put

rx := 1

2distd(x, ∂Ω),

where d is an L-gauge function. For every u ∈ L1loc(Ω), define

T (u)(x) := Mrx (u)(x), (5.150)

where Mr is the average operator defined in (5.50f).Prove the following statements:(i) T (u) is continuous in Ω ,

(ii) u ≤ v ⇒ T (u) ≤ T (v),(iii) u ∈ L∞(Ω)⇒ supΩ |T (u)| ≤ ess supΩ |u|.

Ex. 19) (The strong maximum principle related to T ). Let T be as in the previousexercise, and let u be a continuous function in Ω such that T (u) ≤ u. If Ω

is connected and u attains its minimum in Ω , then u = constant in Ω .Ex. 20) (The weak maximum principle related to T ). Let T be as in the previous

exercises, and let u be a continuous function in Ω such that{

T (u) ≤ u in Ω,

lim infx→y u(x) ≥ 0 ∀ y ∈ ∂Ω.

Prove that u ≥ 0 in Ω .Ex. 21) Let u be an L-harmonic function in an open set Ω ⊆ G, Ω �= G. Assume

u ∈ Lp(Ω), 1 ≤ p ≤ ∞. Then

|u(x)| ≤ c

distd(x, ∂Ω)‖u‖Lp(Ω),

where c is independent of u and Ω , and d is an L-gauge function.(Hint: Formula (5.52) may prove useful.)


Ex. 22) (Another Koebe-type result). Let Ω ⊆ G be open, and let u ∈ C(Ω, R).Suppose that one of the following conditions is satisfied:(i) Mρ(u)(x) = Mr (u)(x) for every ρ, r > 0 such that 0 < ρ ≤ r and

Bd(x, r) ⊂ Ω ,(ii) Mρ(u)(x) = Mr (u)(x) for every ρ, r > 0 such that 0 < ρ ≤ r and

Bd(x, r) ⊂ Ω .Show that u ∈ C∞(Ω, R) and Lu = 0 in Ω .

Ex. 23) Prove the following Harnack inequality on d-spheres.

Theorem 5.16.5. There exists a constant C > 1 such that

sup∂Bd(0,r)

h ≤ C inf∂Bd(0,r)

h (5.151)

for every 0 < r ≤ 1/2 and every L-harmonic non-negative function h onBd(0, 1) \ {0}.

Ex. 24) Let d be any homogeneous norm on G, smooth on G \ {0}. Prove that

∫

d(x)=r

dHN−1(x)

|∇d(x)| = cd rQ−1 for every r > 0,

where cd = QHN(Bd(0, 1)). (Hint: By the coarea formula and the δλ-homogeneity of d , we have

∫ r

0

(∫

{d(x)=ρ}dHN−1(x)

|∇d(x)|)

dρ =∫

{d(x)<r}dHN = rQ HN(Bd(0, 1)).

Then, take the derivative w.r.t. r on both sides.)Ex. 25) In this exercise, we follow the arguments from [HK00] in Lemma 11.1 and

Proposition 11.2, jointly with some of our results in Section 5.2.

Lemma 5.16.6. Let X = {X1, . . . , Xm} be a system of locally Lipschitz-continuous vector fields on R

N .Let BE(x,R) = {z ∈ R

N : |z−x| < R} denote the Euclidean ball centeredat x with radius R. With reference to (5.8a) (page 233) set

M(x,R) := supz∈BE(x,R)

{n∑

j=1

|XjI (z)|}

.

Suppose γ : [0, T ] → RN is X-subunit, γ (0) = x and T < R/M(x,R).

Then γ ([0, T ]) ⊆ BE(x,R).

(Hint: Suppose to the contrary that γ ([0, T ]) � BE(x,R). Hence, thereexists a least t ∈ (0, T ], such that y := γ (t) /∈ BE(x,R). From the mini-mality of t and the continuity of γ , we certainly have y ∈ ∂BE(x,R), i.e.


|y− x| = R. Notice also that dX(x, y) ≤ t ≤ T . Hence, (5.8b) on page 233gives

R = |x − y| ≤ M(x, |x − y|) dX(x, y) ≤ M(x,R)T < R.

This is absurd.)Deduce the following result.

Corollary 5.16.7. Let X = {X1, . . . , Xm} be a system of locally Lipschitz-continuous vector fields on R

N .Then, for every bounded set D ⊂ R

N , there exist positive numbers R0 * 1,R1 % 1 (depending only on D and X) such that BdX

(x, R) ⊆ BE(0, R1)

for every x ∈ D and every R ∈ [0, R0]. In particular, for every x0 ∈ RN ,

every Carnot–Carathéodory ball BdX(x0, R) with small radius (dependently

on x0) is bounded in the Euclidean metric.

(Hint: With the notation of Lemma 5.16.6, let

M := sup{M(x, 1) : x ∈ D}.Obviously, M < ∞, for the Xj ’s are continuous. Set R0 := (4 M)−1. Letx ∈ D and R ≤ R0. (Note that M ≥ M(x, 1), since x ∈ D.) Let alsoy ∈ BdX

(x, R), i.e. dX(x, y) < R. Then there exists a X-subunit curveγ : [0, T ] → R

N such that γ (0) = x, γ (T ) = y and

T < dX(x, y)+ (4M)−1 < 2 R0 = (2M)−1 < 1/M(x, 1).

Then, we can apply Lemma 5.16.6 with R = 1 and derive that γ ([0, T ]) ⊆BE(x, 1). In particular, this gives |y − x| = |γ (T ) − x| < 1, i.e. y ∈BE(x, 1). Due to the arbitrariness of y ∈ BdX

(x, R), this proves

BdX(x, R) ⊆ BE(x, 1) ⊆ D1 :=

⋃

x∈D

BE(x, 1) � RN.

This ends the proof, by finding a suitable R1 % 1 such that D1 ⊆BE(0, R1).)

Proposition 5.16.8. Let X = {X1, . . . , Xm} be a system of locally Lipschitz-continuous vector fields on R

N .Let K be a compact subset of R

N . Then there exists a constant c =c(K,X) > 0 such that

dX(x, y) ≥ c |x − y| for every x, y ∈ K.

Proof. If dX(x, y) = ∞, there is nothing to prove. Hence, we can supposedX(x, y) < ∞. Let γ : [0, T ] → R

N be any X-subunit curve such thatγ (0) = x, γ (T ) = y. Let R := |x − y|. Set


M := sup

{n∑

j=1

|XjI (z)| : z ∈ BE(x, 1+ diam(K)), x ∈ K

}.

Note that M = M(K,X) < ∞, since K is compact and the Xj ’s arecontinuous. Obviously, we do not have γ ([0, T ]) ⊆ BE(x,R), because|γ (T )− x| = |y − x| = R is not < R. Hence, by Lemma 5.16.6,

T ≥ R/M(x,R) ≥ R/M =: c R = c|x − y|.(Indeed, M(x,R) ≤ M .) Passing to the infimum over the above γ ’s, we getdX(x, y) ≥ c |x − y|. ��Derive the following assertion from Corollary 5.16.7 and Proposition 5.16.8.

Proposition 5.16.9. Let X = {X1, . . . , Xm} be locally Lipschitz-continuousvector fields on R

N . Suppose RN is X-connected. Let dX be the relevant

Carnot–Carathéodory distance. Then the map

id : (G, dX)→ (G, dE)

is continuous.

(Hint: Dealing with metric spaces, we can prove sequential continuity. Letxn → x0 in (G, dX). We aim to prove that xn → x0 in (G, dE). By Corol-lary 5.16.7, there exists ε = ε(x0) > 0 such that BdX

(x0, ε) is boundedin (G, dE). By definition of limit, there exists n = n(ε) ∈ N such thatxn ∈ BdX

(x0, ε) whenever n ≥ n. Hence the set {xn : n ≥ n} ∪ {x0} iscontained in a compact subset of R

N . We can thus apply Proposition 5.16.8to derive that, for a suitable c > 0,

c |xn − x0| ≤ dX(xn, x0) for all n ≥ n.

Letting n→∞, we get xn → x0 in (G, dE).)Now, consider a system X of Hörmander vector fields in R

N and therelevant dX. Recall that, by the Carathéodory–Chow–Rashevsky theorem(see Chapter 19 for suitable references), R

N is X-connected. Moreover, byknown results (see, e.g. [NSW85]) the inequalities in Proposition 5.15.1hold (for a suitable r > 0) when d(y−1 ◦ x) is replaced by dX(x, y). Inparticular, given a compact subset K of R

N , there exist r, c > 0 (dependingon K,X) such that

dX(x, y) ≤ c |x − y|1/r ∀ x, y ∈ K.

Arguing as in the first paragraph of the proof of Corollary 5.15.3, derive thefollowing assertion:Given a system X of Hörmander vector fields in R

N , the map id : (G, dE)→(G, dX) is continuous (indeed, it is a homeomorphism, see Proposi-tion 5.16.9).


Ex. 26) Consider the vector fields in R2 defined by

X1 = ∂x1 , X2 = max{0, x1} ∂x2 .

Consider the relevant dX, where X = {X1, X2}. Is R2 X-connected? Prove

that the identity map id : (R2, dE)→ (R2, dX) is not continuous.(Hint: It may be useful to notice that the integral curves of X1 are the linesparallel to the x1 axis; the integral curve of X2 through (x0, y0) is

t �→{

(x0, y0) if x0 ≤ 0,

(x0, y0 + x0 t) if x0 > 0,

i.e. the single point (x0, y0) if x0 ≤ 0 or the lines parallel to the x2 axis ifx0 > 0.)

Ex. 27) Consider the vector fields in R2 defined by

X1 = ∂x1 , X2 = (1+ x22) ∂x2 .

Verify that the system X = {X1, X2} satisfies the Hörmander condition.Consider the relevant dX. Is R

2 X-connected? Prove that there exists a setΩ ⊂ R

2 which is bounded w.r.t. dX but unbounded in the Euclidean metric.Hint: Observe that the integral curve of X2 (hence, a X-subunit path!) start-ing from (x0, y0) is

t �→ (x0, tan(t + arctan(y0))

).

Hence, for example,

dX

((0, 0),

(0, tan

(π

2− 1

n

)))≤ π/2− 1/n < π/2,

but

dE

((0, 0),

(0, tan

(π

2− 1

n

)))−→∞ as n→∞.

Ex. 28) Prove the following linear algebra result.

Lemma 5.16.10. Let v1, . . . , vm be vectors in RN . Suppose w ∈ R

N is suchthat

(�) 〈w, x〉2 ≤m∑

j=1

〈vj , x〉2 ∀ x ∈ RN.

Then there exist scalars α1, . . . , αm such that

w =m∑

j=1

αj vj andm∑

j=1

α2j ≤ 1.


Proof. Let us first prove that w ∈ span{v1, . . . , vm} =: V . From the de-composition R

N = V ⊕ V ⊥ it follows

w = a+ b, a ∈ V, b ∈ V ⊥.

If we choose x := b in (�), we get

‖b‖4 = 〈a+ b, b〉2 ≤m∑

j=1

〈vj , b〉2 = 0,

whence b = 0, i.e. w = a ∈ V .Up to a permutation of the vj ’s, it is not restrictive to suppose that (v1,

. . . , vq) is a basis of V . Then we have

vj =q∑

k=1

βj,k vk ∀ j = 1, . . . , m.

The m× q matrix B whose (j, k)-th entry is βj,k has the block form

B =(

Iq

B

),

where Iq is the identity matrix of order q. Let γ1, . . . , γq ∈ R be such thatw =∑q

k=1 γk vk . We characterize all the scalars α1, . . . , αm ∈ R such thatw =∑m

j=1 αj vj . From the identity

q∑

k=1

γk vk = w =q∑

k=1

(m∑

j=1

αj βj,k

)vk

and the linear independence of v1, . . . , vq we infer

(•) γk =m∑

j=1

αj βj,k ∀ k = 1, . . . , q.

Let us suppose that, besides (•), there also exists a solution x ∈ RN to the

m-equation system

(SL)

{〈vj , x〉 = αj ,

j = 1, . . . , m.

This will give(

m∑

j=1

α2j

)2

=(

m∑

j=1

αj 〈vj , x〉)2

=(⟨

m∑

j=1

αj vj , x

⟩)2

= 〈w, x〉2 ≤m∑

j=1

〈vj , x〉2 =m∑

j=1

α2j ,

whence∑m

j=1 α2j ≤ 1, and the proof is complete.


We remark that, in order to (SL) to be solvable, it is necessary and sufficientthat the rank of the coefficient-matrix of (SL) (i.e. q) equals the rank of thecomplete-matrix of (SL): thanks to the dependence of vq+1, . . . , vm w.r.t.v1, . . . , vq , this is equivalent to

(••) αj =q∑

k=1

βj,k αk ∀ j = q + 1, . . . , m.

Hence, the proof is complete if we show that there exists (α1, . . . , αm) ∈ Rm

satisfying (•) and (••), i.e. a solution to the m-equation and m-indeterminatesystem α1, . . . , αm

(SL)’

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

m∑

j=1

αj βj,k = γk, k = 1, . . . , q,

q∑

k=1

βj,k αk − αj = 0, j = q + 1, . . . , m.

The coefficient-matrix of (SL)’ is(

Iqt B

B −Im−q

).

We show that this matrix is invertible, whence (SL)’ is solvable.In general, if A is a real q × (m− q) matrix, the block matrix

P :=(

Iq AtA −Im−q

)satisfies P 2 = Im +

(A 00 tA

)·(

tA 00 A

).

Observe that P 2 is the sum of a positive-definite matrix plus a positive semi-definite one. In particular, P 2 is positive-definite, hence it is not singular.This gives | det P | = √det P 2 > 0, so that P is not singular too. This endsthe proof. ��By means of Lemma 5.16.10, derive the following equivalent characteriza-tion of X-subunit curve.

Proposition 5.16.11. Let X = {X1, . . . , Xm} be a system of locally Lips-chitz-continuous vector fields on an open set Ω ⊆ R

N .Let γ : [0, T ] → Ω be an absolutely continuous curve. Then γ is X-subunit if and only if there exist measurable functions cj : [0, T ] → R,j = 1, . . . , m, such that, almost everywhere on [0, T ],

γ (t) =m∑

j=1

cj (t)Xj I (γ (t)), andm∑

j=1

(cj (t)

)2 ≤ 1. (5.152)


Proof. Suppose γ : [0, T ] → Ω , satisfies (5.152). The Cauchy-Schwartzinequality in R

m immediately yields

⟨γ (t), ξ

⟩2 ≤(

m∑

j=1

(cj (t))2

)·(

m∑

j=1

⟨Xj(γ (t)), ξ

⟩2)≤

m∑

j=1

⟨Xj(γ (t)), ξ

⟩2,

i.e. γ is a X-subunit curve. Vice versa, let γ be X-subunit. If we applyLemma 5.16.10 with the choice

vj := Xj(γ (t)), j = 1, . . . , m, w := γ (t),

then (5.152) is satisfied for suitable scalar functions cj ’s defined a.e. on[0, T ]. It is easy to prove16 the measurability of the cj ’s. ��

Ex. 29) Let u be an L-harmonic function in an open set Ω ⊆ G. Let Bd(x0, r) ⊂ Ω .Then, for every multi-index α, there exists a constant Cα > 0 (independentof u, x0 and r) such that

∣∣Xαu(x0)∣∣ ≤ Cα r−|α|G sup

Bd(x0,r)

|u|.

(Hint: Since u is L-harmonic, the representation formula (5.50a) (see also(5.50d)) gives

u(x0) =∫

RN

u(z) φr (x−1 ◦ z) dz =

∫

RN

u(z) φr (x−1 ◦ z) dz,

where φr (ζ ) = φr(ζ−1). Now, derive the integral.)

16 Indeed, notice that γ is measurable (since γ is a absolutely continuous), t �→ Xj (γ (t)) iscontinuous and the operations which provide the components of a vector w.r.t. a system ofvectors are continuous. We leave the details to the reader.

Part II

Elements of Potential Theoryfor Sub-Laplacians

6

Abstract Harmonic Spaces

In this chapter, we present some topics from the theory of abstract harmonic spaces.Although this theory is extremely vast, we only focus on the results which are crucialfor the scopes of the next chapters.

In this abstract setting, we shall only deal with the Perron–Wiener–Brelot methodfor the Dirichlet problem, the study of harmonic minorants and majorants and bal-ayage theory.

Our aim here is to furnish the background material for an exhaustive potentialtheory for the sub-Laplacians L, which will be developed through the rest of Part II.The lack of explicit Poisson integral formulas for L forced us to follow the abstractapproach to this theory.

In the subsequent chapters, we shall use this basic results, together with the par-ticular structure of the fundamental solutions, to develop the deepest part of the po-tential theory for L.

The following scheme describes our approach to the abstract potential theory forsub-Laplacians. Notation and definitions will be explained in due course.

Harmonic Spaces (PWB theory)

S-Harmonic Spaces (Wiener Resolutivity Theorem)

S∗-Harmonic Spaces (Bouligand’s Theorem)

L-Harmonic Spaces(Capacity, Polarity, etc.)

338 6 Abstract Harmonic Spaces

Convention. Throughout this chapter (E, T ) will denote a topological Hausdorffspace, locally connected and locally compact. We also assume that the topology Thas a countable basis.

6.1 Preliminaries

Let A ⊆ E and u : A → [−∞,∞]. If x is a point of A, we define

lim infy→x

u(y) := supV ∈Ux

(inf

V ∩Au),

lim supy→x

u(y) := infV ∈Ux

(supV ∩A

u),

where Ux denotes the family of the neighborhoods of x. (Note the slight differencew.r.t. the usual definition of lim sup and lim inf in that, for example, in the lim infdefinition, the inner infV ∩A is usually replaced by inf(V \{x})∩A. Our choice will besoon motivated.)

The function u is called lower semicontinuous (l.s.c., in short) at x ∈ A ifu(x) = lim infy→x u(y). If u(x) = lim supy→x u(y), then u will be called uppersemicontinuous (u.s.c., in short) at x. If u is l.s.c. (u.s.c.) at any point of A, then u

will be said l.s.c. (u.s.c.) on A.We now list some elementary properties of semicontinuous functions, whose

proofs will be left as exercises.

(P1) Suppose that f, g are l.s.c. (u.s.c.) on A. Then f +g, λf with λ ≥ 0, max{f, g},min{f, g} are l.s.c. (u.s.c.) on A. Moreover, λf is u.s.c. if f is l.s.c. and λ < 0.

(P2) f : A → [−∞,∞] is l.s.c. on A if and only if

{f > t} := {x ∈ A : f (x) > t}is open for every t ∈ R. Similarly, f is u.s.c. on A if and only if

{f < t} := {x ∈ A : f (x) < t}is open for every t ∈ R.

(P3) Let (uα)α∈A be a family of l.s.c. (u.s.c.) functions on A. Then supα∈A uα

(infα∈A uα , respectively) is l.s.c. (u.s.c., respectively).(P4) If A is compact and u : A →]−∞,∞] (u : A → [−∞,∞[, respectively) is

l.s.c. (u.s.c., respectively), then u attains its minimum (maximum, respectively)on A.

(P5) If A is compact and u : A →]−∞,∞] (u : A → [−∞,∞[, respectively) isl.s.c. (u.s.c., respectively), then there exists a sequence (fj ) of real continuousfunctions on A such that

fj ≤ fj+1 and fj → u pointwise on A

(fj ≥ fj+1 and fj → u pointwise on A, respectively).

6.1 Preliminaries 339

If u : A → [−∞,∞], one defines the lower regularization u of u as follows

u : A → [−∞,∞], u(x) = lim infy→x

u(y).

Similarly, one defines the upper regularization u of u as follows

u : A → [−∞,∞], u(x) = lim supy→x

u(y).

The functions u and u are l.s.c. and u.s.c., respectively, and

u ≤ u ≤ u.

We now state and prove some propositions on the envelopes of families of functionswhich will play a crucial rôle in our subsequent exposition. Our proofs are simpleadaptation to an abstract setting of Lemmas 3.7.1 and 3.7.4 in [AG01].

Proposition 6.1.1 (Envelopes I). Let (uα)α∈A be a family of l.s.c. functions from A

(⊆ E) to ]−∞,∞]. Then there exists a countable set B ⊆ A such that

supα∈B

uα = supα∈A

uα.

Proof. Let u := supα∈A uα . Then u is l.s.c. (see (P3)) and

{u > t} =⋃

α∈A{uα > t}.

Since (E, T ) is locally compact and endowed with a countable basis of open sets,there exists a countable set At ⊆ A such that

{u > t} =⋃

α∈At

{uα > t}. (6.1)

Define B := ⋃t∈Q

At and v := supα∈B uα . Let us prove that v = u. Assume bycontradiction v = u. Since v ≤ u, this implies the existence of a point y ∈ A suchthat v(y) < u(y). Let τ ∈ Q satisfy v(y) ≤ τ < u(y). By using (6.1), we can findατ ∈ Aτ such that τ < uατ (y). Then

v(y) = supα∈B

uα(y) ≥ uατ (y) > τ ≥ v(y).

This contradiction completes the proof. ��Proposition 6.1.2 (Envelopes II: Choquet). Let (uα)α∈A be a family of functionsfrom A (⊆ E) to [−∞,∞], and let u := infα∈A uα . Then there exists a countableset B ⊆ A such that

u = v, where v = infα∈B

uα.


Proof. By replacing uα with arctan uα , we may assume that u is bounded from be-low. Let (Bn)n∈N be a sequence of open sets such that {Bn : n ∈ N} forms a basisof the topology T and such that, for every m ∈ N,

Nm := {n ∈ N : Bn = Bm}is infinite. For every n ∈ N, we now choose xn ∈ Bn and αn ∈ A such that

u(xn) < infBn∩A

u + 1

nand uαn(xn) < u(xn) + 1

n.

Let us define B := {αn : n ∈ N} and v := infα∈B uα . Then, for every m ∈ N andn ∈ Nm, we have

infBm∩A

v = infBn∩A

v ≤ uαn(xn) < u(xn) + 1

n< inf

Bn∩Au + 2

n= inf

Bm∩Au + 2

n,

hence, being Nm infinite,

infBm∩A

v ≤ infBm∩A

u for every m ∈ N.

As a consequence, for every x ∈ A,

v(x) = sup{Bm : x∈Bm}

(inf

Bm∩Av)

≤ sup{Bm : x∈Bm}

(inf

Bm∩Au)

= u(x),

so that v ≤ u. Since the reverse inequality is obvious, we are done. ��

6.2 Sheafs of Functions. Harmonic Sheafs

We begin with the main definition of this section.

Definition 6.2.1 (Sheaf of functions. Harmonic sheaf). Suppose we are given, forevery open set V ∈ T , a family F(V ) of extended real-valued functions u : V →[−∞,∞]. We say that the map

F : V → F(V )

is a sheaf of functions on E if the following properties hold:

if V1, V2 ∈ T with V1 ⊆ V2 and u ∈ F(V2), then u|V1 ∈ F(V1), (6.2a)

if (Vα)α∈A ⊆ T and u : ⋃α∈A Vα → [−∞,∞]

is such that u|Vα ∈ F(Vα) for every α ∈ A, then u ∈ F(⋃

α∈A Vα).(6.2b)

A sheaf of functions F on E is called harmonic if F(V ) is a linear subspace ofC(V, R), the vector space of the real continuous functions defined on V . When Fis a harmonic sheaf on E and V is an open set in T , a function u ∈ F(V ) will becalled F-harmonic.

6.2 Sheafs of Functions. Harmonic Sheafs 341

6.2.1 Regular Open Sets. Harmonic Measures. Hyperharmonic Functions

Definition 6.2.2 (H-regular set). Let H be a harmonic sheaf on E. We say that anopen set V ∈ T is H-regular if the following conditions are satisfied:

(R1) V is compact and ∂V = ∅;(R2) for every continuous function f : ∂V → R, there exists a unique H-harmonic

function in V , denoted by HVf , such that

limx→y

HVf (x) = f (y) for every y ∈ ∂V ;

(R3) if f ≥ 0, then HVf ≥ 0.

When V is H-regular, from the linearity of F(V ) and the uniqueness assumptionin condition (R2) it follows that

HVf +g = HV

f + HVg , HV

λf = λHVf

for every f, g ∈ C(∂V, R) and for every λ ∈ R. Then, also keeping in mind (R3),for every H-regular open set V and for every x ∈ V , the map

C(∂V, R) � f → HVf (x) ∈ R

is linear and positive. Hence, the following definition is well posed.

Definition 6.2.3 (H-harmonic measure. I). Let H be a harmonic sheaf on E. LetV ∈ T be an H-regular set. Then there exists a Radon measure μV

x on C(V, R) suchthat

HVf (x) =

∫

∂V

f (y) dμVx (y) ∀ f ∈ C(∂V, R).

The measure μVx is called the H-harmonic measure related to V and x.

We provide a definition which will be used throughout the sequel.

Definition 6.2.4 (H-hyperharmonic function). Let H be a harmonic sheaf on(E, T ). Let Ω ∈ T . A function u : Ω →]−∞,∞] is called H-hyperharmonicin Ω if:

(i) u is lower semi-continuous;(ii) for every H-regular open set V ⊂ V ⊆ Ω , one has

u(x) ≥∫

∂V

u(y) dμVx (y) ∀ x ∈ V.

We shall denote by H∗(Ω) the set of the H-hyperharmonic functions in Ω .

Since∫

∂V

u dμVx = sup

{ ∫

∂V

ϕ dμVx | ϕ ∈ C(∂V, R), ϕ ≤ u

},

condition (ii) can be rewritten as follows


(ii)’ for every H-regular open set V ⊆ V ⊆ Ω and for every ϕ ∈ C(∂V, R) suchthat ϕ ≤ u|∂V , one has

HVϕ ≤ u|V .

A function v : Ω → [−∞,∞[ will be called H-hypoharmonic if −v ∈ H∗(Ω).We denote by

H∗(Ω) := −H∗(Ω)

the family of the H-hypoharmonic functions in Ω .

Exercise 6.2.5. Let Ω ⊆ E be open, and let F ⊆ H∗(Ω). Suppose that

u0 := infF

is l.s.c. in Ω . Then u0 ∈ H∗(Ω). (Hint: if V ⊆ V ⊆ Ω is H-regular, then u(x) ≥∫∂V

u dμVx ≥ ∫

∂Vu0 dμV

x for every x ∈ V and u ∈ F .)

We will see later that H∗(Ω) is a sheaf of functions. Now, we state a propositionwhose easy proof will be left as an exercise.

Proposition 6.2.6. Let Ω ⊂ E be open, and let u, v ∈ H∗(Ω). Then:

(i) u + v ∈ H∗(Ω),(ii) λ u ∈ H∗(Ω) for every λ ≥ 0,

(iii) min{u, v} ∈ H∗(Ω).

An analogous proposition also holds, obviously, for H∗(Ω).

6.2.2 Directed Families of Functions

In this section we shall show some results on directed families of functions, extend-ing well-known theorems related to monotone sequences of functions. The results weare going to show will be applied in the subsequent section to families of harmonicand hyperharmonic functions.

Let F be a family of functions from A ⊆ E to [−∞,∞].Definition 6.2.7 (Up (down) directed family). We say that F is up directed, and weshall write

F ↑,

if, for every u, v ∈ F , there exists w ∈ F such that

u ≤ w and v ≤ w.

Analogously, if, for every u, v ∈ F , there exists w ∈ F such that u ≥ w, v ≥ w, wesay that F is down directed, and we shall write F ↓.

6.2 Sheafs of Functions. Harmonic Sheafs 343

Theorem 6.2.8 (Dini–Cartan). Let A ⊆ E be compact, and let F be a down di-rected family of upper semicontinuous functions. Assume

infF := infu∈F

u = 0.

Then F is uniformly convergent to zero, that is, for every ε > 0, there exists u ∈ Fsuch that u(x) < ε for every x ∈ A.

Proof. Let ε > 0. Since infF = 0, for every x ∈ A, there exists ux ∈ F such thatux(x) < ε. The upper semicontinuity of ux implies the existence of a neighborhoodVx of x such that ux(y) < ε for every y ∈ Vx . Since A is compact, we can find

x1, . . . , xp ∈ A such that A ⊆p⋃

j=1

Vxj.

On the other hand, since F is down directed, there exists u ∈ F such that u ≤ uxj

for every j ∈ {1, . . . , p}. As a consequence, given any point y ∈ A, there existsj ∈ {1, . . . , p} such that y ∈ Vxj

, so that u(y) ≤ uxj(y) < ε, and the assertion

follows. ��Corollary 6.2.9. Let F be an up directed (respectively, down directed) family of l.s.c.(respectively, u.s.c.) functions on a compact set A ⊆ E. Assume that

v := supF (respectively, v := infF)

is continuous. Then F is uniformly convergent to u, that is, for every ε > 0, thereexists u ∈ F such that v − u < ε (respectively, u − v < ε).

Proof. Apply the previous theorem to the family

Fv := {v − u : u ∈ F} (respectively Fv := {u − v : u ∈ F}). ��Now, we prove one of the main results of this section.

Theorem 6.2.10 (Levi–Cartan). Let A ⊆ E be compact, and let μ be a Radonmeasure in A. Suppose we are given an up directed family F of l.s.c. functions. Then

supu∈F

∫

A

u dμ =∫

A

(supu∈F

u)

dμ. (6.3)

Analogously, if F is a down directed family of u.s.c. functions, then

infu∈F

∫

A

u dμ =∫

A

(infu∈F

u)

dμ.

Proof. We only prove the first part of the theorem. The second one will follow byapplying (6.3) to the family −F := {−u : u ∈ F}. The function


v := supu∈F

u

is l.s.c. since the functions in F are l.s.c. Therefore,∫

A

v dμ = sup

{ ∫

A

φ dμ : φ ∈ C(A, R), φ ≤ v

}. (6.4)

Since u ≤ v for any u ∈ F , the inequality

supu∈F

∫

A

u dμ ≤∫

A

v dμ (6.5)

readily follows. To prove the reverse one, let us pick φ ∈ C(A, R) such that φ ≤ v

and considerFφ := {min{u, φ} : u ∈ F}.

This family is up directed and

supFφ = min{v, φ} = φ.

Then, we can apply Corollary 6.2.9: for every ε > 0, there exists u0 ∈ F such thatφ − min{u0, φ} < ε. As a consequence,

∫

A

φ dμ ≤ εμ(A) +∫

A

min{u0, φ} dμ ≤ εμ(A) +∫

A

u0 dμ

≤ εμ(A) + supu∈F

∫

A

u dμ.

Noticing that μ(A) < ∞, since A is compact, and letting ε tend to zero, we get∫

A

φ dμ ≤ supu∈F

∫

A

u dμ.

Thanks to (6.4) this inequality implies∫

A

v dμ ≤ supu∈F

∫

A

u dμ.

This equality, together with (6.5), completes the proof. ��We close the section by proving the following theorem, an easy consequence of

Proposition 6.1.1.

Theorem 6.2.11 (Cornea). Let F be an up directed family of continuous functionsin an open set Ω ∈ T . Suppose F has the following property:

for every increasing sequence (fn)n in F,

one has supn∈N fn ∈ C(Ω, R). (6.6)

Then the functionv := sup

u∈Fu

is continuous in Ω .

6.3 Harmonic Spaces 345

Proof. Proposition 6.1.1 implies the existence of a sequence (gn) in F such thatv = supn gn. Since F is up directed, we can construct a sequence (fn) in F such that

f1 = g1, f2 ≥ max{g2, f1}, . . . , fn+1 ≥ max{gn+1, fn}, . . . .Then F � fn ↑ v. Assumption (6.6) implies that v is continuous. ��

6.3 Harmonic Spaces

Definition 6.3.1 (Harmonic space). Let H be a harmonic sheaf on E. We say that(E,H) is a harmonic space if the following axioms are satisfied:

(A1) (Positivity). For every relatively compact open set Ω ⊆ E, there exist h0 ∈H∗(Ω) and k0 ∈ H∗(Ω) satisfying1:

infΩ

h0, infΩ

k0 > 0 and h0(x) < ∞ ∀ x ∈ Ω.

(A2) (Convergence). If {un}n∈N is a monotone increasing sequence of H-harmonicfunctions on an open set Ω ∈ T such that

{x ∈ Ω : sup

n∈N

un(x) < ∞}

is dense in Ω , thenu := lim

n→∞ un

is H-harmonic in Ω .(A3) (Regularity). The family Tr of the H-regular open sets is a basis for the topol-

ogy T .(A4) (Separation). For every x, y ∈ E with x = y, there exist u, v ∈ H∗(E) such

thatu(x) v(y) = u(y) v(x).

The following remark is often useful in applying the abstract harmonic theory tothe linear partial differential equations without zero order terms.

Remark 6.3.2. If the constant functions are H-harmonic in E, then the positivity ax-iom is satisfied. Moreover, the separation assumption (A4) can be restated as follows.

(A4)’ For every x, y ∈ E with x = y, there exists u ∈ H∗(E) such that

u(x) = u(y).

Indeed, (A4)’ imply (A4) by taking in the latter v ≡ 1.

1 An H-hyperharmonic function taking real values in a dense subset of its domain willbe called H-superharmonic, see Section 6.5. Then we can say that h0, −k0 are H-superharmonic.


If Ω ∈ T , V ⊆ V ⊆ Ω is H-regular and u ∈ H(Ω), it readily follows from theproperty (R2) of the regular sets that

u|V = HVu|∂V

.

Hence

u(x) =∫

∂V

u dμVx ∀ x ∈ V.

An important consequence of the regularity axiom is that this property characterizesthe H-harmonic functions.

Proposition 6.3.3 (Characterization of H-harmonicity). Let (E,H) be a har-monic space. Let Ω ∈ T and u ∈ C(Ω, R) be such that

u(x) =∫

∂V

u dμVx for every x ∈ V (6.7)

and for every H-regular open set V ⊆ V ⊆ Ω . Then u ∈ H(Ω).

Proof. Let us defineV := {V ∈ Tr |V ⊆ Ω}.

We claim thatΩ =

⋃

V ∈VV. (6.8)

Taking this claim for granted for a moment, from (6.7) we get

u|V = HVu|∂V

∀ V ∈ V .

Then u is H-harmonic in V for every V ∈ V . Since H is a sheaf, (6.8) now impliesthat u ∈ H(Ω).

So we are left with the proof of (6.8). For every x ∈ Ω , there exists a regularopen set W contained in Ω and containing x (by (A3)). Since ∂W is compact, thereexists V ∈ Tr such that

V ⊆ W, x ∈ V, V ∩ ∂W = ∅(we are again using (A3), together with the Hausdorff separation property of thetopology T ). Consequently, x ∈ V ⊆ V ⊆ W ⊆ Ω . This shows (6.8) and completesthe proof of the proposition. ��Exercise 6.3.4. For every open set Ω ⊆ E, one has

H∗(Ω) ∩ H∗(Ω) = H(Ω).

Corollary 6.3.5. Let (E,H) be a harmonic space. Let {un}n∈N be a sequence inH(Ω), Ω ∈ T , such that

un → u, as n → ∞,

uniformly on every compact subset of Ω . Then u ∈ H(Ω).

6.3 Harmonic Spaces 347

Proof. First of all, u is continuous in Ω , since it is the limit of a sequence of contin-uous functions, locally uniformly convergent. Moreover, for every regular open setV ⊆ V ⊆ Ω and for every x ∈ V , we have

u(x) = limn→∞ un(x) = lim

n→∞

∫

∂V

un dμVx (un ∈ H(Ω))

=∫

∂V

limn→∞ un dμV

x (un is uniformly convergent on V )

=∫

∂V

u dμVx .

Then, by Proposition 6.3.3, u ∈ H(Ω). ��

6.3.1 Directed Families of Harmonic and Hyperharmonic Functions

The results proved in the previous section are the ad hoc tools to show the followingtwo theorems, which are consequences of the convergence axiom (A2) in Section 6.3.

Theorem 6.3.6 (Up directed families in H(Ω)). Let (E,H) be a harmonic space,and let Ω be an open subset of E. Suppose we are given a family F ⊆ H(Ω) suchthat:

(i) F is up directed,(ii) v := supu∈F u is finite in a dense subset of Ω .

Then v is H-harmonic in Ω .

Proof. First, we prove that v is continuous in Ω by using Cornea’s Theorem 6.2.11.Let (fn) be a monotone increasing sequence in F . Obviously,

f := supn∈N

fn ≤ supu∈F

u,

so that, by hypothesis (ii), f < ∞ in a dense subset of Ω . The convergence ax-iom (A2) implies that f ∈ H(Ω), hence it is continuous. This shows that F satisfiesthe assumptions of Theorem 6.2.11, so that v ∈ C(Ω, R). Let us now take an H-regular open set V ⊆ V ⊆ Ω and a point x ∈ Ω . Since ∂V is compact and μV

x is aRadon measure in ∂V , by Levi–Cartan’s Theorem 6.2.10 we obtain

v(x) = supu∈F

u(x) = supu∈F

∫

∂V

u dμVx =

∫

∂V

(supu∈F

u)

dμVx =

∫

∂V

v dμVx .

Then, by Proposition 6.3.3, u ∈ H(Ω). ��Corollary 6.3.7. In the hypotheses of the previous theorem, let F ⊆ H(Ω) be suchthat F ↓ and infF > −∞ in a dense subset of Ω . Then infF ∈ H(Ω).

Proof. Apply Theorem 6.3.6 to the family

−F := {−u : u ∈ F}. ��


Theorem 6.3.8 (Up directed families in H∗(Ω)). Let (E,H) be a harmonic space,and let Ω ⊆ E be open. If F ⊆ H∗(Ω) and F ↑, then

v := supu∈F

u ∈ H∗(Ω).

Proof. The function v is l.s.c. Moreover, for every H-regular open set V ⊆ V ⊆ Ω

and for every x ∈ V , by using Theorem 6.2.10, we obtain

v(x) = supu∈F

u(x) ≥ supu∈F

∫

∂V

u dμVx =

∫

∂V

(supu∈F

u)

dμVx ,

so that v(x) ≥ ∫∂V

v dμVx . This proves the theorem. ��

Exercise 6.3.9. Let V ⊆ E be an H-regular open set, and let f : ∂V → R bel.s.c. and such that ∫

∂V

f dμVx < ∞ ∀ x ∈ V.

Then x → ∫∂V

f dμVx is H-harmonic in V .

(Hint:∫∂V

f dμVx = sup{HV

ϕ (x) : ϕ ∈ C(∂V, R), ϕ ≤ f }.)

Exercise 6.3.10. Let V ⊆ E be an H-regular open set, and let

f : ∂V → R

be μVx -summable for every x ∈ V . Then x → ∫

∂Vf dμV

x is H-harmonic in V . (Hint:Use Ex. 6.3.9 and Corollary 6.3.7.)

6.4 B-hyperharmonic Functions. Minimum Principle

Definition 6.4.1 (B-hyperharmonic function). Let (E,H) be a harmonic spaceand let Ω be an open subset of E. Assume that, for every x ∈ Ω , we are given abasis B(x) of H-regular neighborhoods of x with closure contained in Ω . A l.s.c.function

u : Ω →]−∞,∞]will be called B-hyperharmonic in Ω if

u(x) ≥∫

∂V

u dμVx

for every x ∈ Ω and every V ∈ B(x). We shall denote by B-H∗(Ω) the family of theB-hyperharmonic functions in Ω .

Obviously, H∗(Ω) ⊆ B-H∗(Ω). We will see in Corollary 6.4.9 that H∗(Ω) =B-H∗(Ω). An application of the Levi–Cartan Theorem 6.2.10 gives the followinglemma.

6.4 B-hyperharmonic Functions. Minimum Principle 349

Lemma 6.4.2. Let K ⊆ Ω be compact, and let μ be a measure supported inside K .Suppose we are given a family K of compact subset of K such that:

(i) K ↓, that is, for every A,B ∈ K, there exists C ∈ K contained in A ∩ B;(ii) supp(μ) ⊆ A for every A ∈ K.

Thensupp(μ) ⊆

⋂

A∈KA.

Proof. Let us put B := ⋂A∈K A. We have to show that μ(Ω \ A) = 0. Obviously,

since μ(Ω \ K) = 0 and B ⊆ K , this is equivalent to prove that μ(B) = μ(K).Observing that F := {χA : A ∈ K} is a down directed family of u.s.c functions, wemay apply the Levi–Cartan Theorem 6.2.10 to obtain

μ(B) =∫

K

χB dμ =∫

K

(inf

A∈KχA

)dμ = inf

A∈K

∫

K

χA dμ = infA∈K

μ(A) = μ(K).

The last equality follows from the assumptions supp(μ) ⊆ A ⊆ K for every A ∈ K.This ends the proof. ��Definition 6.4.3 (B-invariant set). Let A ⊆ Ω be relatively closed in Ω , that isA = Ω ∩ A0 with A0 closed in E. We say that A is B-invariant if

for every y ∈ A and every V ∈ B(y), one has supp(μVy ) ⊆ A.

We now show several lemmas needed for proving a minimum principle for B-hyperharmonic functions, which will be the main achievement of this section. Thefirst one is a trivial corollary of Lemma 6.4.2.

Lemma 6.4.4. Let K ⊆ Ω be compact, and let K be a family of closed B-invariantsubset of K . Assume K ↓. Then

B =⋂

A∈KA

is B-invariant.

The second lemma shows that the zero set of a non-negative B-hyperharmonicfunction is a B-invariant set.

Lemma 6.4.5. Let u ∈ B-H∗(Ω), u ≥ 0. Then

u−1(0) := {x ∈ Ω : u(x) = 0}is B-invariant.

Proof. First of all, since u is l.s.c. and

u−1(0) = {x ∈ Ω : u(x) ≤ 0},


the set u−1(0) is relatively closed in Ω . Moreover, if y ∈ u−1(0) and V ∈ B(y), wehave

0 = u(y) ≥∫

V

u dμVy (for u ∈ H∗(Ω))

≥ 0 (recall that u ≥ 0).

Then u = 0 μVy -almost everywhere on ∂V . That is, since supp(μV

y ) ⊆ ∂V ⊆ Ω , wehave

0 = μVy (∂V \ u−1(0)) = μV

y (Ω \ u−1(0)).

This means that supp(μVy ) ⊆ u−1(0), and the lemma is proved. ��

The argument used in this proof can be used for proving the following assertion.

Lemma 6.4.6. Let A ⊆ Ω be B-invariant, and let u ∈ B-H∗(Ω) such that u ≥ 0 inA. Then

A0 := {x ∈ A : u(x) = 0}is B-invariant.

The next lemma shows that every non-empty compact B-invariant set contains anon-empty minimal B-invariant set. Indeed, we have the following result.

Lemma 6.4.7 (Minimal B-invariant set). Let K ⊆ Ω be a non-empty compactB-invariant set. Then there exists A0 ⊆ K such that:

(i) A0 is non-empty and B-invariant,(ii) A0 is minimal, that is,

A ⊆ A0, A = ∅, A B-invariant ⇒ A = A0.

Proof. Define

E := {A ⊆ K : A closed, A = ∅, A B-invariant}.We have to prove that (E,⊆) has a minimal element. We will use Zorn’s lemmastating the existence of such an element if every linearly ordered family of elementsof E has a minorant in E . Then, let E0 ⊆ E be linearly ordered with respect to theinclusion ⊆. Define

B :=⋂

A∈E0

A.

B is compact and non-empty, since every finite intersection of elements of E0 is non-empty, for (E0,⊆) is linearly ordered. Moreover, by Lemma 6.4.4, B is invariant.Then B ∈ E0 and, obviously, B ⊆ A for every A ∈ E0. Thus E0 has a minorant in E .The lemma is proved. ��

Finally, we are ready to prove our minimum principle.

6.4 B-hyperharmonic Functions. Minimum Principle 351

Theorem 6.4.8 (Minimum principle for B-H∗ functions). Let Ω be a relativelycompact open set in a harmonic space (E,H), and let u ∈ B-H∗(Ω). Assume

lim infx→y

u(x) ≥ 0 ∀ y ∈ ∂Ω. (6.9)

Then u(x) ≥ 0 for every x ∈ Ω .

Proof. Argue by contradiction and assume the existence of a point x ∈ Ω such thatu(x) < 0. Let h0 ∈ H∗(Ω) be such that infΩ h0 > 0 and h0 < ∞ at any pointof Ω (see the positivity axiom (A1)). Let us prove the following claim: there exists astrictly positive real constant α such that

u + αh0 ≥ 0 in Ω and K := {x ∈ Ω : u(x) + αh0(x) = 0} is compact.

Consider the function

h := 1

h0max{0,−u}.

Condition (6.9), together with the upper semicontinuity of h, the compactness of Ω ,and the condition u(x) < 0, implies the existence of maxΩ h =: α, with 0 < α < ∞.Moreover:

(i) u + αh0 ≥ u + hh0 = u + max{0,−u} ≥ 0 in Ω .(ii) K := {x ∈ Ω : u(x) + αh0(x) = 0} is non-empty, since the points of K are

the maximum points of h.(iii) K = K . Indeed, first notice that K is relatively closed in Ω being the null set

of a non-negative l.s.c. function. Then K = Ω ∩ K . If we prove that K ⊆ Ω ,we are done. For this it is enough to show that if y ∈ ∂Ω , then y /∈ K . Now,condition (6.9) implies

lim infx→y

(u(x) + αh0) ≥ α infΩ

h0 > 0,

so that u + αh0 > 0 in a suitable neighborhood of y. This, obviously, impliesthat y /∈ K .

(iv) K is compact, since K = K ⊆ Ω and Ω is compact.

By Lemma 6.4.5, K is B-invariant and, by Lemma 6.4.6, there exists A0 ⊆ K ,compact, non-empty, B-invariant and minimal.

We claim thatA0 = {x0} (6.10)

for a suitable x0 ∈ Ω . Assuming this claim for true for a moment, we show howto reach a contradiction, thus proving the theorem. Let V ∈ B(x0). Since A0 is B-invariant, it has to be supp(μV

x0) ⊆ A0 = {x0}. On the other hand, again by the

positivity axiom, we can find k0 ∈ H∗(Ω) such that k0(x0) > 0. It follows that

0 < k0(x0) ≤∫

∂V

k0 dμΩx0

,


so that μVx0

(∂V ) > 0. This is impossible, since supp(μVx0

) ⊆ {x0} and x0 /∈ ∂V . Then,we are left with the proof of (6.10). By contradiction, assume it is false. Then, sinceH∗(E) separates different points of E, there exists w ∈ H∗(E) such that w ≡ ∞.Fix such a hyperharmonic function and choose a real β > 0 such that βu + w < 0somewhere in A0 (recall that if u + αh0 ≡ 0 in A0 and α > 0, then u < 0 in A0).Letting

−γ := minA0

βu + w

h0,

we have γ > 0, βu + w + γ h0 ≥ 0 in A0 and

A1 := {x ∈ A0 : (βu + w + γ h0)(x) = 0} = ∅.

By Lemma 6.4.6, A1 is B-invariant. Then, since A1 ⊆ A0 and A0 is minimal, it hasto be A1 = A0, so that

βu + w + γ h0 ≡ 0 in A0.

On the other hand, being A0 ⊆ K , u + αh0 ≡ 0 in A0. It follows that

w = (βα − γ )h0 := λh0 in A0.

Summing up: for every w ∈ H∗(E), w ≡ ∞ in A0, there exists λ ∈ R such that w =λh0. This implies that H∗(E) cannot separate different points of A0, in contradictionwith the separation axiom (A4). Then A0 has to be a singleton. This proves (6.10)and completes the proof of the theorem. ��

We now show some consequences of the previous theorem. First of all, we showthat the B-hyperharmonicity is equivalent to the hyperharmonicity.

Corollary 6.4.9. Let Ω be an arbitrary open set in E, and let u ∈ B-H∗(Ω) for asuitable B. Then u ∈ H∗(Ω).

Proof. Let V be an H-regular set with the closure contained in Ω . We have to provethat

u(x) ≥∫

∂V

u dμVx ∀ x ∈ V. (6.11)

For every z ∈ V , define

BV (z) = {W ∈ B(z) : W ⊆ V }.Let us now choose φ ∈ C(∂V, R), φ ≤ u|∂V . The function u − HV

φ is BV -hyperharmonic in V . Moreover,

lim infV �x→y

(u − HVφ )(x) ≥ u(y) − φ(y) ≥ 0 ∀ y ∈ ∂V .

Then, by Theorem 6.4.8, u − HVφ ≥ 0 in V , that is,

u(x) ≥∫

∂V

φ dμVx ∀ x ∈ V (6.12)

and for every φ ∈ C(∂V, R), φ ≤ u|∂V . Taking the supremum at the right-hand sidewith respect to these functions φ’s, we obtain (6.11). ��

6.5 Subharmonic and Superharmonic Functions. Perron Families 353

Corollary 6.4.10. Let (E,H) be a harmonic space, and let Ω ⊆ E be open. Themap

Ω → H∗(Ω)

is a sheaf of functions.

Proof. Property (6.2a) (page 340) is trivial. Let us prove (6.2b). Let (Ωα)α∈A be afamily of open subsets of E, and let

u : Ω → [−∞,∞], Ω :=⋃

α∈AΩα,

be a function such that u|Ωα ∈ H∗(Ωα) for every α ∈ A. Then u is l.s.c and u > −∞at any point. For every x ∈ Ω , define

B(x) := {V ∈ Tr : x ∈ V ⊆ V ⊆ Ωα for a suitable α ∈ A}.One readily verifies that u ∈ B-H∗(Ω). Then, by the previous corollary, we inferthat u ∈ H∗(Ω). ��Exercise 6.4.11. Let (E,H) be a harmonic space. Let Ω ⊆ E be open, and letx0 ∈ Ω . If a function u : Ω →]−∞,∞] satisfies:

(i) u ∈ H∗(Ω \ {x0}),(ii) lim supx→x0

u(x) = ∞,

then u ∈ H∗(Ω).

6.5 Subharmonic and Superharmonic Functions. Perron Families

Definition 6.5.1 (H-super- and H-sub-harmonic function). Let (E,H) be a har-monic space, and let Ω ⊆ E be open. A function u ∈ H∗(Ω) will be said H-superharmonic if, for every H-regular open set V ⊆ V ⊆ Ω , the function

V � x →∫

∂V

u dμVx

is H-harmonic in V . The set of the H-superharmonic functions in Ω will be denotedby

S(Ω).

A function v : Ω → [−∞,∞[ will be said H-subharmonic in Ω if −v ∈ S(Ω). Weshall denote by

S(Ω) := −S(Ω)

the set of the H-subharmonic functions in Ω .

From Theorem 6.3.6 we readily obtain the following characterization of the H-superharmonic functions.


Theorem 6.5.2 (Characterization of S(Ω)). Let u ∈ H∗(Ω). Then u ∈ S(Ω) ifand only if

D := {x ∈ Ω : u(x) < ∞}is dense in Ω , that is, D ⊇ Ω .

Proof. By contradiction, suppose D is not dense in Ω . Then there exists an H-regular open set V such that

V ⊆ {x ∈ Ω : u(x) = ∞}. (6.13)

Invoking the positivity axiom, we may assume∫∂V

dμVx > 0 for every x ∈ V . In-

deed, by this axiom, one can suppose the existence of a strictly positive H∗ functionk0 in an open set containing V . This implies

∫

∂V

k0 dμVx ≥ k0(x) > 0,

hence μVx (∂V ) > 0 for every x ∈ Ω . Then, from (6.13) we get

∫

∂V

u dμVx = ∞ ∀ x ∈ V,

showing that x → ∫∂V

u dμVx is not H-harmonic in V . Vice versa, suppose D ⊇ Ω ,

and let V ⊆ V ⊆ Ω be an H-regular set. Consider

F := {HVφ : φ ∈ C(∂V, R), φ ≤ u|∂V }.

F is an up directed family of H-harmonic functions in V such that

supF ≤ u < ∞ in D,

since u ∈ H∗(Ω). Theorem 6.3.6 implies that v := supF is H-harmonic in V . Thiscompletes the proof, since

v(x) =∫

∂V

u dμVx , x ∈ V. ��

A useful criterion for subharmonicity is given by the next corollary.

Corollary 6.5.3. Let (E,H) be a harmonic space and let Ω ⊆ E be open. Let u ∈H∗(Ω). Assume there exists v ∈ S(Ω) such that u ≤ v. Then u ∈ S(Ω).

Proof. Notice that

{x ∈ Ω : u(x) < ∞} ⊇ {x ∈ Ω : v(x) < ∞}and apply Theorem 6.5.2. ��

We also have the following result.


Proposition 6.5.4. Let u, v ∈ S(Ω). Then:

(i) λu ∈ S(Ω) for every λ ≥ 0,(ii) u + v ∈ S(Ω),

(iii) min{u, v} ∈ S(Ω).

Proof. Exercise. ��Let us now introduce the following crucial definition.

Definition 6.5.5 (Perron-regularization). Let (E,H) be a harmonic space, and letΩ ⊆ E be open. Given u ∈ H∗(Ω) and an H-regular set V ⊆ V ⊆ Ω , define

uV : Ω →]−∞,∞], uV (x) :={

u(x), x /∈ V ,∫∂V

u dμVx , x ∈ V .

The function uV will be called the Perron-regularization of u related to V .

The main properties of the Perron-regularization are given in by the followingtheorem.

Theorem 6.5.6 (Properties of the Perron-regularization). Suppose that u ∈H∗(Ω) and V ⊆ V ⊆ Ω is an open H-regular set, then:

(i) uV ≤ u,(ii) uV ∈ H∗(Ω),

(iii) uV ≤ vV if u, v ∈ H∗(Ω) and u ≤ v.

Moreover, if u ∈ S(Ω), then

(iv) uV ∈ S(Ω) and u|V ∈ H(V ).

Proof. (i) It simply follows from the very definition of hyperharmonic function.(ii) We split the proof of this property in two steps.

(ii-a) uV is lower semicontinuous. For this, we have to prove that

lim infx→y

uV (x) ≥ uV (y) for every y ∈ Ω.

The statement holds true if y ∈ V , since, in this case, uV = u in a neighborhoodof y. On the other hand, for y ∈ V , we have

uV (y) =∫

∂V

u dμVy = sup{HV

ϕ |ϕ ∈ C(∂V, R), ϕ ≤ u|∂V }. (6.14)

Then uV is H-hyperharmonic in V (see Theorem 6.3.8), hence lower semicontinuousat any point y ∈ V .

Let us now suppose y ∈ ∂V . In this case uV (y) = u(y), so that


lim infV �x→y

uV (x) = lim infV �x→y

u(x) ≥ u(y) = uV (y).

Then we only have to prove that

lim infV �x→y

uV (x) ≥ u(y). (6.15)

Let ϕ ∈ C(∂V, R), ϕ ≤ u|∂V . Then uV ≥ HVϕ in V , so that

lim infV �x→y

uV (x) ≥ lim infV �x→y

HVϕ (x) = ϕ(y).

Since u|∂V = sup{ϕ ∈ C(∂V, R) : ϕ ≤ u|∂V }, (6.15) follows from the previousinequality.

(ii-b) uV ∈ B-H∗(Ω) with B defined as follows

B(x) =

⎧⎪⎨

⎪⎩

{B : B H-regular open set, x ∈ B ⊆ B ⊆ V } if x ∈ V ,

{B : B H-regular open set, x ∈ B ⊆ B ⊆ Ω \ V } if x /∈ V ,

{B : B H-regular open set, x ∈ B ⊆ B ⊆ Ω} if x ∈ ∂V .

Since uV is H-hyperharmonic in Ω \ V and in V (see (6.14)), we only have to provethat

uV (y) ≥∫

∂B

uV dμBy ∀y ∈ ∂B, ∀ B ∈ B(y).

We have

uV (y) = u(y) (since y ∈ ∂V )

≥∫

∂B

u dμBy (u ∈ H∗(Ω))

≥∫

∂B

uV dμBy (u ≥ uV ).

This shows that uV ∈ B-H∗(Ω). Thus, by Corollary 6.4.9, u ∈ H∗(Ω).(iii) This is quite obvious.(iv) Since uV ∈ H∗(Ω) and uV ≤ u, if u ∈ S(Ω) then uV ∈ S(Ω) (see

Corollary 6.5.3). The harmonicity of uV in V follows from the very definition ofsubharmonic function. ��

We are now ready to give another crucial definition.

Definition 6.5.7 (Perron family). Let (E,H) be a harmonic space, and let Ω ⊆ E

be open. A family of functions F ⊆ H∗(Ω) will be said a Perron family if thefollowing conditions are satisfied:

(i) F ↓, that is, F is down directed,(ii) F has an H-subharmonic minorant, i.e. there exists v0 ∈ S(Ω) such that

v0 ≤ u for every u ∈ F ,


(iii) uV ∈ F for every u ∈ F and for every H-regular open set V ⊆ V ⊆ Ω ,(iv) F contains at least one H-superharmonic function, i.e. there exists u0 ∈ F such

that u0 ∈ S(Ω).

The following important result holds.

Theorem 6.5.8 (The fundamental theorem on Perron families). Let (E,H) be aharmonic space, and let F be a Perron family on an open set Ω ⊆ E. Then

u := infF ∈ H(Ω).

Proof. Let V ⊆ V ⊆ Ω be an arbitrary fixed H-regular set. It is enough to provethat u is H-harmonic in V . Define

FV = {uV |V : u ∈ F , u ∈ S(Ω)}and prove the following list of claims:

(C1) FV = ∅. This follows from assumption (iv) in Definition 6.5.7.(C2) FV ⊆ H(V ). This is granted by Theorem 6.5.6-(iv).(C3) FV ↓. Indeed, let u, v ∈ F ∩ S(Ω). Since F is down directed, there exists

w ∈ F such that w ≤ min{u, v}. The function w is H-superharmonic, for itis bounded from above by min{u, v} which is H-superharmonic (see Corol-lary 6.5.3). It follows that wV ∈ FV and wV ≤ min{uV , vV }, proving thatFV ↓.

(C4) FV has an H-subharmonic minorant, so that infFV > −∞ in a dense subsetof Ω . This directly follows from assumptions (ii) and (iii) in Definition 6.5.7.

(C5) u|V = infFV . Indeed, the inequality

u|V ≤ infFV

is trivial for FV ⊆ {u|V : u ∈ F}. On the other hand, let u0 ∈ F ∩ S(Ω) bethe function given by assumption (iv) in Definition 6.5.7. For every u ∈ F , takev ∈ F satisfying v ≤ min{u, u0} (F ↓). Since u0 ∈ S(Ω), also v ∈ S(Ω). Itfollows that v|V ∈ FV and that

infFV ≤ vV ≤ v|V ≤ u|V .

Therefore,infFV ≤ u|V ∀u ∈ F ,

and the inequality infFV ≤ u|V follows.

Summing up: (C1)–(C4) and Corollary 6.3.7 imply infFV ∈ H(V ). Hence, from(C5) we get u|V ∈ H(V ). The proof is complete. ��


6.6 Harmonic Majorants and Minorants

From the fundamental theorem on Perron families, one easily obtains conditions forthe existence of least harmonic majorants of subharmonic functions and of greatestharmonic minorants of superharmonic functions.

Theorem 6.6.1 (The least H-superharmonic majorant). Let (E,H) be a har-monic space, and let Ω ⊆ E be open. Let v ∈ S(Ω) be such that

v ≤ u0

for a suitable u0 ∈ S(Ω). Then v has a least H-superharmonic majorant h on Ω ,and h ∈ H(Ω).

Proof. Consider the family

F := {u ∈ S(Ω) : v ≤ u ≤ u0}.If u ∈ F and V is an H-regular open set with V ⊆ Ω , we have v ≤ vV ≤ uV ≤ u ≤u0. Hence uV ∈ F . It follows that F is a Perron family, with the H-subharmonicminorant v and the H-superharmonic majorant u0. Then

h := infF ∈ H(Ω),

by Theorem 6.5.8, and h ≥ v in Ω . Hence h is the least element of F and h isH-harmonic in Ω . ��Corollary 6.6.2. In a harmonic space (E,H), every H-superharmonic function withan H-subharmonic minorant in an open set Ω has a greatest H-subharmonic mino-rant which is H-harmonic in Ω .

The following proposition holds.

Proposition 6.6.3. Let Ω ⊆ E be open, and let v1, v2 ∈ S(Ω). Assume v1, v2 havean H-superharmonic majorant. Then v1 +v2 has a least H-superharmonic majorantgiven by

h1 + h2,

where hi is the least H-(super)harmonic majorant of vi .

Proof. Let h be the least H-(super)harmonic majorant of v1 + v2. Obviously, h ≤h1 + h2. On the other hand, being h ≥ v1 + v2, the function h − v1 is an H-superharmonic majorant of v2. Hence

h − v1 ≥ h2.

It follows that h − h2 ≥ v1 and h − h2 is H-harmonic in Ω . Then h − h2 ≥ h1, thatis h ≥ h1 + h2. This completes the proof. ��

6.7 The Perron–Wiener–Brelot Operator 359

6.7 The Perron–Wiener–Brelot Operator

Throughout this section, Ω will denote an open set, with compact closure and non-empty boundary, in a harmonic space (E,H). We shall construct the Perron–Wiener–Brelot operator

f → HΩf

from a suitable linear set of real extended functions f : ∂Ω → [−∞,∞] to the lin-ear space H(Ω) of the H-harmonic functions in Ω . When the H-Dirichlet problem

(H-D)

{u ∈ H(Ω),

limx→y u(x) = f (y) ∀ y ∈ ∂Ω ,

has a solution u, it will turn out that u = HΩf . For this reason, sometimes we shall

call HΩf the generalized solution to (H-D) in the sense of Perron–Wiener–Brelot.

Our construction, in the case of classic harmonic functions, i.e. of the solutions to the“usual” Laplace equation Δu = 0, goes back to O. Perron, N. Wiener and M. Brelot.

Definition 6.7.1 (Upper and lower functions). Let (E,H) be a harmonic space,and let Ω ⊆ E be open and such that Ω is compact and ∂Ω = ∅. Given a functionf : ∂Ω → [−∞,∞], we set2

UΩ

f :={u ∈ H∗(Ω) : lim inf

∂Ωu ≥ f, inf u > −∞

}

andUΩ

f :={v ∈ −H∗(Ω) : lim sup

∂Ω

v ≤ f, sup v < ∞}.

The families UΩ

f and UΩf will be called, respectively, the family of the upper functions

and of the lower functions related to f and Ω .

The function u ≡ ∞ (v ≡ −∞, respectively) is an upper function (lower func-

tion, respectively). Therefore, UΩ

f = ∅ (UΩf = ∅, respectively).

Definition 6.7.2 (Upper and lower solutions). With the hypotheses and notation ofthe previous definition, the real extended functions

HΩ

f := infUΩ

f , HΩf := supUΩ

f

will be called the upper solution and the lower solution, respectively, to the problem(H-D).

2 The notationlim inf

∂Ωu ≥ f

meanslim infx→y

u(x) ≥ f (y) ∀ y ∈ ∂Ω.

Analogous meaning for lim sup∂Ω u ≤ f .


It has to be noticed thatHΩ

f = −HΩ

−f ,

since UΩf = −UΩ

−f . We also have the following proposition, an easy consequence ofthe minimum principle for the H-hyperharmonic functions.

Proposition 6.7.3. In the hypotheses of Definition 6.7.1, for every f : ∂Ω →[−∞,∞], one has

HΩf ≤ H

Ω

f . (6.16)

Proof. For any u ∈ UΩ

f and v ∈ UΩf , one has u − v ∈ H∗(Ω) and

lim inf∂Ω

(u − v) ≥ 0.

Then, from Theorem 6.4.8 (the minimum principle), u − v ≥ 0 in Ω . Thus

u ≥ v ∀ u ∈ UΩ

f , ∀ v ∈ UΩf ,

and (6.16) readily follows. ��The next proposition is a straightforward consequence of the definitions of upper

and lower solution.

Proposition 6.7.4. Let f, g : ∂Ω → [−∞,∞], α ∈ R, α > 0. Then:

(i) f ≤ g ⇒ HΩ

f ≤ HΩ

g , HΩf ≤ HΩ

g ,

(ii) HΩ

f +g ≤ HΩ

f + HΩ

g , HΩf +g ≥ HΩ

f + HΩg , whenever the sums are defined.

(iii) HΩ

αf = αHΩ

f , HΩαf = αHΩ

f , HΩ

−αf = −αHΩf ,

(iv) f ≥ 0 ⇒ HΩ

f ≥ 0, HΩf ≥ 0.

Proof. It is left as an exercise. (Hint: to prove (iv) use the minimum principle andthe fact that u ≡ 0 is H-harmonic, hence both H-hyperharmonic and H-hypoharmonic.) ��

In Section 6.9, we will use the following Beppo Levi-type property of the map

f → HΩ

f , which we leave here as an exercise.

Exercise 6.7.5. Let Ω ⊆ E be a relatively compact open set, and let (fn) be a se-quence of real extended functions fn : ∂Ω → [−∞,∞] such that

fn ↗ f and HΩ

fn∈ H(Ω) ∀n ∈ N.

ThenH

Ω

fn↗ H

Ω

f .

Hint: h = supn HΩ

fn∈ H∗(Ω), see Theorem 6.3.8. For any fixed x ∈ Ω and ε > 0

choose un ∈ UΩ

fnsuch that un(x) < H

Ω

fn(x) + ε

2n . Then

6.7 The Perron–Wiener–Brelot Operator 361

u = h +∞∑

n=1

(un − HΩ

fn) ∈ UΩ

f .

Definition 6.7.6 (Resolutive function). Let (E,H) be a harmonic space, and letΩ ⊆ E be an open set with compact closure and non-empty boundary. A real ex-tended function f : ∂Ω → [−∞,∞] will be said resolutive if:

(i) HΩ

f = HΩf ,

(ii) HΩ

f ∈ H(Ω).

In this case, we setHΩ

f := HΩ

f (= HΩf ),

and we say that HΩf is the generalized solution, in the sense of Perron–Wiener–

Brelot, to the problem (H-D). We also call HΩf the PWB function related to Ω and

f . The set of the resolutive functions f : ∂Ω → [−∞,∞] will be denoted byR(∂Ω),

R(∂Ω) := {f : ∂Ω → [−∞,∞] | f is resolutive}. (6.17)

The connection between HΩf and the H-Dirichlet problem (H-D) is showed by

the following proposition.

Proposition 6.7.7. Let the hypotheses of Definition 6.7.6 hold. Let f : ∂Ω →[−∞,∞] be a bounded function. Then the following statements are equivalent:

(i) f is resolutive and limx→y HΩf (x) = f (y) for every y ∈ ∂Ω .

(ii) There exists u ∈ H(Ω) such that limx→y u(x) = f (y) for every y ∈ ∂Ω .

In this latter case, u = HΩf .

Proof. (i) ⇒ (ii). This is trivial.(ii) ⇒ (i). Just remark that u is bounded from above and below, and that u ∈

UΩ

f ∩ UΩf . ��

We would like to explicitly remark that R(∂Ω) is not empty, since it contains thenull function. From Proposition 6.7.4 it follows that:

(i) if f, g ∈ R(∂Ω) and f + g is defined, then f + g ∈ R(∂Ω) and HΩf +g =

HΩf + HΩ

g ,

(ii) if f ∈ R(∂Ω) and c ∈ R, then c f ∈ R(∂Ω) and HΩcf = cHΩ

f .

In particular,R∞(∂Ω) := R(∂Ω) ∩ L∞(∂Ω)

is a real vector space, and the map

R∞(∂Ω) � f → HΩf ∈ C(Ω, R)

is linear and monotone.


We want to show that R(∂Ω) contains the restrictions to the boundary of Ω

of the continuous superharmonic functions. We first show a proposition having anindependent interest, an easy consequence of the fundamental theorem on the Perronfamilies (Theorem 6.5.8, page 357).

Proposition 6.7.8. Let Ω ⊆ E be an open set with compact closure and non-emptyboundary. Let f : ∂Ω → [−∞,∞]. Assume that there exist

u0 ∈ UΩ

f ∩ S(Ω) and v0 ∈ UΩf ∩ S(Ω). (6.18)

ThenH

Ω

f ,HΩf ∈ H(Ω).

Proof. Using (6.18), we recognize that UΩ

f is a Perron family. Hence, by Theo-rem 6.5.8,

HΩ

f ∈ H(Ω).

From (6.18) it also follows that UΩ

−f is a Perron family, thus

HΩf = supUΩ

f = − infUΩ

−f ∈ H(Ω).

This ends the proof. ��Theorem 6.7.9 (Resolutivity. I). Let (E,H) be a harmonic space, and let Ω ⊆ E

be an open set with compact closure and non-empty boundary.If u ∈ C(Ω, R) and

u|Ω ∈ S(Ω),

thenf := u|∂Ω ∈ R(∂Ω).

Proof. By the positivity axiom (A1), there exists h0 ∈ S(Ω) such that infΩ h0 > 0.Denote

λ := max{0,−u}infΩ h0

.

Then u ≥ −λh0 in Ω and −λh0 ∈ S(Ω) ∩ UΩf . On the other hand,

u|Ω ∈ S(Ω) ∩ UΩ

f .

By Proposition 6.7.8,

HΩ

f ,HΩf ∈ H(Ω).

Moreover, HΩ

f ≤ u|Ω so that, since u is continuous up to ∂Ω ,

HΩ

f ∈ UΩf .

6.8 S-harmonic Spaces: Wiener Resolutivity Theorem 363

Thus HΩ

f ≤ HΩf . The opposite inequality always being true, this implies

HΩ

f = HΩf . Summing up,

HΩ

f = HΩf ∈ H(Ω),

that is, f is resolutive. ��This last theorem would achieve a full strength if we knew that the family of con-

tinuous subharmonic functions is sufficiently rich. This happens in the S-harmonicspaces introduced in the following section.

6.8 S-harmonic Spaces: Wiener Resolutivity Theorem

Definition 6.8.1 (S-harmonic space). A harmonic space (E,H) will be said S-harmonic if the family

S+c (E) := {u ∈ S(E) ∩ C(E, R) : u ≥ 0}

separates the points of E, that is,

for every x, y ∈ E, x = y, there exists

u, v ∈ S+c (E) such that u(x)v(y) = u(y)v(x). (6.19)

In a S-harmonic space, the positivity property of the axiom (A1) takes a strongerform.

Proposition 6.8.2 (Positivity axiom in a S-harmonic space). Let (E,H) be a S-harmonic space. For every compact set K ⊆ E, there exists w ∈ S+

c (E) such that

infK

w > 0.

Proof. Since S+c (E) separates the points of E, for every x ∈ K , there exists

ux ∈ S+c (E) such that ux(x) > 0. By the lower continuity of u, we can find an

open neighborhood Vx of x such that infV ux > 0. Since K is compact, there existx1, . . . , xp ∈ K such that

K ⊆p⋃

j=1

Vxj.

Then, if we define

w :=p∑

j=1

uxj,

we have w ∈ S+c (E) and infK w > 0. ��


A crucial property of the S-harmonic spaces is given by the next proposition.

Proposition 6.8.3. Let (E,H) be a S-harmonic space. For every compact set K ⊆E, for every f ∈ C(K, R) and for every ε > 0, there exist u, v ∈ S+

c (E) such that

supK

|f − (u − v)| < ε.

Proof. DefineA := {u − v : u, v ∈ S+

c (E)}.It is immediate to recognize that A is a linear subspace of C(E, R). Moreover, since(E,H) is S-harmonic, A separates the points of E. Finally, if p = u − v ∈ A, thenalso p+ := max{p, 0} belongs to A, since

p+ = u − min{u, v}.Then, by the Stone–Weierstrass theorem,3 for every f ∈ C(K, R) and ε > 0, thereexists p ∈ A such that

supK

|f − p| < ε.

This ends the proof. ��We are now ready to prove the resolutivity of any continuous function.

Theorem 6.8.4 (Wiener resolutivity theorem). Let (E,H) be a S-harmonic space,and let Ω ⊆ E be an open set with compact closure and non-empty boundary. Everycontinuous function f : ∂Ω → R is resolutive.

Proof. By Theorem 6.7.9, if u ∈ S+c (E), then u|∂Ω is resolutive. As a consequence,

every functionp = (u − v)|∂Ω, u, v ∈ S+

c (E),

is resolutive. By Proposition 6.8.2, for every ε > 0, there exist uε, vε ∈ S+c (E) such

that|f − p| < ε on ∂Ω, p = pε := (uε − vε)|∂Ω. (6.20a)

By Proposition 6.8.2, there exists w ∈ S+c (E) such that inf∂Ω w > 1. Then, from

(6.20a) we havep − εw < f < p + εw on ∂Ω,

so that, letting v := w|∂Ω , we have

HΩp−εv ≤ HΩ

f ≤ HΩ

f ≤ HΩ

p+εv. (6.20b)

On the other hand, we know that p − εv, p + εv ∈ R(∂Ω). It follows that

3 See the Appendix to this section.

6.8 S-harmonic Spaces: Wiener Resolutivity Theorem 365

HΩp−εv = HΩ

p−εv = HΩp − εHΩ

v ,

HΩ

p+εv = HΩp+εv = HΩ

p + εHΩv .

Using these identities in (6.20b), we obtain

0 ≤ HΩ

f − HΩf ≤ 2εHΩ

v ≤ 2εw, (6.20c)

where, in the last inequality, we used the fact that w is an upper function related tow|∂Ω and Ω . Letting ε tend to zero in (6.20c), we get

HΩ

f = HΩf . (6.20d)

Inequalities (6.20b) also give,

0 ≤ HΩ

f − HΩp−εv ≤ HΩ

p+εv − HΩp−εv = 2εHΩ

v ≤ 2εw ≤ 2ε supΩ

w.

Then, as ε → 0,

HΩp−εv = HΩ

pε−εv → HΩ

f uniformly on Ω.

Since HΩp−εv ∈ H(Ω), from this limit and Corollary 6.3.5 we obtain

HΩ

f ∈ H(Ω).

Together with (6.20d), this implies the resolutivity of f . ��Exercise 6.8.5. Let Ω ⊆ E be open with Ω compact and ∂Ω = ∅. Let w ∈ S+

c (E)

be such that infΩ w =: λ > 0. Then, for every bounded function f : ∂Ω → R, wehave

−λ sup∂Ω

|f | ≤ HΩf ≤ H

Ω

f ≤ λ sup∂Ω

|f |.In particular, if f is resolutive,

supΩ

|HΩf | ≤ λ sup

∂Ω

|f |.

(Hint: Note that w supΩ |f | ∈ UΩ

f .)

Exercise 6.8.6 (R(Ω) = R(Ω)). Let Ω ⊆ E be an open set as in the previousexercise. Let (fn)n∈N be a sequence of bounded resolutive functions such that

sup∂Ω

|fn − f | → 0 as n → ∞.

Then f is resolutive.

Exercise 6.8.7. Let Ω ⊆ E be an open set as in the previous exercise, and letf : ∂Ω → R be a bounded function. Then

HΩ

f ,HΩf ∈ H(Ω).


6.8.1 Appendix: The Stone–Weierstrass Theorem

The aim of this section is to prove the following well-known density result.

Theorem 6.8.8 (The Stone–Weierstrass theorem). Let (Y, T ) be a compact topo-logical space, and let A ⊆ C(Y, R) satisfy the following conditions:

(i) A is a real vector space,(ii) if p ∈ A, then p+ := max{0, p} ∈ A,

(iii) A separates the points of E, that is,

for every x, y ∈ Y, x = y, there exists p, q ∈ A such that p(x)q(y) = p(y)q(x).

Then, for every f ∈ C(Y, R) and for every ε > 0, there exists p ∈ A such that

supY

|f − p| < ε.

Proof. Assumptions (i) and (ii) imply

|u| = u+ + (−u)+ ∈ A for every u ∈ A.

As a consequence, max{u, v} = u + v + |u − v| and min{u, v} = u + v − |u − v|belong to A for any u, v ∈ A. Let us now fix u ∈ C(Y, R). For every x, y ∈ Y , dueto assumption (iii), there exist α, β ∈ R and u, v ∈ A such that

{αu(x) + βv(x) = f (x),

αu(y) + βv(y) = f (y).(6.21)

Obviously, ux,y := αu + βv ∈ A, and, for every ε > 0, there exists a neighborhoodVy of y such that

ux,y(z) < f (z) + ε ∀z ∈ Vy.

Since Y is compact, we can choose y1, . . . , yp ∈ Y such that⋃p

j=1 Vyj= Y . Letting

ux := min{ux,y1 , . . . , ux,yp }, we have

ux ∈ A and ux(z) < f (z) + ε ∀z ∈ Y. (6.22)

On the other hand, by the first identity in (6.21), ux(x) = f (x) for every x ∈ Y .Then, since ux is continuous, for every x ∈ Y , there exists a neighborhood Wx of x

such thatux(z) > f (z) − ε ∀ z ∈ Wx.

Since Y is compact, there exist Wx1 , . . . ,Wxq such that⋃q

j=1 Wxj= Y . As a con-

sequence, letting u := max{ux1 , . . . , uxq }, we get u ∈ A and u(z) > f (z) − ε forevery y ∈ Y . On the other hand, keeping in mind (6.22), we also have

u = max{ux1 , . . . , uxq } < f + ε in Y.

This completes the proof. ��

6.9 H-harmonic Measures for Relatively Compact Open Sets 367

6.9 H-harmonic Measures for General Relatively Compact OpenSets. A Characterization of the Resolutive Functions

As in the previous section, Ω will denote a relatively compact (i.e. with compactclosure in E) open set with non-empty boundary, contained in a S-harmonic space(E,H).

Let us fix a point x ∈ Ω . By the Wiener resolutivity theorem (Theorem 6.8.4)and Proposition 6.7.4, the map

C(∂Ω, R) � f → HΩf (x) ∈ R

is well defined, linear and positive. Then by the Riesz representation theorem, thefollowing definition is well posed.

Definition 6.9.1 (H-harmonic measure. II). Let Ω be a relatively compact open setwith non-empty boundary, contained in the S-harmonic space (E,H). There existsa unique Radon measure, denoted by μΩ

x , such that

HΩf (x) =

∫

∂Ω

f dμΩx for every f ∈ C(∂Ω, R).

We shall call μΩx the H-harmonic measure related to Ω and x.

Exercise 6.9.2. For every relatively compact open set Ω ⊆ E, one has μΩx (∂Ω) <

∞ for every x ∈ Ω . (Hint: Note that μΩx (∂Ω) = HΩ

1 (x) and use the a priori estimateof Exercise 6.8.5.)

When Ω is an H-regular open set, this definition gives back the one given inDefinition 6.2.3. Our aim here is to show the following theorem.

Theorem 6.9.3 (Characterization of resolutivity). Let (E,H) be a S-harmonicspace and let Ω ⊆ E be a relatively compact open set with non-empty boundary.Given f : ∂Ω → [−∞,∞], the following statements are equivalent:

(i) f ∈ R(∂Ω),(ii) f ∈ L1(∂Ω,μΩ

x ) for every x ∈ Ω .

In this case

HΩf (x) =

∫

∂Ω

f dμΩx for every x ∈ Ω.

The proof of this theorem requires some preliminary results having an indepen-dent interest.

Notation. Given u : Ω →]−∞,∞], lower semicontinuous and bounded below,we set

u∗ : Ω →]−∞,∞], u∗(x) :={

u(x) if x ∈ Ω ,

lim infy→x u(y) if x ∈ ∂Ω .


It is quite easy to recognize that u∗ is lower semicontinuous.Similarly, given u : Ω → [−∞,∞[, upper semicontinuous and bounded above,

we set

u∗ : Ω → [−∞,∞[, u∗(x) :={

u(x) if x ∈ Ω ,

lim supy→x u(y) if x ∈ ∂Ω .

The function u∗ is upper semicontinuous.

Lemma 6.9.4. With the previous notation, if u ∈ H∗(Ω) and infΩ u > −∞, then:

(i) u(x) ≥ ∫∂Ω

u∗ dμΩx > −∞ for every x ∈ Ω ,

(ii) x → ∫∂Ω

u∗ dμΩx belongs to H∗(Ω).

An analogous result also holds if u ∈ H∗(Ω) and supΩ u < ∞.

Proof. For every φ ∈ C(∂Ω, R), φ ≤ u∗|∂Ω , we have

HΩφ ≤ u,

since u ∈ UΩ

φ . Then

u(x) ≥ supφ

HΩφ (x) = sup

φ

∫

∂Ω

φ dμΩx =

∫

∂Ω

u∗ dμΩx .

(The suprema are taken with respect to φ ∈ C(∂Ω, R), φ ≤ u∗|∂Ω .) On the otherhand, since inf∂Ω u∗ > −∞ and μx(Ω) < ∞ for every x ∈ Ω (see Exercise 12 atthe end of the chapter), we have

∫

∂Ω

u∗ dμΩx > −∞.

This proves (i). To obtain (ii), we just have to use Theorem 6.3.8 keeping in mindthat HΩ

φ ∈ H(Ω). ��Lemma 6.9.5. Let f : ∂Ω →]−∞,∞] be a l.s.c. function. Then:

(i) for every x ∈ Ω , we have

HΩ

f (x) =∫

∂Ω

f dμΩx ,

(ii) HΩ

f ∈ H(Ω) if and only if

HΩ

f (x) < ∞ for every x ∈ Ω.

6.9 H-harmonic Measures for Relatively Compact Open Sets 369

Proof. (i) Let (fn) be a sequence of continuous functions on ∂Ω such that fn ↗ f .

Since HΩ

fn= HΩ

fn∈ H(Ω) for every n ∈ N, we have (see Exercise 6.7.5, page 360)

HΩfn

↗ HΩ

f , (6.23)

so that, by the Beppo Levi monotone convergence theorem,

HΩ

f (x) = limn→∞ HΩ

fn(x) = lim

n→∞

∫

∂Ω

fn dμΩx =

∫

∂Ω

f dμΩx

for every x ∈ Ω .(ii) The only if part is trivial. The if part follows from (6.23) and Theorem 6.2.8,

page 343. ��Proof (of Theorem 6.9.3). (i) ⇒ (ii). Let f ∈ R(∂Ω). Then, for every x ∈ Ω and

ε > 0, there exist u ∈ UΩ

f and v ∈ UΩf such that

u(x) − ε < HΩf (x) < v(x) + ε.

By Lemma 6.9.4,

−∞ <

∫

∂Ω

u∗ dμΩx − ε < HΩ

f (x) <

∫

∂Ω

v∗ dμΩx + ε < ∞,

and we have ∫

∂Ω

(u∗ − v∗) dμΩx < 2ε,

so that, since v∗ ≤ f ≤ u∗, u∗ is l.s.c., v∗ is u.s.c., and ε is an arbitrary positivenumber,

f ∈ L1(∂Ω, dμΩx ) ∀ x ∈ Ω

and

HΩf (x) =

∫

∂Ω

f dμΩx for every x ∈ Ω.

(ii) ⇒ (i). It is not restrictive to assume f ≥ 0. Define

F := {g : ∂Ω → [−∞,∞[ | g u.s.c., 0 ≤ g ≤ f

}

andF := {

h : ∂Ω →]−∞,∞] |h l.s.c., h ≥ f}.

Let us fix x ∈ Ω and ε > 0. By Vitali–Carathéodory’s theorem, there exist g ∈ Fand h ∈ F such that ∫

∂Ω

(h − g) dμΩx < ε.

Then h, g ∈ L1(∂Ω,μΩx ) and, by Lemma 6.9.5-(i),


−∞ <

∫

∂Ω

g dμΩx = HΩ

g (x) ≤ HΩf (x) ≤ H

Ω

f (x)

=∫

∂Ω

h dμΩx ≤

∫

∂Ω

g dμΩx + ε < ∞.

This proves that −∞ < HΩf (x) ≤ H

Ω

f (x) < ∞ for every x ∈ Ω and that

supg∈F

HΩg = HΩ

f = HΩ

f = infh∈F

HΩ

h . (6.24)

On the other hand, by Lemma 6.9.5-(ii),

HΩg ∈ H(Ω) for every g ∈ F ,

since g ≥ 0. Hence HΩg > −∞ at any point of Ω . Thus, from (6.24) and Theo-

rem 6.3.6 we finally obtain

HΩf = H

Ω

f ∈ H(Ω).

This completes the proof. ��Exercise 6.9.6. Prove that if f : ∂Ω → R is l.s.c. and bounded from above, then f

is resolutive.

6.10 S∗-harmonic Spaces: Bouligand’s Theorem

We now give a definition introducing a new property which is not usually assumed inthe abstract potential theory. However, as we will see in Chapter 7, this assumptiondoes not affect at all the possibility to apply our theory to the sub-Laplacians.

Definition 6.10.1 (S∗-harmonic space). A S-harmonic space (E,H) will be saidS∗-harmonic if the following property holds:

(A*) For every x0 ∈ E there exists sx0 ∈ S+c (E) such that sx0(x0) = 0 and

infE\V sx0 > 0

for every neighborhood V of x0. (See Definition 6.8.1 for the definition ofS+

c (E).)

This property will make transparent the rôle of the barrier functions in studyingthe continuity up to the boundary of the Perron–Wiener–Brelot functions.

Convention. Throughout this section, Ω will denote an open set in a S∗-harmonicspace (E,H). We will always assume, without any further comments, that Ω iscompact and ∂Ω = ∅.

6.10 S∗-harmonic Spaces: Bouligand’s Theorem 371

We know from the Wiener resolutivity theorem that every continuous functionf : ∂Ω → R is resolutive, so that the Perron–Wiener–Brelot function HΩ

f is H-

harmonic in Ω . However, in general, we cannot expect a “good” behavior of HΩf at

the boundary points of Ω .

Definition 6.10.2 (H-regular point). A point y ∈ ∂Ω will be called H-regular if

limΩ�x→y

HΩf (x) = f (y) ∀ f ∈ C(∂Ω, R).

Obviously, the function HΩf is the solution (the unique solution, thanks to the

minimum principle!) of the H-Dirichlet problem

(H-D)

{u ∈ H(Ω),

limx→y u(x) = f (y) ∀ y ∈ ∂Ω,

for every f ∈ C(∂Ω, R), if and only if all the boundary points of Ω are H-regularpoints.

Unfortunately, as we said before, we have to expect that, in general, ∂Ω containsboundary points which are not H-regular.4

The notion of H-barrier function will allow us to give a necessary and sufficientcondition for the H-regularity.

Definition 6.10.3 (H-barrier function). Let y ∈ ∂Ω . A H-barrier function for Ω

at y is a function w defined in Ω ∩ V , being V a suitable open neighborhood of y,such that (see also Fig. 6.1):

(i) w ∈ S(Ω ∩ V ),(ii) w(x) > 0 for every x ∈ Ω ∩ V ,

(iii) limx→y w(x) = 0.

The link between H-regularity and H-barrier functions is given by the followingtheorem.

Theorem 6.10.4 (Bouligand’s theorem). Let (E,H) be a S∗-harmonic space, andlet Ω ⊆ E be a relatively compact open set with non-empty boundary. A pointx0 ∈ ∂Ω is H-regular for Ω if and only if there exists an H-barrier function forΩ at x0.

The proof of this theorem is quite long. It is convenient to premise some lemmas.

Lemma 6.10.5. Let V ⊆ E be a regular open set, and let U be a relatively opensubset of ∂V . For every x0 ∈ V and ε > 0, there exists a compact set K ⊆ U and anon-negative function h ∈ H(V ) such that:

4 See, for instance, Exercise 6.10.7 (page 375) related to the classical S. Zaremba’s exampleof non-regular boundary points.


Fig. 6.1. A H-barrier function

(i) h(x0) < ε,(ii) lim infV �x→y h(x) ≥ 1 for every y ∈ U \ K .

Proof. Let K ⊆ U be such that μVx0

(U \ K) < ε, and let f : ∂V → R be thecharacteristic function of U \ K . Since f is bounded and l.s.c., it is resolutive (seeEx. 6.9.6, page 370). Let us put h := HV

f . Then h ∈ H(V ), h ≥ 0, since f ≥ 0 and

h(x0) =∫

∂V

f dμVx0

= μVx0

(U \ K) < ε.

We now choose a sequence (fn) of continuous functions on ∂V such that 0 ≤ fn ≤ f

and fn ↗ f . Since V is regular, for every y ∈ U \ K and n ∈ N, we have

lim infx→y

h(x) ≥ limx→y

HVfn

(x) = fn(y).

By taking the supremum with respect to n at the last right-hand side, we get

lim infx→y

h(x) ≥ 1 for every y ∈ U \ K.

This completes the proof. ��Lemma 6.10.6. Let Ω ⊆ E be a relatively compact open set, and let x0 ∈ ∂Ω .Define f : ∂Ω → R, f (x) = sx0(x), where sx0 is the continuous subharmonicfunction of assumption (A∗). Define

sΩx0

:= HΩf .

Then:

(i) sΩx0

is harmonic in Ω ,(ii) infΩ\U sΩ

x0> 0 for every neighborhood U of x0,

(iii) limΩ�x→x0 sΩx0

(x) = 0 if x0 is H-regular for Ω ,(iv) limΩ�x→x0 sΩ

x0(x) = 0 if there exists a barrier function for Ω at x0.

6.10 S∗-harmonic Spaces: Bouligand’s Theorem 373

Proof. (i) This is obvious, since f is continuous, hence resolutive.(ii) Since sx0 |Ω ∈ UΩ

f , one has sx0 |Ω ≤ HΩf . Thus, the assertion follows from

the positivity property of sx0 .(iii) Since sΩ

x0= HΩ

f , if x0 is H-regular for Ω , then one has

limx→x0

sΩx0

(x) = f (x0) = 0.

(iv) Suppose that there exists a barrier function b at x0. Then there exists a neigh-borhood B of x0 such that b ∈ S(Ω ∩ B) and b(x) → 0 as x → x0.

Let w ∈ S+c (E) be such that w ≥ 1 in Ω (see Proposition 6.8.2). The maximum

principle of Ex. 6.8.5 implies the existence of a positive constant M such that

0 ≤ sΩx0

≤ M in Ω.

Let ε > 0 be fixed. Choose a regular neighborhood V of x0 such that

V ⊆ B and supV ∩∂Ω

f ≤ ε.

Let K ⊆ Ω ∩ ∂V be compact, and let h ∈ H(V ) be such that h ≥ 0,

h(x0) ≤ ε

Mand lim inf

x→yh(x) ≥ 1 ∀y ∈ Ω ∩ ∂V \ K.

(see the previous lemma). Since b > 0 in B ∩ Ω and B ⊇ V , we have m :=infK b > 0. For a given function u ∈ UΩ

f , define

u0 := u − εw − M

(b

m+ h

)

in Ω ∩V . Then u0 ∈ S(Ω), and lim sup∂(Ω∩V ) u0 ≤ 0. Indeed, since u ≤ HΩf ≤ M ,

we have

y ∈ K ⇒ lim supx→y

u0(x) ≤ supΩ

u − M

minfK

b ≤ M − M = 0, (6.25a)

y ∈ Ω ∩ ∂V \ K ⇒lim sup

x→yu0(x) ≤ sup

Ω

u − M lim infx→y

h(x) ≤ M − M = 0, (6.25b)

y ∈ V ∩ ∂Ω ⇒lim sup

x→yu0(x) ≤ lim sup

x→yu(x) − εw(y) ≤ f (y) − ε ≤ 0. (6.25c)

The maximum principle for subharmonic functions implies u0 ≤ 0 in Ω ∩V , that is,

u ≤ εw + M

(b

m+ h

)in Ω ∩ V.

This inequality holds for every u ∈ UΩ

f , so that


sΩx0

≡ HΩf ≤ εw + M

(b

m+ h

)in Ω ∩ V.

Then, since b(x) → 0 as x → x0, and Mh(x0) ≤ ε,

0 ≤ lim supx→x0

sΩx0

(x) ≤ εw(x0) + ε for every ε > 0.

This completes the proof. ��We are now ready to prove Bouligand’s theorem.

Proof (of Theorem 6.10.4). If x0 is H-regular for Ω , then sΩx0

is a barrier function forΩ at x0, see Lemma 6.10.6-(i)–(iii). Vice versa, suppose that there exists a barrierfunction for Ω at x0. Let us prove that

limΩ�x→x0

HΩϕ (x) = ϕ(x0) (6.26)

for every ϕ ∈ C(∂Ω, R). Then sΩx0

has properties (i)–(ii) and (iv) in Lemma 6.10.6.Let w ∈ S+

c (Ω) be such that infΩ w ≥ 1 (see Proposition 6.8.2), and let V be aneighborhood of x0. Define

ωϕ(V ) := sup{|ϕ(z)w(x0) − ϕ(x0)w(z)| : z ∈ V ∩ ∂Ω

},

Mϕ := sup{|ϕ(z)w(x0) − ϕ(x0)w(z)| : z ∈ ∂Ω

},

m := inf{sΩx0

(x) : x ∈ Ω \ V},

and

u :=(

ϕ(x0)

w(x0)+ ωϕ(V )

)w + Mϕ

sΩx0

m.

Then u ∈ S(Ω). Moreover,

lim infΩ�x→z

u(x) ≥ ϕ(z) ∀ z ∈ ∂Ω.

Indeed, if z ∈ V ∩ ∂Ω , the very definition of ωϕ(V ) and the inequality w ≥ 1 in Ω

imply

lim infx→z

u(x) ≥(

ϕ(x0)

w(x0)+ ωϕ(V )

)w(z) ≥ ϕ(z).

On the other hand, if z ∈ ∂Ω \ V , keeping in mind the definitions of m and Mϕ , onehas

lim infx→z

u(x) ≥ ϕ(x0)

w(x0)w(z) + Mϕ ≥ ϕ(z).

We also have infΩ u > −∞. Thus, u ∈ UΩ

ϕ . Hence HΩϕ ≤ u and

lim supΩ�x→x0

HΩϕ (x) ≤ lim

Ω�x→x0u(x) = ϕ(x0) + ωϕ(V )w(x0),

6.11 Reduced Functions and Balayage 375

since sΩx0

(x) → 0 as x → x0. By taking the infimum with respect to V at the lastright-hand side, we get

lim supΩ�x→x0

HΩϕ (x) ≤ ϕ(x0) (6.27)

for every ϕ ∈ C(∂Ω, R). As a consequence,

lim infx→x0

HΩϕ (x) = − lim sup

Ω�x→x0

HΩ−ϕ(x) ≥ ϕ(x0).

This inequality, together with (6.27), implies (6.26). ��Exercise 6.10.7 (S. Zaremba’s counterexample). Let x0 ∈ Ω . Consider the setΩ = Ω \ {x0}. Assume there exists u ∈ S+(Ω) such that

u(x0) = ∞, 0 ≤ u(x) < ∞ ∀ x ∈ Ω.

Define

f : ∂Ω → R, f (x) ={

0 if x ∈ ∂Ω ,

1 if x = x0.

Then HΩf ≡ 0. In particular, x0 is not H-regular for Ω . (Hint: the function ε u

belongs to U Ω

f for all ε > 0.)

6.11 Reduced Functions and Balayage

The notions of reduced function and balayage are crucial in the potential theory.Before giving the basic definitions, we need a general preliminary result.

Theorem 6.11.1 (Envelopes in S(Ω)). Let (E,H) be a harmonic space and letΩ ⊆ E be open. Given F ⊆ S(Ω), assume that

u0 := infF

is locally bounded from below. Then

u0 ∈ S(Ω).

Remark 6.11.2. In the setting of the harmonic spaces related to the sub-Laplacians,the statement of Theorem 6.11.1 can be strengthened (see Theorem 9.5.6, page 449).

Proof. The function u0 is l.s.c. in Ω and > −∞ at any point. Moreover, for a givenH-regular set V ⊆ V ⊆ Ω , we have

∫

∂V

u0 dμVx ≤

∫

∂V

u dμVx ≤ u(x) (6.28)

at any point x ∈ V for every u ∈ F . Then

376 6 Abstract Harmonic Spaces∫

∂V

u0 dμVx ≤ u0(x) ∀ x ∈ V. (6.29)

On the other hand, since u ∈ S(Ω), x → ∫∂V

u dμVx is H-harmonic in Ω , and the

first inequality in (6.28) implies that also

x →∫

∂V

u0 dμVx

is H-harmonic, hence continuous, in V (see Exercise 6.3.9). Therefore, from inequal-ity (6.29) we obtain ∫

∂V

u0 dμVx ≤ u0(x),

so that u0 ∈ S(Ω). ��Exercise 6.11.3. Let F ⊆ H∗(Ω), for an open set Ω ⊆ E. Assume that

u0 := infF

is locally bounded from below. Then u0 ∈ H∗(Ω).(Hint: Apply the previous theorem to the families

Fn := {min{u, n} : u ∈ F}, n ∈ N.

Then use Theorem 6.3.8.)

Definition 6.11.4 (H-reduced function and H-balayage). Let f : E → [0,∞] bea non-negative real extended function, and let A ⊆ E. Define

RfA := inf

{v ∈ H∗(E) : v ≥ 0 in E, v ≥ f in A

}

and

RfA :=

(RfA).

RfA and Rf

A are called, respectively, the H-reduced function and the regularized H-reduced function of f on A.

RfA is also called the H-balayage of f on A.

We, obviously, have0 ≤ Rf

A ≤ RfA.

Moreover, by Exercise 6.11.3,

RfA ∈ H∗(E).

If the function f ∈ H∗(E), f ≥ 0, from the very definition of reduced function weget Rf

A ≤ f . So that, in this case,

0 ≤ RfA ≤ Rf

A ≤ f,

and RfA is H-superharmonic if f is H-superharmonic. Moreover,

RfA(x) = Rf

A(x) = f (x) for every x ∈ Int(A). (6.30)

6.11 Reduced Functions and Balayage 377

Exercise 6.11.5. Prove the claimed (6.30). Prove also that if f ∈ S(E), f ≥ 0, then

RfE = inf

{v ∈ S(E) : v ≥ 0 in E, v ≥ f in A

},

and RfE = f in A. ��

An application of the fundamental theorem on Perron families gives the follow-ing result.

Theorem 6.11.6 (Properties of H-reduction and H-balayage. I). Let (E,H) be aharmonic space. Let f ∈ S(E), f ≥ 0, and let A ⊆ E. Then

(i) RfA is H-superharmonic in E,

(ii) RfA = f in A and Rf

A = f in int(A),

(iii) RfA and Rf

A are equal and H-harmonic in Ω := E \ A.

An improvement of (iii) will be given in the setting of sub-Laplacians.

Proof. We already know that (i) and (ii) hold true.To prove (iii), consider the family

F := {v|Ω : v ∈ S(E), v ≥ 0 in E, v ≥ f in A

}.

It is easy to see that F is a Perron family in Ω and that f and g ≡ 0 are, respec-tively, an H-superharmonic majorant and an H-subharmonic minorant of F . Then,by Theorem 6.5.8,

(RfA)|Ω = infF ∈ H(Ω).

In particular, RfA is continuous in Ω , hence

RfA = Rf

A in Ω.

Then (iii) follows. ��The following theorem states some properties of the reduced function and the

balayage. Their easy proofs are left as an exercise.

Theorem 6.11.7 (Properties of H-reduction and H-balayage. II). Let (E,H) bea harmonic space, and let A,B ⊆ E and f, g ∈ S(E), f, g ≥ 0. Then

RfA ≤ Rf

B if A ⊆ B, (6.31a)

RfA ≤ Rg

A if f ≤ g, (6.31b)

RλfA = λRf

A if λ ≥ 0. (6.31c)

The same assertions hold for the operator R.


Some other crucial properties of the reduced function and the balayage will beproved in the context of the harmonic spaces associated to the sub-Laplacians. Theuse of the average operators Mr and Mr will make the proofs much easier than inthe general abstract setting.

Bibliographical Notes. In this chapter, we presented some topics from the theoryof abstract harmonic spaces mainly inspired by the ones developed by H. Bauer in[Bau66] and C. Costantinescu and A. Cornea in [CC72]. See also J.L. Doob [Doo01].


Unlike the other chapters, several exercises of Chapter 6 are distributed throughoutthe text. For the reading convenience, we gather below the list of these exercises.

Ex. 1) Prove property (P1) in Section 6.1.Ex. 2) Prove property (P2) in Section 6.1 by recognizing that u : A → [−∞,∞]

is l.s.c. if and only if it is continuous from the topological space (A, T |A)

and ([−∞,∞], I), where I is the topology on [−∞,∞] generated by theintervals ]t,∞], t ∈ R.

Ex. 3) Prove property (P3) in Section 6.1.(Hint: Use (P2).)

Ex. 4) Prove property (P4) in Section 6.1.(Hint: Use Ex. 2, first recognizing that K ⊆ ]−∞,∞] is compact w.r.t. thetopology I if and only if K has real minimum.)

Ex. 5) Prove property (P5) in Section 6.1.(Hint: For every a ∈ A and ε > 0, there exists Va,ε ∈ Ua such that u(a) −ε < u(x) for every x ∈ Va,ε. Let m := minA u. Urysohn’s lemma impliesthe existence of a continuous function fa,ε : A → R such that, m − ε ≤f ≤ u(a) − ε, f (a) = u(a) − ε and f (x) = m − ε for every x ∈ A \ Va,ε.Since {fa,ε : a ∈ A, ε > 0} ⊆ F := {f ∈ C(A, R) : f ≤ u}, one hassupf ∈F f = u. To conclude the proof, use Proposition 6.1.1.)

Ex. 6) Solve Ex. 6.3.4, page 346, and prove Proposition 6.2.6, page 342.Ex. 7) Solve Ex. 6.2.5, page 342.Ex. 8) Solve Exercises 6.3.9 and 6.3.1, page 348.Ex. 9) Solve Ex. 6.4.11, page 353.

Ex. 10) Prove Proposition 6.5.4, page 355.Ex. 11) Prove Proposition 6.7.4, page 360.Ex. 12) Solve Exercises 6.8.5, 6.8.6, 6.8.7, page 365.Ex. 13) Solve Ex. 6.10.7, page 375.Ex. 14) Solve Ex. 6.11.3, page 376.


Ex. 15) Solve Ex. 6.11.5, page 377.Ex. 16) Prove Theorem 6.11.7.Ex. 17) Prove directly that if f is any function, then f is l.s.c. Moreover, if g is l.s.c.

and g ≤ f , then g ≤ f .Ex. 18) Let Ω be a relatively compact open set in a harmonic space (E,H). Let

u ∈ S(U), where U is an open set containing Ω . Show that the function

x →∫

∂Ω

u dμΩx =: h(x)

is H-harmonic in Ω .(Hint: F = {HΩ

ϕ : ϕ ∈ C(∂Ω, R), ϕ ≤ u|∂Ω} is an up directed family ofharmonic functions such that supF = h and supF ≤ u|Ω .)

Ex. 19) Let (E,H) be a S∗-harmonic space, and let Ω ⊆ E be open and relativelycompact. Let z ∈ ∂Ω , and let U be an open neighborhood of z. Prove that

z is H-regular for Ω ⇔ z is H-regular for Ω ∩ U.

Ex. 20) Let (E,H) be a S∗-harmonic space, and let Ω1 ⊆ Ω2 ⊆ E be open andrelatively compact. Let z ∈ ∂Ω1 ∩ ∂Ω2. Assume z is H-regular for Ω2.Prove that z is H-regular for Ω1.

Ex. 21) Let Ω be a relatively compact open set in a S-harmonic space (E,H). Let(fn) be a sequence of resolutive functions from ∂Ω into [−∞,∞]. Supposethat

supn

HΩ

|fn|(x) < ∞ for every x ∈ Ω.

Assume (fn) is pointwise convergent to f : ∂Ω → [−∞,∞]. Prove thatf is resolutive.

Ex. 22) Let Ω be a relatively compact open set in a S-harmonic space (E,H). Letf : ∂Ω → [−∞,∞[ be bounded above. Prove that if z ∈ ∂Ω is H-regular,one has

lim supΩ�x→z

HΩ

f (x) ≤ lim sup∂Ω�y→z

f (y).

(Hint: For every fixed β ∈ R, β > lim sup∂Ω�y→z f (y), choose a function

ϕ ∈ C(∂Ω, R) such that ϕ ≥ f and ϕ(z) = β. Then HΩ

f ≤ HΩ

ϕ = HΩϕ .)

Ex. 23) Let (E,H) be a harmonic space such that the constant functions are H-harmonic. Let ϕ : R → R be a convex function. Prove the following state-ments:(i) if Ω ⊆ E is open and u ∈ H(Ω), then ϕ(u) ∈ S(Ω),

(ii) if we also assume ϕ monotone increasing, then ϕ(u) ∈ S(Ω) wheneveru ∈ S(Ω).

(Hint: Since 1 ∈ H(E), μVx (∂V ) = 1 for every H-regular set V and every

x ∈ V . Use Jensen’s inequality.)Ex. 24) Let Ω1 and Ω be open subsets of E with Ω1 ⊆ Ω . Given u ∈ H∗(Ω) and

v ∈ H∗(Ω1) define


w : Ω →]−∞,∞], w ={

u in Ω \ Ω1,

min{u, v} in Ω1.

Prove that, if w is l.s.c., then w ∈ H∗(Ω). Moreover, w ∈ S(Ω) if u ∈S(Ω).

Ex. 25) Let Ω1 and Ω be open subsets of E with Ω1 ⊆ Ω . Given u ∈ S(Ω) andv ∈ S(Ω1) define w as in the previous exercise. Assume that

lim infΩ1�x→y

v(x) ≥ u(y) for every y ∈ Ω ∩ ∂Ω1.

Then w ∈ S(Ω1).Ex. 26) Let Ω be a relatively compact open set in a S∗-harmonic space (E,H) and

let x0 ∈ ∂Ω . Prove that x0 is H-regular for Ω iff x0 is H-regular for everyconnected component O of Ω such that x0 ∈ ∂O.

7

The L-harmonic Space

The aim of this short chapter is to prove that the theory of abstract harmonic spacesdeveloped in Chapter 6 can be applied to the setting of (homogeneous) Carnotgroups G. More precisely, fixed a sub-Laplacian L on G, the L-harmonic functionsprovide a harmonic sheaf LH on G. We shall prove that (G,LH) turns out to be aS∗-harmonic space.

The ingredients we use are the structure of the fundamental solution for L and theHarnack-type theorem established in Chapter 5. The more involved task is to showthe existence of a basis of the topology of R

N formed by L-regular sets, i.e. by setsfor which the Dirichlet problem for L is solvable in the usual classical sense.

For reading convenience, in the second part of the chapter, we recall somebasic definitions and results from the abstract theory of Chapter 6, just to showhow they read in the sub-Laplacian setting. Moreover, we prove a criterion of L-subharmonicity for C2-functions, we show that the gauge balls are L-regular setsand we write an explicit formula for the density of the L-harmonic measures of thegauge balls at their center.

Convention. The same convention as in the incipit of Chapter 5 applies. In otherwords, the potential theory developed in this chapter can be straightforwardly gen-eralized and rephrased in the context of the abstract stratified groups. It suffices toconsider the notion of L-harmonic function on a stratified group in Definition 2.2.27on page 146 and to keep in mind Remark 2.2.28. That remark ensures that, if H is anabstract stratified group, G is a “homogeneous copy” of H and Ψ : G → H the Liegroup isomorphism in Remark 2.2.26 (page 145) the harmonic (hence the hyperhar-monic, subharmonic, etc.) functions in H simply “pull back” via Ψ to the harmonic(hyperharmonic, subharmonic, etc.) functions in G.

7.1 The L-harmonic Space

Let G = (RN, ◦, δλ) be a homogeneous Carnot group in RN and let L be one of its

sub-Laplacians. We know that

382 7 The L-harmonic Space

L = div(A(x)∇) =N∑

i,j=1

ai,j (x) ∂xi ,xj+

N∑

i=1

bi(x) ∂xi,

where A(x) = (ai,j )i,j≤N is a symmetric non-negative matrix with smooth entriesand constant a1,1 > 0.

For every open set Ω ⊆ G denote

LH(Ω) := {u ∈ C∞(Ω, R) : Lu = 0

}.

A function u ∈ LH(Ω) will be called L-harmonic in Ω . The map

LH : Ω → LH(Ω)

is a harmonic sheaf. The aim of this section is to prove that

(RN,LH) is a S∗-harmonic space.

We split the proof in four steps, corresponding to the axioms we have to verify.

(A1) (Positivity). This axiom is trivially satisfied for the constant functions are L-harmonic.

(A2) (Convergence). This readily follows from the Harnack theorem proved in The-orem 5.7.10, page 268.

We postpone to the end of the section the proof of (A3), which is our main task.Now we proceed with (A4) and (A∗).

(A4) (Separation). We shall show that S+c (G), the family of continuous non-

negative LH-superharmonic functions, separates the points of G. Since Lis left translation invariant and the constant functions are L-harmonic, it isenough to prove the following statement:

for every x0 �= 0, there exists u ∈ S+c (G) : u(x0) �= u(0).

Let Γ be the fundamental solution of L with pole at the origin. Since Γ isL-harmonic in G \ {0} and

limx→0

Γ (x) = ∞ =: Γ (0),

one has Γ ∈ S(G) (see Exercise 7.3). For k ∈ N, define

Γk = min{Γ, k}.Obviously, Γk ∈ S+

c (G). Moreover, Γk(0) ↗ ∞ as k ↗ ∞ while, as x0 �= 0,Γk(x0) ↗ Γ (x0) ∈ ]0,∞[. Then there exists k ∈ N such that

Γk(x0) �= Γk(0),

and we are done.

7.1 The L-harmonic Space 383

(A∗) (The function sx0 ). For every fixed x0 ∈ G, let us define

sx0(x) = d(x−10 ◦ x), x ∈ G,

where d is an L-gauge function (see Definition 5.4.1). Then sx0 is continuousin G, sx0(x0) = 0 and

infG\V sx0 > 0

for every neighborhood V of x0. Moreover, in G \ {x0},

Lsx0 = (Q − 1)

d|∇Ld|2 ≥ 0

(see (5.34)). Then sx0 ∈ S(G \ {x0}). Since sx0 ≥ 0 and sx0(x0) = 0, it followsthat sx0 ∈ S(G) (see Ex. 7.3.4).

(A3) (Regularity). This is the most difficult part of our task. We shall use someresults from the theory of linear second order partial differential equations ofelliptic type.

From now on, for a fixed ε > 0, we shall denote

Lε := L + εΔ, (7.1)

where Δ = ∑Nj=1 ∂2

xjis the Laplace operator.

The characteristic form of Lε

qLε(x, ξ) := 〈A(x)ξ, ξ 〉 + ε |ξ |2, ξ ∈ R

N,

is strictly positive definite at any point of G. Moreover, for every compact set K ⊆ G

there exists λ = λ(K) > 0 such that

λ−1 |ξ |2 ≤ qLε(x, ξ) ≤ λ|ξ |2 ∀ x ∈ K, ξ ∈ R

N.

This means that Lε is uniformly elliptic on every bounded open set Ω , so that if ∂Ω

also satisfies some mild regularity condition, the Dirichlet problem{Lεu = 0 in Ω,

u|∂Ω = ϕ, ϕ ∈ C(∂Ω, R),(7.2)

has a solution1

uε ∈ C∞(Ω) ∩ C(Ω)

such that uε|∂Ω = ϕ.A condition ensuring this existence result is the following one2:

(R) For every x0 ∈ ∂Ω , there exists a function h(x0, ·) of class C2 in an open setcontaining Ω such that:

1 Lε is hypoelliptic for it is elliptic and its coefficients are smooth. Then u ∈ C∞ if Lεu = 0.2 See the monograph by D. Gilbarg and N.S. Trudinger [GT77, Chapter 6].


(a) h(x0, ·) > 0 in Ω \ {x0}, h(x0, x0) = 0.(b) Lεh(x0, ·) ≤ 0 in Ω .

A geometric condition ensuring (R) is a particular form of the Poincaré exteriorball condition.

Definition 7.1.1 (Non-characteristic exterior ball condition). We say that ∂Ω sat-isfies the non-characteristic exterior ball condition at a point x0 ∈ ∂Ω if there existsz ∈ G \ Ω such that

(i) D(z, r) ∩ Ω = {x0}, r = |x0 − z|,(ii) qL(x, z − x) := 〈A(x)(z − x), z − x〉 > 0 ∀ x ∈ Ω .

Here D(z, r) denotes the standard Euclidean ball with center z and radius r . Thestandard Euclidean norm is denoted by | · |.Note 7.1.2. If (i) and (ii) hold, then the boundary of Ω is non-characteristic at x0,since ν := z − x0 is orthogonal to ∂Ω at x0 and

qL(x0, ν) = ⟨A(x0) ν, ν

⟩> 0.

Lemma 7.1.3. Let Ω ⊆ G be a bonded open set satisfying the non-characteristicexterior ball condition at a point x0 ∈ ∂Ω . Define

h(x0, x) = exp(−μ r2) − exp(−μ |z − x|2), x ∈ G,

where

μ = maxx∈Ω

{ |〈b(x), z − x〉|qL(x, z − x)

: x ∈ Ω

}

(as before, | · | stands for the usual Euclidean norm). Then h ∈ C∞(G, R) andsatisfies (a) and (b) of condition (R).

Proof. The smoothness of h is obvious. Condition (R)-(a) readily follows from (i) inDefinition 7.1.1. Moreover, by (ii) in Definition 7.1.1 and the very definition of μ,Lεh(x0, x) equals

− exp(−μ |x − x0|2)(4μ2qL(x, z − x) + 2μ〈b(x), z − x〉 + 4εN),

which is non-negative in Ω . ��We now prove another crucial lemma for our purposes.

Lemma 7.1.4. Let Ω and h be as in Lemma 7.1.3, and let uε be the solution of theDirichlet problem (7.2). Then

|uε(x) − ϕ(x0)| ≤ 2 hϕ(x0, x) ∀ x ∈ Ω. (7.3)


The function hϕ is defined as follows. For t > 0, we let

ωϕ(x0, t) := t + sup{|ϕ(x) − ϕ(x0)| : x ∈ ∂Ω, |x − x0| ≤ t},and

m(x0, t) := inf{h(x0, x) : |x − x0| ≥ t, |x − z| ≥ r}.The function t → ωϕ(x0, t)m(x0, t) is continuous and strictly increasing. Denote byTϕ(x0, ·) its inverse function. We define

hϕ(x0, x) := Tϕ

(x0, sup

∂Ω

|ϕ|h(x0, x)).

It should be noted that hϕ is independent of ε. Moreover,

limx→x0

hϕ(x, x0) = 0.

Proof (of Lemma 7.1.4). Let t > 0 be fixed. Consider the function

w(x) := ϕ(x0) + ωϕ(x0, t) + sup∂Ω

|ϕ|h(x0, x)

m(x0, t).

One readily verifies that

limx→y

w(x) ≥ ϕ(y) = limx→y

uε(x) ∀ y ∈ ∂Ω.

Then, since Lew ≤ 0, by the maximum principle we get uε ≤ w in Ω , that is

uε(x) − ϕ(x0) ≤ ωϕ(x0, t) + sup∂Ω

|ϕ| h(x0, x)

m(x0, t)

for every x ∈ Ω and t > 0. The same inequality holds by replacing in it uε with −uε

and ϕ with −ϕ. Then

|u(x) − ϕ(x0)| ≤ ωϕ(x0, t) + sup∂Ω

|ϕ| h(x0, x)

m(t), x ∈ Ω, (7.4)

for every t > 0. Now we choose t > 0 such that

ω(x0, t) = sup∂Ω

|ϕ|h(x0, x)

m(t),

that is t = T (x0, sup∂Ω |ϕ|h(x0, x)). With this choice of t , from (7.4) we ob-tain (7.3). This completes the proof. ��

From Lemma 7.1.3 and Lemma 7.1.4 we obtain a resolutivity result for L.

Proposition 7.1.5. Let Ω ⊆ G be a bounded open set satisfying the non-characteris-tic exterior ball condition at any point of ∂Ω . Then Ω is LH-regular, that is: for everyϕ ∈ C(∂Ω, R) there exists a unique function u ∈ C∞(Ω, R) such that

{Lu = 0 in Ω,

limx→y u(x) = ϕ(y) ∀ y ∈ ∂Ω.


Moreover, u ≥ 0 if ϕ ≥ 0.

Proof. First of all we prove the existence. Let ϕ ∈ C(∂Ω, R) and let uε, 0 < ε < 1,be the solution of the Dirichlet problem (7.2). By the maximum principle

supΩ

|uε| ≤ sup∂Ω

|ϕ|,

so that

sup0<ε<1

∫

Ω

|uε|2 dx < ∞.

Then there exists u ∈ L2(Ω) and a sequence εj ↘ 0 such that

limj→∞ uεj

= u

in the weak topology of L2(Ω). As a consequence, for every test function ψ ∈C∞

0 (Ω),

∫

Ω

uL∗ψ dx = limj→∞

∫

Ω

uεjL∗ψ dx

= limj→∞

(∫

Ω

uεjL∗

εψ dx − εj

∫

Ω

uεjΔψ dx

)

= limj→∞

∫

Ω

ψLεuεjdx = 0.

Therefore Lu = 0 in the weak sense of distributions. Since L is hypoelliptic, wemay assume

u ∈ C∞(Ω, R) and Lu = 0 in Ω.

On the other hand, by Lemma 7.1.4, for every x0 ∈ ∂Ω we have

ϕ(x0) − 2 hϕ(x0, x) ≤ uεj(x) ≤ ϕ(x0) + 2 hϕ(x0, x) ∀ x ∈ ∂Ω (7.5)

and for every j ∈ N. Then, since the weak limits in L2(Ω) of non-negative functionsare non-negative, inequalities (7.5) imply

|uε(x) − ϕ(x0)| ≤ hϕ(x0, x) ∀ x ∈ Ω, x0 ∈ ∂Ω.

Then, keeping in mind that hϕ(x0, x) → 0 as x → x0,

limx→x0

u(x) = ϕ(x0) ∀ x0 ∈ ∂Ω.

This completes the proof of the existence. The uniqueness and the positivity ofu if ϕ ≥ 0 straightforwardly follow from the weak maximum principle of Theo-rem 5.13.4 (page 295). ��


Finally, we are ready to complete the proof of (A3). We will construct a ba-sis (Vλ)λ>0 of L-regular neighborhoods of the origin. It will follow that, for everyx0 ∈ G,

{x0 ◦ Vλ}λ>0

is a basis of L-regular open sets containing x0.Let e1 = (1, 0, . . . , 0) be the first element of the canonical basis of G. Then

qL(0, e1) = 〈A(0) e1, e1〉 = a1,1 > 0.

As a consequence, there exists ρ > 0 such that qL(x, ν) > 0 for every x, ν ∈ G,|x| < ρ, |ν − e1| < ρ.

Therefore, for R > 0 sufficiently large and ε > 0 sufficiently small,

V1 := D(R e1, R + ε) ∩ D(−R e1, R + ε) (7.6)

satisfies the non-characteristic exterior ball condition at any point of its boundary.(See also Fig. 7.1.)

Fig. 7.1. Figure of (7.6)

Then, by Proposition 7.1.5, V1 is LH-regular. Since L commutes with the dila-tions δλ, λ > 0, if we put

Vλ := δλ ◦ V,

the family {Vλ}λ>0 is a basis of LH-regular neighborhoods of the origin. This com-pletes the proof of (A3). ��


7.2 Some Basic Definitions and Selecta of Properties

Throughout this section, Ω will be an open subset of a homogeneous Carnotgroup G.

Notation. For the sake of brevity, we shall simply denote by H the harmonic sheafLH introduced in the previous section.

In Section 7.1, we have shown that

(G,H)

is a S∗-harmonic space. Hence the abstract potential theory developed throughoutChapter 6 applies to the present setting. Some of the basic definitions and resultsfrom that theory will be recalled along this section, for reading convenience.

A bounded open set V ⊂ G will be called L-regular if the boundary value prob-lem {

Lu = 0 in V ,

u|∂V = ϕ(7.7)

has a (unique) solution u := HVϕ for every continuous function ϕ : ∂V → R. We

say that u solves (7.7) if u is L-harmonic in V and

limx→y

u(x) = ϕ(y) ∀ y ∈ ∂V .

The weak maximum principle of Theorem 5.13.4 implies the uniqueness of the so-lution to the boundary value problem (7.7). Moreover,

HVϕ ≥ 0 in V whenever ϕ ≥ 0 on ∂V .

Then, if V is L-regular, for every fixed x ∈ V the map

C(∂V, R) � ϕ → HVϕ (x) ∈ R

defines a linear positive functional on C(∂V, R). As a consequence (by the classicalRiesz representation theorem for linear positive functionals), there exists a Radonmeasure μV

x supported in ∂V such that

HVϕ (x) =

∫

∂V

ϕ(y) dμVx (y) ∀ ϕ ∈ C(∂V, R).

We shall call μVx the L-harmonic measure related to V and x.

The family of L-regular sets is not empty. Indeed, in Section 7.1 we have provedthe following proposition.

Proposition 7.2.1. The family of the L-regular open sets is a basis of the Euclideantopology of G.

The following definition of L-subharmonic function is equivalent to the abstractone given in Definition 6.5.1 (page 353), in force of Theorem 6.5.2 and the definitionof H-hyperharmonic function given in Definition 6.2.4 (page 341).

7.2 Some Basic Definitions and Selecta of Properties 389

Definition 7.2.2 (L-subharmonic function). Let Ω be an open subset of G. A func-tion u : Ω → [−∞,∞[ will be called L-subharmonic in Ω if:

(i) u is upper semi-continuous and u > −∞ in a dense subset of Ω ,(ii) for every L-regular open set V with closure V ⊂ Ω and for every x ∈ V ,

u(x) ≤∫

∂V

u(y) dμVx (y). (7.8)

The family of the L-subharmonic functions in Ω will be denoted by

S(Ω).

Remark 7.2.3. As we have already noticed in the abstract harmonic spaces (see Sec-tion 6.2.1), condition (ii) in Definition 7.2.2 is equivalent to the following one:

(ii)’ HVϕ ≥ u|V ∀ ϕ ∈ C(∂V, R): ϕ ≥ u|∂V ,

for every L-regular open set V such that V ⊂ Ω .

We shall denote by S(Ω) the family of the L-superharmonic functions, i.e. theset of functions u : Ω →]−∞,∞] such that −u ∈ S(Ω). Then, keeping in mindthe previous definition, u ∈ S(Ω) if and only if

(i) u is lower semi-continuous and u < ∞ in a dense subset of Ω ,(ii) for every L-regular open set V with closure V ⊂ Ω and for every x ∈ V ,

u(x) ≥∫

∂V

u(y) dμVx (y). (7.9)

Proposition 7.2.4. A function u : Ω → R is L-harmonic in Ω if and only if u ∈S(Ω) ∩ S(Ω). In other words,

H(Ω) = S(Ω) ∩ S(Ω).

Proof. See Exercise 6.3.4 on page 346. ��By using the weak maximum principle of Theorem 5.13.4 (page 295), we obtain

the following criterion of L-subharmonicity for functions of class C2.

Proposition 7.2.5 (Smooth L-subharmonic functions). Let u be a function inC2(Ω, R). Then u is L-subharmonic if and only if

Lu ≥ 0 in Ω.

Proof. Let V ⊂ V ⊂ Ω be an L-regular open set and let ϕ ∈ C(∂V, R), ϕ ≥ u. Onehas

390 7 The L-harmonic Space{L(u − HV

ϕ ) ≥ 0 in V ,

lim supx→y(u(x) − HVϕ (x)) ≤ u(y) − ϕ(y) ≤ 0 for every y ∈ ∂V .

Then, by the maximum principle, u − HVϕ ≤ 0 in V , so that u is L-subharmonic.

Vice versa, let u ∈ S(Ω) and assume, by contradiction, Lu(x0) < 0 at some pointx0 ∈ Ω . It follows that Lu < 0 in a neighborhood Ω0 of x0. Then, by the firstpart of the proof, u ∈ S(Ω0). On the other hand, u ∈ S(Ω0) since u ∈ S(Ω). ByProposition 7.2.4, it follows that u ∈ H(Ω0), i.e. Lu = 0 in Ω0, a contradiction. ��Remark 7.2.6. The previous proof can be easily adapted to prove the following state-ment. Let Ω ⊆ G be open and let x0 ∈ Ω . Let u ∈ C2(Ω \ {x0}, R) be such that

Lu ≥ 0 in Ω \ {x0}, lim infx→x0

u(x) = −∞.

Then, if we extend u at x0 letting u(x0) := −∞, we have u ∈ S(Ω).Indeed, let V ⊂ V ⊂ Ω be a L-regular open set and let ϕ ∈ C(∂V, R), ϕ ≥ u|∂V .

Proceeding as in the first part of the previous proof, we see that

u|V ≤ (HVϕ )|V if x0 /∈ V

andu|V \{x0} ≤ (HV

ϕ )|V \{x0} if x0 ∈ V .

Then, in any case, u|V ≤ (HVϕ )|V , so that u ∈ S(Ω). ��

Example 7.2.7. Let d be an L-gauge and let x0 ∈ G be fixed. Define

γx0(x) :={

(d(x−10 ◦ x))2−Q if x �= x0,

∞ if x = x0.

Then, γx0 is L-superharmonic in G. Indeed, since d is an L-gauge, γx0 is L-harmonic(hence, L-superharmonic) in G \ {x0}. ��

Let Ω ⊂ G be a bounded open set. By the Wiener resolutivity Theorem 6.8.4in S-harmonic spaces (page 364), we know that any function ϕ ∈ C(∂Ω, R) isresolutive. Then there exists the Perron–Wiener–Brelot generalized solution HΩ

ϕ ∈H(Ω) to the boundary value problem

{Lu = 0 in Ω,

u|∂Ω = ϕ.(7.10)

As it is well-known, even in the classical case of Laplace operator, we cannot expectthat

limx→y

HΩϕ (x) = ϕ(y) (7.11)

for every y ∈ ∂Ω . If (7.11) holds for every ϕ ∈ C(∂Ω, R), we say that y is aL-regular point of ∂Ω .

7.2 Some Basic Definitions and Selecta of Properties 391

Obviously, problem (7.10) is solvable for every continuous boundary datum ϕ

(i.e. Ω is an L-regular set) if and only if every point of ∂Ω is L-regular. Bouligand’sTheorem 6.10.4 in S∗-harmonic spaces (page 371) implies that

a point y ∈ ∂Ω is L-regular if there exists an L-barrier function at y in Ω ,i.e. a function w ∈ S(Ω) such that

w(x) > 0 for every x ∈ Ω and w(x) → 0 as x → y.

By using this result, one easily proves the following proposition.

Proposition 7.2.8 (The d-balls are L-regular). Let d be an L-gauge. Then, the d-balls

Bd(x0, r), x0 ∈ G, r > 0

are L-regular open sets.

Proof. The function

v(x) :={

(d(x−10 ◦ x))2−Q − r2−Q if x �= x0,

∞ if x = x0

is L-superharmonic in Bd(x0, r) (see Example 7.2.7). Moreover, v > 0 in Bd(x0, r)

and v(x) → 0 as x → y for every y ∈ ∂Bd(x0, r). Then v is an L-barrier function forBd(x0, r) at any point of its boundary. Bouligand’s Theorem implies that Bd(x0, r)

is L-regular. ��The surface mean value formula (5.45) in Theorem 5.5.4 (page 256) provides an

explicit “density” of the L-harmonic measures related to Bd(x0, r) and x0:

Theorem 7.2.9 (Density). Let d be an L-gauge. Given x0 ∈ G and r > 0, we have

dμBd(x0,r)x0

(y) = βd (Q − 2)

rQ−1KL(x0, y) · dσ(y), (7.12)

where

KL = |∇Ld|2(x−1 ◦ y)

|∇(d(x−1 ◦ ·))|(y)

and dσ denotes the Hausdorff (N − 1)-dimensional measure on ∂Bd(x0, r).

Proof. Let 0 < ρ < r and ϕ ∈ C(∂Bd(x0, r), R). Since

h := HBd(x0,r)ϕ

is L-harmonic in Bd(x0, r), by using the surface mean value formula (5.45) (page 256),we have ∫

∂Bd(x0,r)

ϕ(y) dμBd(x0,r)x0

(y) = h(x0) = Mρ(h)(x0).


On the other hand, since h(y) → ϕ(z) as y → z for every z ∈ ∂Bd(x0, r),

limρ→r

Mρ(h)(x0) = Mr (h)(x0)

= βd (Q − 2)

rQ

∫

∂Bd(x0,r)

ϕ(y)KL(x, y) dσ(y).

Then∫

∂Bd(x0,r)

ϕ(y) dμBd(x0,r)x0

(y) = βd (Q − 2)

rQ

∫

∂Bd(x0,r)

ϕ(y)KL(x0, y) dσ(y)

for every ϕ ∈ C(∂Bd(x0, r), R) and the assertion follows. ��We refer the interested reader to the papers [UL97,UL02] for some estimates of

L-Poisson kernels on more general domains of Carnot groups.

Bibliographical Notes. In showing the existence of a basis of the topology of G

formed by L-regular sets, we followed an idea by J.-M. Bony [Bon69].For other presentations of potential theory, linear and non-linear and in sub-

Riemannian settings, see [BT03,Bir95,HKM93,Kil94,TW02b,VM96].For some topics in potential theory in an elliptic degenerate context and estimates

of the Poisson kernel, see [BAKS84,CGN02,HH87,UL97,UL02].


Ex. 1) Prove that Γ ∈ S(G).(Hint: Use Ex. 4.)

Ex. 2) Complete the proof of the following result, which is a consequence of Har-nack’s Theorem for sub-Laplacians.

Theorem 7.3.1. Let Ω ⊆ G be a bounded, open and connected set such that0 ∈ Ω . For all x ∈ Ω , there exists hx ∈ L1(∂Ω,μΩ

0 ) such that

dμΩx (ξ) = hx(ξ) dμΩ

0 (ξ).

Moreover, if K is a compact subset of Ω , there exists a positive constant c(K)

depending only on K such that

c(K)−1 ≤ ‖hx; L∞(∂Ω,μΩ0 )‖ ≤ c(K) for every x ∈ K .

Proof (Sketch). In this proof, for brevity, for any given x ∈ Ω we agree to setμx := μΩ

x . Let E ⊆ ∂Ω be closed. Then χE is u.s.c., and we have (why?)

μx(E) =∫

∂Ω

χE dμx = HΩχE

(x).


Now, HΩχE

is a non-negative L-harmonic function on the connected openset Ω , so that (why?) for every compact set K ⊂ Ω there exists c =c(K) > 0 such that

HΩχE

(x) ≤ cHΩχE

(y) for every x, y ∈ K.

Hence, μx(E) ≤ c μy(E) for every closed E ⊆ ∂Ω . Consequently, we ob-tain (why?)

c−1μy(E) ≤ μx(E) ≤ c μy(E) for every Borel set E ⊆ ∂Ω.

This proves that μy � μx � μy for every couple of points x, y ∈ Ω .Hence, by Lebesgue decomposition theorem, we infer the existence of hx,y ∈L1(∂Ω,μy) as in the assertion. We claim that

c−1 ≤ ‖hx,y(·); L∞(∂Ω,μy)‖ ≤ c.

Indeed, suppose first to the contrary that there exists a Borel set E ⊆ ∂Ω

such that μy(E) > 0 and hx,y(ξ) > c, μy-almost-everywhere on E. Thiswould give the contradiction

c≥μx(E)

μy(E)= 1

μy(E)

∫

E

hx,y(ξ) dμy(ξ) � c

∫E

dμy(ξ)

μy(E)= c.

Then take y = 0 and set hx := hx,0. ��Ex. 3) Let f ∈ Lp(∂Bd(0, 1), μ), 1 ≤ p < ∞, where

μ := μBd(0,1)0

is the L-harmonic measure of Bd(0, 1) at x0 = 0. Then:(i) f is resolutive,

(ii) letting u := HBd(0,1)f , we have

sup0≤λ≤1

∫

∂Bd(0,1)

|u(δλ(x))|p dμ(x) = ‖f ‖Lp(Bd(0,1)).

Ex. 4) Let Ω ⊆ G be open and let x0 ∈ Ω . Suppose we are given a function

u ∈ C(Ω, R) ∩ S(Ω \ {x0})such that u ≥ 0 and u(x0) = 0. Then

u ∈ S(Ω).

(Hint: For every y ∈ Ω , there exists a basis of regular neighborhoods of y

such that

u(y) ≤∫

∂V

u dμVy .

Then use Corollary 6.4.9.)


Ex. 5) (Asymptotic Koebe Theorem). Let Ω ⊆ G be open and let u ∈ C(Ω, R).Assume that

ALu = 0 in Ω or ALu = 0 in Ω.

Prove that u ∈ C∞(Ω, R) and Lu = 0 in Ω . AL and AL denote the asymp-totic L Laplacians introduced in Ex. 8 of Chapter 5.(Hint: Use the maximum principles of Ex. 10 of Chapter 5 to show thatu|V = HV

u|∂Vfor every L-regular open set V ⊆ V ⊆ Ω .)

Ex. 6) (M. Picone’s Maximum Principle). Consider the following differential op-erator

L =N∑

i,j=1

ai,j (x) ∂xi ,xj+

N∑

j=1

bj (x) ∂xj,

where ai,j = aj,i , bj are real functions in a bounded open subset Ω of G.Assume the following facts hold:(i) qL(x, ξ) := ∑N

i,j=1 ai,j (x)ξiξj ≥ 0 ∀ x ∈ Ω, ∀ ξ ∈ RN ,

(ii) there exists h ∈ C2(Ω, R) such that Lh < 0 and h > 0 in Ω .Then L satisfies the following weak maximum principle:if u ∈ C2(Ω, R) verifies

{Lu ≤ 0 in Ω,

lim supx→y u(x) ≤ 0 ∀ y ∈ ∂Ω

then u ≤ 0 in Ω .(Hint: See the proof of Theorem 5.13.4.)

Ex. 7) Let L be the differential operator introduced in Ex. 6. Together with the hy-potheses (i) and (ii) assumed in that exercise, suppose that

(iii) L is hypoelliptic in Ω ,(iv) for every x0 ∈ Ω , there exists ξ0 ∈ R

N such that qL(x0, ξ0) > 0,(v) for every x0 ∈ Ω , there exists a positive real function γx0 of class C2 in

Ω \ {x0}) such that

Lγx0 = 0 in Ω \ {x0}, limx→x0

γx0(x) = ∞.

For every open set U ⊆ Ω define

LH(U) = {u ∈ C∞(U, R) : Lu = 0 in U

}.

Prove that (Ω, LH) is a S∗-harmonic space.(Hint: Just follow the lines of Section 7.1. Take as sx0 the function 1/γx0 .)

Ex. 8) Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Ω ⊆ G bean L-regular connected open set. Show that

supp μΩx = ∂Ω


for every x ∈ Ω

(Hint: Use the strong maximum principle.)Note: This result shows that the L-harmonic space on G is an “elliptic spacein the sense of Brelot” (see [CC72]).

Ex. 9) (The Exterior L-Ball Regularity Condition). Let G be a homogeneousCarnot group and let d be a L-gauge, L being a sub-Laplacian on G. In thefollowing statement, we say that Ω has at y the property of the exterior L-ballif there exists a d-ball Bd(z, ρ) such that

G \ Ω ⊇ Bd(z, ρ) and y ∈ ∂Bd(z, ρ).

Prove the following result:

Let Ω ⊆ G be a bounded open set and let y ∈ ∂Ω . Assume Ω has at y theproperty of the exterior L-gauge ball. Then y is an L-regular point for Ω .

8

L-subharmonic Functions

Throughout the chapter, G = (RN, ◦, δλ), L = ∑N1j=1 X2

j and d will denote, respec-tively, a homogeneous Carnot group, a sub-Laplacian on G and an L-gauge on G.Our main task is to provide some characterizations of the L-subharmonic functionsu in terms of suitable sub-mean properties w.r.t. the mean value operator Mru andMru (introduced in Chapter 5), in terms of the monotonicity w.r.t. the radius of theseoperators and in terms of the sign of Lu in the weak sense of distributions.

We also provide some maximum principles for L-subharmonic functions andsmoothing approximation theorems. Finally, in Section 8.3, we furnish a brief inves-tigation on the continuous convex functions on G.

Throughout the chapter S(Ω) and S(Ω) will denote the sheafs of theL-subharmonic and the L-superharmonic functions, respectively, in the open setΩ ⊆ G.

8.1 Sub-mean Functions

In this section, we shall denote by Mr and Mr the average operators defined in (5.46)and (5.50f), respectively (pages 256 and 259).

Definition 8.1.1 (Solid and surface sub-mean function). If Ω ⊆ G is an open set,we say that an u.s.c. function u : Ω → [−∞,∞[ satisfies the local surface (localsolid) sub-mean property if, for every x ∈ Ω , there exists rx > 0 such that

u(x) ≤ Mr (u)(x)(u(x) ≤ Mr (u)(x)

)for 0 < r < rx . (8.1)

If (8.1) holds for any r > 0 such that Bd(x, r) ⊂ Ω , we shall say that u satisfies theglobal surface (global solid) sub-mean property.

The next theorem shows that solid sub-mean functions satisfy weak and strongmaximum principles.

398 8 L-subharmonic Functions

Theorem 8.1.2 (Maximum principles for sub-mean functions). Let Ω ⊆ G be anopen set. Let u : Ω → [−∞,∞[ be an u.s.c. function satisfying the local solidsub-mean property. The following statements hold:

(i) if Ω is connected and there exists x0 ∈ Ω such that u(x0) = maxΩ u, thenu ≡ u(x0) in Ω ,

(ii) if Ω is bounded and lim supΩy→x u(y) ≤ 0 for every x ∈ ∂Ω , then u(y) ≤ 0in Ω .

Proof. (i) Let x0 ∈ Ω be such that u(x0) is the maximum of u in Ω . We may supposethat u(x0) �= −∞. Since u is L-sub-mean, we have

0 ≤ md

rQ

∫

Bd(x0,r)

ΨL(x−10 ◦ y)

(u(y) − u(x0)

)dy (8.2)

for some r = rx0 > 0. Thus, since ΨL ≥ 0 and u(y) ≤ u(x0),

ΨL(x−10 ◦ y)

(u(y) − u(x0)

) = 0 almost everywhere in Bd(x0, r).

On the other hand,

ΨL > 0 in a dense open subset of Bd(x0, r)

(see the first step in the proof of Lemma 5.7.1, page 262) and u is u.s.c. This impliesu = u(x0) on the whole Bd(x0, r). The assertion follows from a connectednessargument.

(ii) Let x0 ∈ Ω be such that

supBd(x0,r)∩Ω

u = supΩ

u ∀ r > 0.

If x0 ∈ ∂Ω , by the hypothesis and the boundary behavior of u, we have

0 ≥ lim supΩy→x0

u = infr>0

supBd(x0,r)∩Ω

u = supΩ

u,

whence u ≤ 0 on Ω . If x0 ∈ Ω , by the upper semicontinuity of u, we have

u(x0) ≥ infr>0

supBd(x0,r)∩Ω

u = infr>0

supΩ

u = supΩ

u.

Hence u(x0) = maxΩ u. From (i) it follows that u ≡ u(x0) on the connected com-ponent of Ω containing x0, so that, for some x ∈ ∂Ω ,

0 ≥ lim supy→x

u(y) ≥ u(x0) = maxΩ

u.

The assertion is proved. ��Theorems 8.1.2 and 7.2.9 (page 391) allow us to show that the solid and surface

sub-mean properties are equivalent.

8.1 Sub-mean Functions 399

Theorem 8.1.3 (Equivalence of sub-mean properties). Let u : Ω → [−∞,∞[ bean u.s.c. function. Then the following statements are equivalent:

(1) u satisfies the local solid sub-mean property,(2) u satisfies the global solid sub-mean property,(3) u satisfies the local surface sub-mean property,(4) u satisfies the global surface sub-mean property.

Proof. (1) ⇒ (4). Let Bd(x, r) ⊂ Ω and ϕ ∈ C(∂Bd(x, r), R) be such that u ≤ ϕ

in ∂Bd(x, r). Since u − HBd(x,r)ϕ is a solid sub-mean function, by the maximum

principle in Theorem 8.1.2, we have u − HBd(x,r)ϕ ≤ 0 in Bd(x, r). In particular,

u(x) ≤ HBd(x,r)ϕ (x) =

∫

∂Bd(x,r)

ϕ(y) dμBd(x,r)x (y)

= (by Theorem 7.2.9)βd(Q − 2)

rQ−1

∫

∂Bd(x,r)


Then, taking the infimum with respect to the continuous functions ϕ ≥ u|∂Ω , we get

u(x) ≤ βd(Q − 2)

rQ−1

∫

∂Bd(x,r)

u(y)KL(x, y) dσ(y) = Mr (u)(x),

and (4) is proved.(4) ⇒ (3). This is trivial.(3) ⇒ (1). Suppose u(x) ≤ Mr (u)(x) for 0 < r < rx . Then

u(x) ≤ Q

rQ

∫ r

0ρQ−1 Mρ(u)(x) dρ = Mr (u)(x).

(4) ⇒ (2). The same proof as the previous one.(2) ⇒ (1). This is trivial. ��

From now on, we shall call sub-mean any function satisfying one of the proper-ties (1)–(4) in Theorem 8.1.3.

Theorem 8.1.4 (L1loc of sub-mean functions). Let Ω be a connected open subset

of G, and let u : Ω → [−∞,∞[ be a sub-mean u.s.c. function.Suppose that u(x0) > −∞ for some x0 ∈ Ω . Then u ∈ L1

loc(Ω).

Proof. We first prove that u > −∞ in a dense subset of Ω . Since Ω is connectedand there exists x0 ∈ Ω such that u(x0) > −∞, it is enough to show that u > −∞in a dense subset of Bd(x, r) for any Bd(x, r) ⊂ Ω such that u(x) > −∞. This laststatement follows from

−∞ < u(x) ≤ Mr (u)(x) = md

rQ

∫

Bd(x,r)

ΨL(x−1 ◦ y) u(y) dy

and the property ΨL > 0 in an open dense set.


Let us now fix z0 ∈ Ω and R > 0 such that Bd(z0, R) ⊂ Ω . We shall prove theexistence of a constant λ > 0 such that

∫

Bd(z0,λR)

u(z) dz > −∞.

From this statement our theorem will follow.Let us choose three positive constants λ0, λ, Λ such that

c (λ0 + λ) ≤ Λ, c (λ0 + Λ) ≤ 1, (8.3)

where c is the constant appearing in the pseudo-triangle inequality for the gauge d

(see (5.55), page 262). Since the set

{x ∈ Ω | ΨL(x−1 ◦ z0) > 0 }is an open and dense subset of Ω and

{x ∈ Ω | u(x) > −∞}is dense in Ω too, there exists x0 ∈ Bd(z0, λ0 R) such that

x0 �= z0, u(x0) > −∞ and ΨL(x−10 ◦ z0) > 0.

Due to the smoothness of ΨL out of the origin, we may refine the choice of λ in such away that, for a suitable m0 > 0, we get ΨL(x−1

0 ◦z) ≥ m0 for every z ∈ Bd(z0, λR).On the other hand, inequalities (8.3) imply the inclusions

Bd(z0, λR) ⊆ Bd(x0,ΛR) ⊆ Bd(z0, R).

Then, if we putU := max

Bd(z0,R)

u (∈ R),

we have (denoting by meas the Lebesgue measure in G)∫

Bd(z0,λR)

u(z) dz

=∫

Bd(z0,λR)

(u(z) − U

)dz + U meas

(Bd(z0, λR)

)

≥ 1

m0

∫

Bd(z0,λR)

ΨL(x−10 ◦ z)

(u(z) − U

)dz + U meas

(Bd(z0, λR)

)

≥ 1

m0

∫

Bd(x0,Λ R)

ΨL(x−10 ◦ z)

(u(z) − U

)dz + U meas

(Bd(z0, λR)

)

= (Λ R)Q

md m0MΛ R(u − U)(x0) + U meas

(Bd(z0, λR)

)

≥ (Λ R)Q

md m0

(u(x0) − U

) + U meas(Bd(z0, λR)

)> −∞.


8.2 Some Characterizations of L-subharmonic Functions 401

We close this section by proving that every (ε, G)-mollifier preserves the sub-mean properties (see Definition 5.3.6 on page 239).

Theorem 8.1.5 (Mollifier of a sub-mean function). Let u : Ω → [−∞,∞[ beu.s.c. and u ∈ L1

loc(Ω). Then, if u is sub-mean in Ω , uε is sub-mean in Ωε too.

Proof. For x ∈ Ωε and r > 0 such that Bd(x, r) ⊆ Ωε, we have

Mr (uε)(x) =∫

Bd(0,ε)

Jε(z)

(md

rQ

∫

Bd(x,r)

ΨL(x−1 ◦ y) u(z−1 ◦ y) dy

)dz

=∫

Bd(0,ε)

Jε(z) Mr (u)(z−1 ◦ x) dz

≥∫

Bd(0,ε)

Jε(z) u(z−1 ◦ x) dz = uε(x),

and the assertion follows. ��

8.2 Some Characterizations of L-subharmonic Functions

We begin by proving the following theorem.

Theorem 8.2.1 (L-hypoharmonicity and sub-mean property). Let Ω be an opensubset of G and u : Ω → [−∞,∞[ be an u.s.c. function. Then u is L-hypoharmonicif and only if u is sub-mean.

Proof. Let u be a sub-mean function. By using the maximum principle in Theo-rem 8.1.2 and arguing as in the first part of the proof of Theorem 8.1.3, it is easy toprove that u is L-hypoharmonic.

Vice versa, let us suppose that u is L-hypoharmonic and prove that u satisfies theglobal surface sub-mean property. Let Bd(x, r) ⊂ Ω . Since Bd(x, r) is an L-regularopen set, for all ϕ ∈ C(∂Bd(x, r), R) such that ϕ ≥ u on ∂Bd(x, r), we have

u(x) ≤ HBd(x,r)ϕ (x) =

∫

∂Bd(x,r)

ϕ(y) dμBd(x,r)x (y).

On the other hand, from Theorem 7.2.9 (page 391) we have∫

∂Bd(x,r)

ϕ(y) dμBd(x,r)x (y) = βd(Q − 2)

rQ−1

∫

∂Bd(x,r)


Hence

βd(Q − 2)

rQ−1

∫

∂Bd(x,r)

ϕ(y)KL(x, y) dσ(y) ≥ u(x) ∀ ϕ ∈ C(∂Bd(x, r), R),

so that

Mr (u)(x) = βd(Q − 2)

rQ−1

∫

∂Bd(x,r)

u(y)KL(x, y) dσ(y) ≥ u(x).

Thus, u has the surface sub-mean property. ��


Corollary 8.2.2. Let Ω be an open subset of G, and let u : Ω → [−∞,∞[ bean u.s.c. function, finite in a dense subset of Ω . Then u ∈ S(Ω) if and only if u issub-mean.

The following result will be used very often.

Corollary 8.2.3 (Characterization of S(Ω)). Let Ω be a connected open subsetof G, and let u : Ω → [−∞,∞[ be an u.s.c. function.

Then u ∈ S(Ω) if and only if u is sub-mean and u > −∞ in at least one pointof Ω .

Proof. If u ∈ S(Ω), then u > −∞ in a dense subset of Ω , and u is sub-mean byTheorem 8.2.1. Vice versa, if u is sub-mean and u > −∞ in at least one point ofΩ , by Theorem 8.1.4, u ∈ L1

loc(Ω). In particular, u > −∞ in a dense subset of Ω .Then, by Theorem 8.2.1, u ∈ S(Ω). ��Corollary 8.2.4. If u ∈ S(Ω), then u ∈ L1

loc(Ω).

Corollary 8.2.5 (A Brelot-type convergence result). Let {un}n∈N be a sequenceof L-subharmonic functions in a connected open set Ω ⊆ G. Assume {un}n∈N ismonotone decreasing.

Then, if we set u := limn→∞ un, we have u ∈ S(Ω) provided there exists x0 ∈ Ω

such that u(x0) > −∞.

Proof. The function u is u.s.c. and sub-mean, since so are the un’s, and un ≥ un+1for every n ∈ N. Then, the assertion follows from Corollary 8.2.3. ��Corollary 8.2.6. Let Ω ⊆ G be open, and let x0 ∈ Ω . Let u ∈ S(Ω \ {x0}) be suchthat

limx→x0

u(x) = −∞.

Then, if we continue u at x0 by letting u(x0) = −∞, we have u ∈ S(Ω).

Proof. The continued function u is u.s.c. and finite in a dense subset of Ω . Moreover,for every x ∈ Ω \ {x0} and r > 0 such that Bd(x, r) ⊂ Ω \ {x0}, we have

u(x) ≤ Mr (u)(x),

since u is L-subharmonic (hence sub-mean) in Ω \ {x0}. On the other hand,

u(x0) = −∞ ≤ Mr (u)(x0) ∀ r > 0 : Bd(x0, r) ⊂ Ω.

Thus, u is locally sub-mean in Ω , hence u ∈ S(Ω). ��From Corollary 8.2.3 and the abstract results on directed families of hyperhar-

monic functions in Section 6.3.1 (page 347), we obtain the following assertion.


Theorem 8.2.7 (Down directed families in S(Ω)). Let F be a down directed familyof L-subharmonic functions in an open connected set Ω ⊆ G. The function

u := infF

is L-subharmonic in Ω if and only if there exists a point x0 ∈ Ω such that

u(x0) > −∞.

Proof. By Theorem 6.3.8 (page 348), u is L-hypoharmonic in Ω . In particular, u isu.s.c. and sub-mean in Ω . Then, by Corollary 8.2.3, u is L-subharmonic if and onlyif there exists a point x0 ∈ Ω such that u(x0) > −∞. ��Corollary 8.2.8. Let F be a down directed family of L-subharmonic functions in anopen set Ω ⊆ G. Then, on every connected component of Ω , the function

u := infF

is L-subharmonic or identically equal to −∞.

The L-subharmonicity can be characterized in terms of the monotonicity withrespect to r of the averaging operators Mr and Mr . We first prove the followinglemma.

Lemma 8.2.9. Let m : [0,∞[→ R be a strictly increasing function. Let f : [0,∞[→R be an increasing function, integrable w.r.t. m, in the sense of Riemann-Stieltjes. Ifr ∈ ]0,∞[, we set

M(r) :=∫ r

0dm(ρ), F (r) := 1

M(r)

∫ r

0f (ρ) dm(ρ).

Then F ≤ f and F is non-decreasing.

Proof. From the monotonicity of f we have

F(r) = 1

M(r)

∫ r

0f (ρ) dm(ρ) ≤ f (r)

1

M(r)

∫ r

0dm(ρ) = f (r).

Finally, if 0 < r1 < r2,

F(r2) − F(r1)

=(

1

M(r2)− 1

M(r1)

) ∫ r1

0f (ρ) dm(ρ) + 1

M(r2)

∫ r2

r1

f (ρ) dm(ρ)

≥(

1

M(r2)− 1

M(r1)

)M(r1) f (r1) + 1

M(r2)

(M(r2) − M(r1)

)f (r1) = 0.



In order to state the next theorem, it is convenient to introduce the followingdefinition: if Ω ⊆ G is an open set and x ∈ Ω , we set

R(x) := d(x,Ω) (= sup{r > 0 : Bd(x, r) ⊆ Ω}).Theorem 8.2.10 (S(Ω) and the monotonicity of Mr and Mr ). Let Ω be an opensubset of G, and let u : Ω → [−∞,∞[ be an u.s.c. function finite in a dense subsetof Ω . Then the following statements are equivalent:

(i) u ∈ S(Ω),(ii) r �→ Mr (u)(x) is monotone increasing for 0 < r < R(x) and

u(x) = limr→0+ Mr (u)(x),

(iii) r �→ Mr (u)(x) is monotone increasing for 0 < r < R(x) and

u(x) = limr→0+ Mr (u)(x).

Proof. (i) ⇒ (ii). Let Bd(x, s) ⊂ Ω and 0 < r < s. Define

F := {ϕ ∈ C(∂Bd(x, s), R) : ϕ ≥ u on ∂Bd(x, s)}.Since Bd(x, s) is an L-regular open set, we have

HBd(x,s)ϕ (y) ≥ u(y) ∀ y ∈ Bd(x, s),

so that, by the surface mean value Theorem 5.5.4,

Mr (u)(x) ≤ Mr (HBd(x,s)ϕ )(x) = HBd(x,s)

ϕ (x) = Ms(HBd(x,s)ϕ )(x)

= Ms(ϕ)(x) = βd(Q − 2)

rQ−1

∫

∂Bd(x,s)

KL(x, y) ϕ(y) dσ(y).

Hence

Mr (u)(x) ≤ infϕ∈F

βd(Q − 2)

rQ−1

∫

∂Bd(x,s)

KL(x, y) ϕ(y) dσ(y)

= βd(Q − 2)

rQ−1

∫

∂Bd(x,s)

KL(x, y) u(y) dσ(y) = Ms(u)(x).

This proves the first part of the statement. In order to prove the second one, we justhave to note that, by the upper semicontinuity of u, for every λ > u(x) there existsr > 0 such that

Mr (u)(x) < λ for 0 < r < r .

On the other hand, u(x) ≤ Mr (u)(x) for every r < R(x). Then

u(x) ≤ Mr (u)(x) < λ ∀ r ∈ ]0, r[.


(ii) ⇒ (iii). Since

Mr (u)(x) = Q

rQ

∫ r

0ρQ−1 Mρ(u)(x) dρ ∀ r ∈ ]0, R(x)[

and ρ �→ Mρ(u)(x) is monotone increasing, by Lemma 8.2.9, r �→ Mr (u)(x)

is monotone increasing too. Moreover, by arguing as in the previous step, u(x) =limr→0+ Mr (u)(x).

(iii) ⇒ (i). Since r �→ Mr (u)(x) is a monotone increasing function and u(x) =limr→0+ Mr (u)(x), we have

u(x) ≤ Mr (u)(x) if 0 < r < R(x).

Then u satisfies the global solid sub-mean property, so that, by Theorem 8.2.1, u ∈S(Ω). ��

We know that a smooth function u is L-subharmonic if and only if Lu ≥ 0. Asimilar characterization holds for non-smooth L-subharmonic functions. More pre-cisely, we have the following result.

Theorem 8.2.11 (S(Ω) and the inequality Lu ≥ 0 in D′(Ω). I). Let Ω be an opensubset of G, and let u : Ω → [−∞,∞[ be an u.s.c. function. Then the followingstatements are equivalent:

(i) u ∈ S(Ω),(ii) u ∈ L1

loc(Ω), u(x) = limr→0+ Mr (u)(x) for every x ∈ Ω , and

Lu ≥ 0 in Ω

in the weak sense of distributions.1

Remark 8.2.12. We remark that in condition (ii) of Theorem 8.2.11 the hypothesisu(x) = limr→0+ Mr (u)(x) cannot be removed. Consider, for example, the functionu = χ{0} (the characteristic function of the set {0}). Then, it is immediately seenthat u is u.s.c., u ∈ L1

loc(G), Lu = 0 in G (in the weak sense of distributions), butu /∈ S(G) for it is not sub-mean at the origin. On the other hand, if, in addition, u iscontinuous then u(x) = limr→0+ Mr (u)(x) always holds true. ��

In order to prove the theorem, we need the following real analysis lemma.

Lemma 8.2.13. Let f : ]0,∞[→ R be a L1loc function such that

∫ ∞

0f (t) h′(t) dt ≤ 0 ∀ h ∈ C1

0(]0,∞[), h ≥ 0. (8.4)

1 The inequality Lu ≥ 0 in the weak sense of distributions means∫

ΩuLφ ≥ 0 ∀ φ ∈ C∞

0 (Ω), φ ≥ 0.


Let α > 1. Suppose that

F(t) := α

tα

∫ t

0sα−1 f (s) ds

is finite for 0 < t < T . Then t �→ F(t) is monotone increasing in ]0, T [.Proof. Let t1, t2 be Lebesgue points of f such that 0 < t1 < t2. For a fixed ε > 0,ε < t2−t1

2 , we choose a sequence {hn}n of positive C10(]0,∞[) functions such that

supp hn ⊆ [t1, t2], supn supt |h′n(t)| < ∞, and, as n → ∞,

h′n(t) →

⎧⎨

⎩

1/ε if t1 < t < t1 + ε,

−1/ε if t2 − ε < t < t2,

0 if t /∈ [t1, t1 + ε] ∪ [t2 − ε, t2].Replacing hn in (8.4) and letting n tend to infinity, we get

1

ε

∫ t1+ε

t1

f (t) dt − 1

ε

∫ t2

t2−ε

f (t) dt ≤ 0.

From this inequality, letting ε → 0+, we obtain f (t1) ≤ f (t2), since t1 and t2 areLebesgue points for f .

Since almost every point of ]0,∞[ is a Lebesgue point for f , we can use thesame argument as in the proof of Lemma 8.2.9 in order to show that F(t1) ≤ F(t2)

for every pair of Lebesgue points t1, t2 ∈ ]0, T [, with t1 ≤ t2. This result, togetherwith the continuity of F , proves the lemma. ��

We are now in the position to prove Theorem 8.2.11.

Proof (of Theorem 8.2.11). (i) ⇒ (ii). If u ∈ S(Ω), then, by Corollary 8.2.4,u ∈ L1

loc(Ω). Moreover, by Theorem 8.2.10, u(x) = limr→0+ Mr (u)(x). By The-orem 8.2.1, we also have that u is sub-mean in Ω , so that, by Theorem 8.1.5, uε issub-mean in Ωε. As a consequence, by Theorem 8.2.1, uε ∈ S(Ωε), so that, sinceuε ∈ C∞(Ωε), Luε ≥ 0 in Ωε. Since uε → u in L1

loc(Ω), as ε ↘ 0, we get Lu ≥ 0in the weak sense of distributions.

(ii) ⇒ (i). The inequality Lu ≥ 0 in the weak sense of distributions means∫

Ω

uLφ ≥ 0 ∀ φ ∈ C∞0 (Ω), φ ≥ 0. (8.5)

We now choose a suitable test function φ. Let us fix a d-ball Bd(x0, R) ⊆ Ω anda non-negative function g ∈ C∞([0, R[, R), constant in a right neighborhood of 0.Define

φ(x) := g(d(x−10 ◦ x)).

Then φ ∈ C∞0 (Bd(x0, R)) and, by Proposition 5.4.3 (page 247),

Lφ = |∇Ld|2(

g′′(d) + Q − 1

dg′(d)

).


Replacing the right-hand side in (8.5), using Federer’s coarea formula, and keepingin mind the definition of the average operator Mt , we get

0 ≤∫ ∞

0

(g′′(t) + Q − 1

tg′(t)

) ( ∫

{d=t}uKL dσ

)dt

= 1

βd(Q − 2)

∫ ∞

0

(tQ−1 g′(t)

)′ Mt (u) dt.

Then, if we choose

g(t) :=∫ ∞

t

h(s)

sQ−1ds,

where h is any non-negative C∞0 (]0, R[; R) function, the previous inequality can be

written as ∫ ∞

0h′(t)Mt (u)(x0) dt ≤ 0.

Then, by Lemma 8.2.13, t �→ Mt (u)(x0) is monotone increasing in ]0, R[. On theother hand, by hypothesis, limr→0+ Mr (u)(x) = u(x). Thus, by Theorem 8.2.10,u ∈ S(Ω). ��Remark 8.2.14. From the previous proof, we have that

t �→ Mt (u)(x), 0 ≤ t < R(x), x ∈ Ω,

is monotone increasing whenever u ∈ L1loc(Ω) and Lu ≥ 0 in the weak sense of

distributions. Here, we denoted

R(x) := sup{r > 0 : Bd(x, r) ⊂ Ω}.On the other hand, if u is just a L1

loc(Ω) function, very standard arguments showthat:

(i) (x, t) �→ Mt (u)(x) is continuous in {(x, t) : x ∈ Ω, 0 ≤ t < R(x)},(ii) limj→∞ Mtj (u)(x) = u(x) almost everywhere in Ω , for a suitable sequence

tj ↓ 0 as j ↑ ∞. ��The above remark, together with Theorems 8.2.10 and 8.2.11, leads to the next

theorem.

Theorem 8.2.15 (S(Ω) and the inequality Lu ≥ 0 in D′(Ω). II). Let u ∈ L1loc(Ω),

with open Ω ⊆ G. Then the following statements are equivalent:

(1) Lu ≥ 0 in Ω , in the weak sense of distributions,(2) t �→ Mt (u)(x) is monotone increasing on ]0, R(x)[ for every x ∈ Ω ,(3) there exists a function u ∈ S(Ω) such that u(x) = u(x) almost everywhere in Ω .

The function in (3) is unique. It is given by

u(x) = limt↓0

Mt (u)(x), x ∈ Ω.


Proof. (1) ⇒ (2). This follows from the previous remark.(2) ⇒ (3). The monotonicity of t �→ Mt (u)(x) implies the existence of

u(x) := limt↓0

Mt (u)(x) ∀ x ∈ Ω.

Remark 8.2.14-(ii) implies u(x) = u(x) almost everywhere in Ω . Then, if 0 < t <

τ , we haveMt (u)(x) = Mt (u)(x) ≤ Mτ (u)(x) = Mτ (u)(x).

Hence, t �→ Mt (u)(x) is monotone increasing. Let us now show that u is u.s.c. Since

u(x) = limt↓0

Mt (u)(x) = limt↓0

Mt (u)(x),

by Theorem 8.2.10, the assertion will follow.Let λ ∈ R and x ∈ Ω be such that u(x) < λ. Then, by the very definition of u, we

have Mt (u)(x) < λ for every t small enough. The continuity of (t, y) �→ Mt (u)(y)

now impliesMt (u)(y) < λ

for every t sufficiently small and for any y close to x. As a consequence, beingu ≤ Mt (u),

u(y) < λ

for every y in a neighborhood of x. Thus, u is u.s.c., and this part of the proof iscomplete.

(3) ⇒ (1). By Theorem 8.2.11, if u ∈ S(Ω), then u ∈ L1loc(Ω) and Lu ≥ 0

in the weak sense of distributions. Since u = u almost everywhere in Ω , theseproperties hold for u too. ��Theorem 8.2.16 (Caccioppoli–Weyl’s lemma for sub-Laplacians). Let u ∈L1

loc(Ω) be a weak solution to

Lu = 0 in Ω .

Then there exists a smooth L-harmonic u in Ω such that

u(x) = u(x) almost everywhere in Ω .

Proof. By the previous theorem, there exists u1 ∈ S(Ω) and u2 ∈ S(Ω) such thatu1(x) = u(x) = u2(x) almost everywhere in Ω . Moreover,

u1(x) = limt↓0

Mt (u)(x) = u2(x).

Hence, u := u1 = u2 ∈ S(Ω)∩S(Ω), so that, by Proposition 7.2.4, u is L-harmonicin Ω . We also obviously have u(x) = u(x) almost everywhere in Ω . This completesthe proof. ��


From Theorem 8.2.11 and Theorem 8.2.15 we also obtain the following charac-terization of L-subharmonic functions.

Theorem 8.2.17 (S(Ω) and the inequality Lu ≥ 0 in D′(Ω). III). Let Ω be anopen subset of G. Then the following statements are equivalent:

(i) u ∈ S(Ω),(ii) u ∈ L1

loc(Ω), Lu ≥ 0 in Ω in the weak sense of distributions, and2

u(x) = ess lim supy→x

u(y) ∀ x ∈ Ω. (8.6)

We first prove the following lemma.

Lemma 8.2.18. Let Ω be an open subset of G. If u ∈ S(Ω), then (8.6) holds.

Proof. Since u is u.s.c., we have u := ess lim sup u ≤ lim sup u ≤ u. Assume bycontradiction that u �= u. Then there exists x ∈ Ω such that u(x) < λ < u(x)

(for some λ ∈ R). In particular u ≤ λ a.e. in a neighborhood of x, so thatMr (u)(x) ≤ λ for any small r > 0. Recall that, by Theorem 8.2.11, we haveu(x) = limr→0+ Mr (u)(x). Hence we obtain u(x) ≤ λ, which gives a contradic-tion. ��Proof (of Theorem 8.2.17). (i) ⇒ (ii) follows from Theorem 8.2.11 and Lem-ma 8.2.18.

(ii) ⇒ (i). According to Theorem 8.2.15, there exists v ∈ S(Ω) coinciding al-most everywhere with u. From Lemma 8.2.18 we infer that v = ess lim sup v. Inorder to obtain u = v, we now only need to observe that v = u almost everywhereimplies ess lim sup v = ess lim sup u and to recall that ess lim sup u = u by hypoth-esis. ��

Further, we explicit state some maximum principles for L-subharmonic func-tions, which immediately follow from Theorems 8.1.2 and 8.2.1.

Theorem 8.2.19 (The strong maximum principle for S(Ω)). Let Ω ⊆ G be anopen set, and let u ∈ S(Ω). The following statements hold:

(i) if Ω is connected and there exists x0 ∈ Ω such that u(x0) = maxΩ u, thenu ≡ u(x0) in Ω ,

(ii) if lim supΩy→x u(y) ≤ 0 for every x ∈ ∂Ω , and lim supΩy→∞ u(y) ≤ 0 if Ω

is unbounded, then u(y) ≤ 0 in Ω .

2 Here we use the following notation

ess lim supy→x

u(y) = inf{

ess supV

u : V is a neighborhood of x},

whereess sup

V

u = inf{m ∈ R ∪ {∞} : m ≥ u(x) for almost every x ∈ V }.


Proof. Due to Theorems 8.1.2 and 8.2.1, we only have to prove (ii) in the case whenΩ is unbounded. From the conditions at the boundary of Ω and at infinity we obtain:for every ε > 0, there exists R > 0 such that

lim supΩ∩Bd(0,R)x→y

u(x) ≤ ε ∀ y ∈ ∂(Ω ∩ Bd(0, R))

andu(x) < ε ∀ x ∈ Ω \ Bd(0, R).

Then, by the maximum principle on bounded domains, u(x) < ε for every x ∈ Ω .Letting ε vanish, we get u ≤ 0 in Ω . ��

We close the section by giving a theorem on superposition of superharmonicfunctions.

Theorem 8.2.20. Let Λ ⊆ Rν and Ω ⊆ G be connected open sets. Let

u : Λ × Ω →]−∞,∞]be a l.s.c. function. Finally, let μ be a Radon measure with compact support con-tained in Λ. Assume that u(z, ·) ∈ S(Ω) for every fixed z ∈ Λ. Then

U(x) :=∫

Λ

u(z, x) dμ(z), x ∈ Ω,

is L-superharmonic in Ω if there exists x0 ∈ Ω such that

U(x0) < ∞. (8.7)

Proof. Since u is l.s.c., we have

lim infx→y

U(x) ≥∫

Λ

lim infx→y

u(z, x) dμ(z)

≥∫

Λ

u(z, y) dμ(z) ∀ y ∈ Ω.

Thus, U is l.s.c. in Ω . Let x ∈ Ω and r > 0 be such that Bd(x, r) ⊂ Ω . ByFubini–Tonelli’s theorem, we obtain

Mr (U)(x) =∫

Λ

Mr (u(z, ·))(x) dμ(z)

(since u(z, ·) is super-mean)

≥∫

Λ

u(z, x) dμ(z) = U(x). (8.8)

Thus, U is super-mean. From Corollary 8.2.3 and condition (8.7) it follows thatU ∈ S(Ω). This completes the proof. ��

8.3 Continuous Convex Functions on GGG 411

Remark 8.2.21. The previous theorem still holds if we replace the condition

supp(μ) compact and contained in Λ

with the following one:

for every compact set K ⊂ Ω ,there exists a function ϕK ∈ L1(Λ, dμ) such that

u(z, x) ≥ ϕK(z) ∀ z ∈ Λ ∀ x ∈ K. (8.9)

Indeed, if this condition is satisfied, we can still apply Fubini–Tonelli’s theoremin (8.8). For example, we explicitly remark that (8.9) holds with ϕK ≡ 0 if u isnon-negative.

8.3 Continuous Convex Functions on GGG

Throughout this section, we shall follow two different approaches in defining convexfunctions in Carnot groups. Our main references will be G. Lu, J.J. Manfredi andB. Stroffolini [LMS04] (see also P. Juutinen and the same authors, [JLMS07]) andD. Danielli, N. Garofalo and D.-M. Nhieu [DGN03].

If an extra hypothesis is added (namely, continuity) these two notions are equiv-alent. Since the aim of this section is only to give an overview on convexity, wedecided to restrict our attention to continuous convex functions on Carnot groups,since the dropping of the continuity hypothesis would lead us beyond our scopes.The reader will be referred to suitable references for more general results (seeNote 8.3.18).

A convex function in RN is subharmonic with respect to every constant coeffi-

cient elliptic operator, and vice versa (see, e.g. [Hor94]). This makes quite naturalthe following (first) extension of the notion of convexity to Carnot groups. To thisend, we introduce the following notation. We shall denote by Sm+ the cone of the realm × m symmetric and strictly positive definite matrices.

Let G be an abstract stratified Lie group with the Lie algebra g. Let also V1 ⊕V2 ⊕· · ·⊕Vr be a stratification of g according to Definition 2.2.3 (page 122). We setV = (V1, . . . , Vr). Let finally X = {X1, . . . , Xm} be a basis of the first layer V1 ofthe stratification (the so-called horizontal layer). Given a matrix A = (aj,k)j,k ∈ S

m+,we denote by LA the sub-Laplacian3

LA =m∑

j,k=1

aj,kXjXk. (8.10)

Definition 8.3.1 (v-convex function). Let G be a stratified group, and let V =(V1, . . . , Vr ) be a given stratification of the algebra of G. Let {X1, . . . , Xm} be abasis of the first layer V1.

A continuous real-valued function u in an open set Ω ⊆ G is v-convex if it isLA-subharmonic in Ω for every A ∈ Sm+, being LA as in (8.10).

3 See Remark 8.3.3 below.


Note 8.3.2. Obviously, the notion of v-convexity is in general well-given for uppersemi-continuous functions. The above definition is one of the equivalent forms ofthe notion of v-convexity in [LMS04], convexity in the viscosity sense. Our choiceto add the extra hypothesis of continuity has already been justified by the sake ofsimplicity.

We explicitly remark the following fact: the notion of v-convexity depends onthe stratification V = (V1, . . . , Vr) of g or, more precisely, by the horizontal layerV1 (since V1 determines all the Vi’s). Hence, another possible notation should be“H-convex function” (where “H” stands for “horizontal”).

Remark 8.3.3. As we shall see in Proposition 16.1.1 in Chapter 16 (page 623), anysub-Laplacian on G is of the form (8.10) for a suitable symmetric positive defi-nite matrix A, and vice versa, any operator LA in that form is a sub-LaplacianLA = ∑m

k=1Y2k with Yk = ∑m

j=1(A1/2)k,j Xj . This immediately proves the fol-

lowing proposition.

Proposition 8.3.4 (Equivalent definition of v-convexity). A continuous real-valuedfunction u in an open set Ω ⊆ G is v-convex if and only if it is L-subharmonic in Ω

for every sub-Laplacian L (related to V ).More explicitly, denoted by V1 the first layer of the stratification V , a continuous

function u is v-convex in Ω if and only if it is L-subharmonic in Ω w.r.t. every L =∑mk=1 Y 2

k , being (Y1, . . . , Ym) any basis of V1.

Note 8.3.5. Observe that the above Proposition 8.3.4 highlights the fact that thenotion of v-convexity in Definition 8.3.1 does not depend on the particular basis(X1, . . . , Xm) of V1.

Remark 8.3.6. We know from Section 8.2 that the LA-subharmonicity of u is equiv-alent to the inequality

LAu ≥ 0 in D′(Ω), (8.11)

i.e. in the weak sense of distributions. It follows that the v-convexity is invariant withrespect to left translations and dilations on G, and that locally uniform limits of v-convex functions are v-convex. Indeed, such properties hold for the solutions to theinequality (8.11). ��

For a function u of class C2, the v-convexity can be characterized in terms of itshorizontal Hessian related to the family X , i.e. of the matrix

X 2u(x) :=(

XjXku(x) + XkXju(x)

2

)

j,k=1,...,m

.

Proposition 8.3.7 (v-convexity and the horizontal Hessian). With all the above no-tation, a function u ∈ C2(Ω, R) is v-convex if and only if

X 2u(x) ≥ 0 for every x ∈ Ω.


Proof. Since u ∈ C2, the distributional inequality (8.11) is equivalent to

LAu(x) ≥ 0

for every x ∈ Ω , so that u is v-convex if and only if

m∑

j,k=1

aj,k XjXku(x) ≥ 0 for every A = (aj,k)j,k in Sm+ and every x ∈ Ω .

Then the assertion follows from the following linear algebra lemma. ��Lemma 8.3.8. Let B = (bi,j )i,j be an m × m real matrix. Then

m∑

j,k=1

aj,k bj,k ≥ 0 (8.12)

for every A = (aj,k)j,k in Sm+ if and only if the symmetric part of B is non-negative

definite, i.e.

Bsym :=(

bj,k + bk,j

2

)

j,k≤m

≥ 0.

Proof. Let ξ = (ξ1, . . . , ξm)T ∈ Rm and ε > 0. Consider the matrix

Aε := ξ · ξT + ε Im = (ξj ξk)j,k≤m + ε Im,

where Im denotes the m × m identity matrix. Clearly, Aε ∈ Sm+ since it is symmetricand

〈Aεη, η〉 = ηT · ξ · ξT · η + ε |η|2 = (ηT · ξ)(ξT · η) + ε |η|2 = 〈ξ, η〉2 + ε |η|2,for every (column-vector) η ∈ R

m. Then, we can use Aε in (8.12) obtaining

0 ≤m∑

j,k=1

bj,k ξj ξk + ε

m∑

j=1

bj,j .

Letting ε → 0, we get

0 ≤m∑

j,k=1

bj,k ξj ξk =m∑

j,k=1

(bj,k + bk,j

2

)ξj ξk ∀ ξ ∈ R

m.

This proves the “only if” part of the lemma.To prove the “if” part, we first remark that inequality (8.12) rewrites as

trace(A · Bsym) ≥ 0.

Then, for given A ∈ Sm+ taking a matrix C ∈ Sm+ such that C2 = A, we have

trace(A · Bsym) = trace(C2 · Bsym) = trace(C · Bsym · C) ≥ 0,

since the matrix C ·Bsym ·C = C ·Bsym ·CT is non-negative definite if Bsym is. Thiscompletes the proof. ��


The following result holds.

Corollary 8.3.9 (v-convexity for C2-functions). A C2(Ω, R) function u is v-convexif and only if one of the following equivalent statements hold:

(1) X 2u ≥ 0 on Ω for one basis X = (X1, . . . , Xm) of V1;(2) X 2u ≥ 0 on Ω for every basis X = (X1, . . . , Xm) of V1;(3) Lu ≥ 0 on Ω for every sub-Laplacian L (related to the stratification V ).

Proof. (1) is a restatement of Proposition 8.3.7.(2) follows from Proposition 8.3.7 and the independence of the notion of v-

convexity w.r.t. the basis X of V1 (see Note 8.3.5).(3) follows from Proposition 8.3.4 and the characterization of L-subharmonicity

for C2-functions. ��The smooth v-convex functions are dense in the set of the continuous v-convex

functions. Indeed, the following result holds (see the definition of mollifier in Defini-tion 5.3.6, page 239; precisely, see the particular case in Example 5.3.7 on page 239).

Theorem 8.3.10 (Smoothing of a v-convex function). Let G be a homogeneousCarnot group. Let V be a fixed stratification of the algebra of G. Let also d be anyfixed homogeneous norm on G.

Suppose u ∈ C(Ω, R) is a v-convex function. For ε > 0, let Ωε be the set{y ∈ G : Bd(y, ε) ⊆ Ω}. For x ∈ Ωε, define as usual

uε(x) =∫

Ω

u(y) Jε(x ◦ y−1) dy, (8.13)

where J ∈ C∞0 (Bd(0, 1)) and Jε(z) = ε−QJ(δ1/ε(z)). Then:

(i) uε ∈ C∞(Ωε) and uε → u as ε → 0 uniformly on the compact sets in Ω ,(ii) uε is v-convex in Ωε.

Proof. (i) is a standard property of the mollifiers.(ii). Since u is v-convex, u is LA-subharmonic for every A ∈ Sm+. It follows from

Theorem 8.1.5 that uε is LA-subharmonic in its domain. Since this holds for everyA ∈ Sm+, we infer that uε is v-convex in Ωε. ��

The characterization of v-convexity in terms of the horizontal Hessian impliesthe convexity in the usual sense along the V -horizontal segments and vice versa. Tomake this statement precise, we first introduce the notion of V -horizontal segmentand of V -horizontal subspace.

Definition 8.3.11 (V -horizontal segment and subspace). Let (G, ◦) be a stratifiedgroup with the Lie algebra g. Let also V = (V1, . . . , Vr) be a given stratificationof g, and let Exp : g → G be the usual exponential map. We set

V := Exp (V1),

and we say that V is the V -horizontal subspace of G.


Let x ∈ G be fixed. If h is any element of V, we say that

[x ◦ h−1, x ◦ h] := {x ◦ Exp (t Log (h)) : −1 ≤ t ≤ 1

}

is a V -horizontal segment (along h and) centered at x.Finally, for a fixed x ∈ G, we say that

Vx := x ◦ V = {x ◦ Exp (X) : X ∈ V1}is the V -horizontal subspace through x.

Remark 8.3.12. Notice that V = Ve, being e the identity of G. We explicitly remarkthat V is a subset of G closed under inversion (for (Exp (X))−1 = Exp (−X)), itcontains the identity of G, but V is not in general a subgroup of G. Indeed

Exp (X) ◦ Exp (Y ) = Exp (X � Y) = Exp

(X + Y + 1

2[X, Y ] + · · ·

),

and, if X, Y ∈ V1, it does not generally hold X � Y ∈ V1. We remark that V is asubmanifold of G of dimension m = dim(V1). ��Theorem 8.3.13 (v-convexity and the horizontal segments). Let u ∈ C2(Ω, R)

with open Ω ⊆ G. Then u is v-convex in Ω if and only if, for every x ∈ Ω and everyh ∈ V such that [x ◦ h−1, x ◦ h] ⊂ Ω , the function

(−1, 1) t �→ u(x ◦ Exp (t X)

)(being X = Log (h))

is convex in the classical sense.

Proof. We split the proof in two parts.Sufficiency. Let u be v-convex. Take any h ∈ V such that the segment along h

centered at x is contained in Ω . Let X := Log (h). Hence, the function

(−1, 1) t �→ f (t) := u(x ◦ Exp (t X)

)

is well posed and, thanks to the C2-regularity assumption on u, f ∈ C2(−1, 1).By (2.36a), page 117, we have

f ′′(t) = (X2u)(x ◦ Exp (t X)

). (8.14)

Recalling that X ∈ V1 by definition of V, we can suppose that X is the first elementof a basis of V1, say X . By Proposition 8.3.7, X 2u(z) ≥ 0 for every z ∈ Ω . Sincez := x ◦ Exp (t X) ∈ Ω , for [x ◦ h−1, x ◦ h] ⊂ Ω , this gives

(X2u)(z) = 〈X 2u(z) e1, e1〉 ≥ 0 (being e1 = (1, 0, . . . , 0) ∈ Rm).

Consequently, (8.14) gives f ′′(t) ≥ 0. Being t ∈ (−1, 1) arbitrary, this proves thatf is convex.


Necessity. By the C2-regularity of u and thanks to Corollary 8.3.9-(3), we haveto prove that, given a basis X = (X1, . . . , Xm) for V1, it holds

∑mj=1 X2

j u(x) ≥ 0for every x ∈ Ω . Fix any j ∈ {1, . . . , m} and any x ∈ Ω . Since Ω is open, thereexists ε > 0 so small that

x ◦ Exp (t εXj ) ∈ Ω for every t ∈ [−1, 1].This means that setting h := Exp (ε Xj ), we have [x ◦ h−1, x ◦ h] ⊂ Ω . Since h

clearly belongs to V, the necessity assumption ensures that the function

(−1, 1) t �→ f (t) := u(x ◦ Exp (t ε Xj )

)

is a convex function in the classical sense. In particular, f ′′(0) ≥ 0. On the otherhand, again by (2.36a), page 117, we have

0 ≤ f ′′(0) = ε2(X2j u)(x),

whence (X2j u)(x) ≥ 0. The arbitrariness of j and x proves that

∑mj=1 X2

j u is non-negative on Ω , and the proof is complete. ��

We explicitly remark that, though Definition 8.3.1 of v-convexity needs someregularity assumption on u to be well posed (for example, upper semicontinuity ora L1

loc assumption), the characterization of v-convexity as stated in Theorem 8.3.13suggests another notion of convexity, which is free from any a priori assumptionon u. We hence give the following definition.

Definition 8.3.14 (Horizontally convex function). Let (G, ◦) be a stratified groupwith a fixed stratification V = (V1, . . . , Vr). Let Ω ⊆ G be open. A function u :Ω → R will be called horizontally convex (or, in short, H-convex) in Ω if thefunction

[−1, 1] t �→ u(x ◦ Exp (t X))

is convex (in the classical sense) for every X ∈ V1 such that (set h = Exp (X)) thehorizontal segment [x ◦ h−1, x ◦ h] is contained in Ω .

Note 8.3.15. The above notion of H-convex function is equivalent (see Proposi-tion 8.3.17) to that of “weakly H-convex” (weakly horizontally convex) functionintroduced by D. Danielli, N. Garofalo and D.-M. Nhieu [DGN03]. The same no-tion is also referred to as “CC-convexity”. L.A. Caffarelli’s 1997 NSF Proposal iscited in [DGN03] as seemingly the first reference for a notion of convexity in theHeisenberg groups. The simpler name “H-convex” is now commonly used.

We now focus our attention on homogeneous Carnot groups. See Remark 8.3.20for the details on the link between the notions of convexity in the abstract and in thehomogeneous setting.

It turns out that for continuous functions v-convexity and horizontal convexityare equivalent. This will be easily seen first noticing the following facts:


(i) Pointwise limits of horizontally convex functions are horizontally convex;(ii) If u ∈ C(Ω, R) is horizontally convex, then so is its mollifier uε defined

in (8.13). Indeed, the change of variable x ◦ y−1 = z in the integral at theright-hand side of (8.13) gives

uε(x) =∫

Ω

u(z−1 ◦ x)Jε(z) dz.

Then, for every X ∈ V1,

t �→ uε(x ◦ Exp (t X)) =∫

Ω

u(z−1 ◦ x ◦ Exp (t X)

)Jε(z) dz

is convex since t �→ u((z−1 ◦ x) ◦ Exp (t X)) is convex for every z ∈ Bd(0, ε)

with ε > 0 small enough.

We are now in the position to easily prove the equivalence between v- and H- con-vexity for continuous functions on homogeneous Carnot groups.

Theorem 8.3.16 (v- and H-convexity for continuous functions). Let G be a homo-geneous Carnot group. Let u ∈ C(Ω, R) with open Ω ⊆ G. Then u is v-convex inΩ if and only if it is horizontally convex in Ω .

Proof. If u is v-convex, then by Theorem 8.3.10, the function uε in (8.13) is smoothand v-convex in Ωε. Then, by Theorem 8.3.13, uε is horizontally convex. On theother hand, uε converges to u as ε → 0, uniformly on the compact subsets of Ω . Theprevious remark (i) implies that u is horizontally convex.

Vice versa, let us assume that u is horizontally convex. The previous remark(ii) tells us that uε has the same property, and Theorem 8.3.13 ensures that uε is v-convex. As ε → 0, we get the v-convexity of u (see Remark 8.3.6). This completesthe proof. ��

In the following result, we compare Definition 8.3.14 to the notion of H-convexfunction in the recent literature. For example, 4) in the proposition below is [DGN03,Definition 5.5], whereas 5) is [Mag06, Definition 3.4]. For the sake of simplicity, weconsider the case Ω = G. For the general case, see V. Magnani [Mag06, Proposi-tion 3.9].

Proposition 8.3.17 (Characterizations of H-convexity). Let (G, ∗) be a stratifiedgroup, and let V = (V1, . . . , Vr ) be a given stratification of g, the Lie algebra of G.For any λ > 0, let Δλ : g → g be the linear map such that (for every i = 1, . . . , r)Δλ acts on Vi as the multiplication times λi . Let δλ := Exp ◦ Δλ ◦ Log . Finally, letu : G → R be a function.

Then the following statements are equivalent:

(1) u is horizontally convex on G;(2) for every x ∈ G, X ∈ V1 and t ∈ (0, 1), it holds

u(x ∗ Exp (t X)

) ≤ (1 − t) u(x) + t u(x ∗ Exp (X)

); (8.15a)


(3) for every x ∈ G, h ∈ V and λ ∈ (0, 1), it holds

u(x ∗ δλ(h)

) ≤ (1 − λ) u(x) + λ u(x ∗ h); (8.15b)

(4) for every x ∈ G, x′ ∈ Vx and λ ∈ (0, 1), it holds

u(x ∗ δλ(x

−1 ∗ x′)) ≤ (1 − λ) u(x) + λ u(x′); (8.15c)

(5) for every x, x′ ∈ G satisfying the geometrical constraint x−1 ∗ x′ ∈ V and everyλ ∈ (0, 1), it holds (8.15c).

Proof. Obviously, (4) and (5) are equivalent, since the condition x−1 ∗ x′ ∈ V isequivalent to x′ ∈ x ∗ V = Vx .

Moreover, (3) is clearly equivalent to (4), by setting h = x−1 ∗ x′.We immediately notice that (2) is a restatement of (3), since the following facts

hold: V = Exp (V1); δλ ◦ Exp = Exp ◦ Δλ; for every X ∈ V1, we have

δλ(Exp (X)) = Exp (Δλ(X)) = Exp (λ X).

We are then left with the proof of the equivalence of (1) and (2). Suppose firstthat (1) holds. Then the function

(�) [−1, 1] t �→ f (t) := u(x ∗ Exp (t X))

is convex for every x ∈ G and every X ∈ V1. In particular, we have

u(x ∗ Exp (t X)) = f (t) = f(t · 1 + (1 − t) · 0

) ≤ t f (1) + (1 − t) f (0)

= t u(x ∗ Exp (X)) + (1 − t) u(x),

and (8.15a) follows. Vice versa, suppose that (2) holds. We have to prove that thefunction f in (�) is convex. To this end, fix t1, t2 ∈ [−1, 1] and θ ∈ [0, 1]. We haveto demonstrate that f (θ t2 + (1 − θ)t1) ≤ θ f (t2) + (1 − θ) f (t1), i.e.

u(x ∗ Exp

(θ t2 X + (1 − θ)t1 X

))

≤ θ u(x ∗ Exp (t2 X)) + (1 − θ) u(x ∗ Exp (t1 X)). (8.16)

Observe that

x ∗ Exp(θ t2 X + (1 − θ)t1 X

) = x ∗ Exp(t1 X + θ (t2 − t1)X

)

= x ∗ Exp((t1 X) � (

θ (t2 − t1)X))

= x ∗ Exp (t1 X) ∗ Exp(θ (t2 − t1)X

). (8.17)

Here we used the Campbell–Hausdorff formula Exp (A � B) = Exp (A) ∗ Exp (B)

(for every A,B ∈ g) jointly with the obvious fact (λ X) � (μ X) = λ X + μ X (forevery λ,μ ∈ R), since it holds


(λ X) � (μ X) = λ X + μ X + 1

2[λ X,μX] + · · · = λ X + μ X.

Now, taking into account (8.17) and applying (8.15a) with x, t and X respectivelyreplaced by x ∗ Exp (t1 X), θ and (t2 − t1)X (which clearly belongs to V1), we infer

u(x ∗ Exp

(θ t2 X + (1 − θ)t1 X

))

≤ (1 − θ) u(x ∗ Exp (t1 X)) + θ u(x ∗ Exp (t1 X) ∗ Exp

((t2 − t1)X

))

= (1 − θ) u(x ∗ Exp (t1 X)) + θ u(x ∗ Exp (t2 X)),

for arguing as above,

Exp (t1 X) ∗ Exp((t2 − t1)X

) = Exp((t1 X) � (

(t2 − t1)X))

= Exp((t1 X) + (t2 − t1)X

)

= Exp (t2 X).

This ends the proof. ��Note 8.3.18 (On the continuity of convex functions on Carnot groups). We recordsome references from the recent literature.

D. Danielli, N. Garofalo and D.-M. Nhieu [DGN03] proved that locally boundedH-convex functions on the Heisenberg groups are locally Lipschitz-continuous w.r.t.the quasi-distance generated by a homogeneous norm.

Z.M. Balogh and M. Rickly [BR02] have improved this result, removing thehypothesis of local bound.

For general Carnot groups, V. Magnani [Mag06] has proved that the H-convexfunctions which are locally bounded from above are locally Lipschitz-continuous (inthe above sense).

Finally, M. Sun and X. Yang [SY06] have recently proved that, in general Carnotgroups of step two, the H-convex functions are locally bounded, so that (by the citedresult in [Mag06]) they are locally Lipschitz-continuous.

Note 8.3.19 (Some other references). We collect some recent references on convex-ity for Carnot groups. Besides the already mentioned D. Danielli, N. Garofalo, D.-M. Nhieu [DGN03] and G. Lu, J.J. Manfredi and B. Stroffolini (and P. Juutinen)[LMS04,JLMS07] and the references in the previous Note 8.3.18, the reader is alsoreferred to C.E. Gutiérrez and A. Montanari [GM04a,GM04b], M. Rickly [Ric06],C. Wang [Wan05].

Remark 8.3.20 (Convexity in the homogeneous Carnot group setting). By Proposi-tion 2.2.22 on page 139, given any abstract stratified group (H, ∗) and any of itsstratifications V = (V1, . . . , Vr ), there exists a homogeneous Carnot group (G, •)

with the following properties:

(1) (G, •) is isomorphic to (H, ∗) via a Lie-group isomorphism Ψ : G → H, whence(denoted by g and h the relevant Lie algebras) dΨ : g → h is a Lie algebra iso-


morphism; Ψ is completely determined by a fixed choice of E = (E1, . . . , EN),a basis of h adapted to the stratification V ;

(2) Set Vi := (dΨ )−1(Vi) for every i = 1, . . . , r , then V = (V1, . . . , Vr ) is astratification of g; more explicitly, V1 is formed by the first m elements of theJacobian basis of g, where m = dim(V1);

(3) Denoting by Exp H : h → H, Exp G : g → G the relevant exponential maps andsetting V = Exp H(V1), V = Exp G(V1), we have Ψ (V) = V.

(4) With the above notation, V has a very explicit expression w.r.t. the usual coordi-nates on G,

V = {(h1, . . . , hm, 0, . . . , 0) : h1, . . . , hm ∈ R

}.

The statements in (3) and (4) deserve explicit proofs:(3). By Theorem 2.1.59 (page 119), we know that

(�) Exp H ◦ dΨ = Ψ ◦ Exp G,

whence

Ψ (V) = Ψ (Exp G(V1)) = Ψ(Exp G((dΨ )−1(V1))

) = Exp H(V1) = V.

(4). Let us denote by ei (i = 1, . . . , N) the i-th element of the standard basisof R

N ≡ G. Also, let us denote by Z1, . . . , ZN the Jacobian basis of g. By Proposi-tion 2.2.22-(2), we know that Exp G is a linear map and that

Exp G(Zi) = ei, dΨ (Zi) = Ei, i = 1, . . . , N.

As a consequence, it holds (use (�) and notice that V1 = span{E1, . . . , Em})V = Ψ −1(V) = Ψ −1(Exp H(V1)

) = (Exp G ◦ (dΨ )−1)(V1)

(notice that Exp G ◦ (dΨ )−1 is linear)

= span{(

Exp G ◦ (dΨ )−1)(Ei) : i = 1, . . . , m}

= span{(

Exp G(Zi) = ei : i = 1, . . . , m}

= {(h1, . . . , hm, 0, . . . , 0) : h1, . . . , hm ∈ R

}. ��

Let Ω be an open subset of H, and let us denote by

V -Convv(Ω), V -ConvH(Ω),

respectively, the v-convex functions and the horizontally convex functions on Ω

(w.r.t. the stratification V ). Analogously, if Ω is an open subset of G, we use thenotation

V -Convv(Ω), V -ConvH(Ω)


(this time w.r.t. the stratification V ). We aim to prove that the following equalitieshold:

V -Convv(Ω) = {u ◦ Ψ −1 : u ∈ V -Convv(Ψ−1(Ω))},

V -ConvH(Ω) = {u ◦ Ψ −1 : u ∈ V -ConvH(Ψ −1(Ω))},V -Convv(Ω) = {u ◦ Ψ : u ∈ V -Convv(Ψ (Ω))},V -ConvH(Ω) = {u ◦ Ψ : u ∈ V -ConvH(Ψ (Ω))}.

The first (hence the third) equality holds true, since the notion of v-convexity onlyinvolve sub-harmonicity w.r.t. the relevant sub-Laplacians (and we know that the sub-Laplacians related to V and to those related to V are Ψ -related). The fourth equality(hence the second) follows by the arguments below.

Let Ω ⊆ G be open, let x ∈ Ω and X ∈ V1 be such that

x • Exp G(t X) ∈ Ω for every t ∈ [−1, 1].This is equivalent to Ψ (x • Exp G(t X)) ∈ Ψ (Ω) for every t ∈ [−1, 1]. But we have(see (�) above)

Ψ(x • Exp G(t X)

) = Ψ (x) ∗ Ψ(Exp G(t X)

) = Ψ (x) ∗ Exp H

(t dΨ (X)

).

So, if u ∈ V -ConvH(Ψ (Ω)), by definition of H-convexity, we have that

[−1, 1] t �→ f (t) := u(Ψ (x) ∗ Exp H(t X)

)

is convex (here we let X := dΨ (X) and we noticed that, by definition of V1 =(dΨ )−1(V1), it holds X ∈ V1). On the other hand, by the above arguments,

f (t) = (u ◦ Ψ )(x • Exp G(t X)

),

and the convexity of f (jointly with the arbitrariness of X ∈ V1) proves that u ◦ Ψ ∈V -ConvH(Ω).

The above remarks demonstrate that the study of v- and H- convexity can beentirely carried out in the homogeneous setting, without lack of generality.

The (more) manageable structure of homogeneous Carnot groups (see, e.g. prop-erty (4) above) may be useful in studying convexity in a simpler way. ��Bibliographical Notes. A comprehensive introduction to the classical potential the-ory and to the study of sub-harmonicity can be found, e.g. in the following mono-graphs: D.H. Armitage and S.J. Gardiner [AG01] and in W.K. Hayman and P.B. Ken-nedy [HK76]. See also H. Aikawa and M. Essén [AE96], S. Axler, P. Bourdon,W. Ramey [ABR92], N. du Plessis [duP70], L.L. Helms [Helm69], N.S. Landkof[Lan72].

For a bibliography concerning with the notions of convexity in Carnot groups,see the references within Section 8.3.

Some of the topics presented in this chapter also appear in [BL03].



Ex. 1) Prove that if u is a continuous function which coincides with an L-sub-harmonic function almost everywhere, then u is L-subharmonic. (Hint: UseTheorem 8.2.11.)

Ex. 2) Prove that two L-superharmonic functions on an open set Ω which are equala.e. in Ω coincide everywhere. (Hint: Use Theorem 8.2.11.)

Ex. 3) Prove that LA in (8.10) is indeed a sub-Laplacian on G.Ex. 4) Let Ω be a connected bounded open set in G and let u : Ω → [−∞,∞[ be

an L-hypoharmonic function. Then u is L-subharmonic in Ω if and only ifthere exists a point x0 ∈ Ω such that

u(x0) > −∞.

Ex. 5) Show that if u ∈ S(Ω), then

u(x) = lim supΩy→x

u(y).

Ex. 6) Let F be a family of L-subharmonic functions in an open set Ω ⊆ G.Assume that

u(x) := sup{v(x) : v ∈ F}is u.s.c. and belongs to L1

loc(Ω). Then

u ∈ S(Ω).

Ex. 7) Let Γ be the fundamental solution of L, and let A be a closed subset of G.Define

u(x) := sup{Γ (y−1 ◦ x) : y ∈ A}.Then u ∈ S(G \ A) and it is locally Lipschitz-continuous with respect to d .

Ex. 8) There exists a function u ∈ S(G) ∩ C(G) such that Ut := {x : u(x) ≤ t}is compact for every t > 0 and

⋃

t>0

Ut = G

(Hint: Use suitable powers of a gauge function.)Ex. 9) Let Ω be any open set in G. Then there exists a function u ∈ S(Ω) ∩ C(Ω)

such that Ut := {x : u(x) ≤ t} is compact for every t > 0 and

⋃

t>0

Ut = Ω.

(Hint: Use the previous exercises.)


Ex. 10) Let u be an L-superharmonic function in an open set Ω ⊆ G. Show that

Mr (u)(x) < ∞for every r > 0 such that Bd(x, r) ⊆ Ω . (Hint: Mr(u)(x) < ∞ sinceu ∈ L1

loc. Then use the integral representation of Mr in terms of Mρ , andthe monotonicity of Mρ , 0 < ρ ≤ r .)

Ex. 11) Let Ω ⊆ G be open, u : Ω →]−∞,∞] be L-superharmonic and α ∈ G.Show that

x �→ u(α ◦ x)

is L-superharmonic in α−1 ◦ Ω .Ex. 12) Let Ω be a connected bounded open set in G, and let f : ∂Ω → [−∞,∞].

Assume HΩ

f �≡ ∞ and HΩf �≡ −∞. Then H

Ω

f and HΩf are L-harmonic

in Ω . (Hint: If HΩ

f �≡ ∞, then UΩ

f contains an L-superharmonic function.

Similarly, if HΩf �≡ −∞, then UΩ

f contains an L-subharmonic function.)

Ex. 13) Let T be as in (5.150) of Ex. 18, Chapter 5 (page 326), and let u ∈ S(Ω).Prove the following statements:(i) T (u) ≤ u,

(ii) T (k)(u) ≥ T (k+1)(u) for every k ∈ N, where

T (k) = T ◦ · · · ◦ T︸︷︷︸k times

.

Ex. 14) Let T be as in the previous exercise, and let u ∈ S(Ω) and v ∈ S(Ω) besuch that v ≤ u. Then:(i) v ≤ T (k)(v) ≤ T (k)(u) ≤ u for every k ∈ N,

(ii) T (k)(v) ↑ v∗, T (k)(u) ↓ u∗ for suitable v∗, u∗ ∈ L1loc(Ω). Moreover,

T (v∗) = v∗ ≤ u∗ = T (u∗).

Ex. 15) Let Ω ⊆ G be a bounded open set, and let T be as in the previous exercises.Let u ∈ S(Ω) ∩ C(Ω) and f = u|∂Ω . Then (T (k)(u))k≥1 is monotonedecreasing and

limk→∞ T (k)(u) = HΩ

f .

(Hint: If h := limk→∞ T (k)(u), then h = T (h). It follows that v ≤ h ≤ w

for every v ∈ UΩf and w ∈ UΩ

f .)

Ex. 16) We keep the notation of the previous exercise. Let F ∈ C(Ω) and f =F |∂Ω . Prove that the sequence (T (k)(F ))k≥1 is convergent and

limk→∞ T (k)(F ) = HΩ

f .

(Hint: For every n ∈ N, there exists un ∈ S(G) ∩ C(G) and vn ∈ S(G) ∩C(G) such that vn− 1

n< F < un+ 1

nin Ω . Then use the previous exercises.)


Ex. 17) Let L be a sub-Laplacian, and let d be an L-gauge. Prove that

dα ∈ S(G) ⇐⇒ α ≥ 0.

Prove also that

dα ∈ S(G) ⇐⇒ 2 − Q ≤ α ≤ 0.

(Hint: Compute L(dα) in G \ {0}. Remind that if dα ∈ S(G), then dα < ∞at any point.)

Ex. 18) Let Ω ⊆ G be open, and let u ∈ C(Ω, R). Prove the equivalence of thefollowing statements:• u ∈ S(Ω),• ALu ≥ 0 in Ω ,• ALu ≥ 0 in Ω .Here AL and AL denote the asymptotic L sub-Laplacians introduced inExercise 8 of Chapter 5.

9

Representation Theorems

The aim of this chapter is mainly to establish some representation theorems ofRiesz-type for L-subharmonic functions, being L a sub-Laplacian on a homogeneousCarnot group G.

We start by introducing the L-Green function GΩ , first for an L-regular do-main Ω , then for general open sets. In order to prove the symmetry of GΩ in thelatter case, we show that any open set can be approximated by L-regular domains.We then come to the core of the chapter, by introducing the L-Green potentials andproving several Riesz-type representation theorems for L-subharmonic functions.

In the rest of the chapter, we give some applications of the above results. InSection 9.5, we prove the Poisson–Jensen formula for L-regular domains. We shallturn back to this formula in Chapter 11, where we shall prove the Poisson–Jensenformula for arbitrary domains (see Theorem 11.7.6 on page 518), by using the theoryof L-polar sets.

In Section 9.7, we prove a monotone approximation theorem for L-subharmonicfunctions via smooth L-subharmonic functions, by using the smoothing operatorsconstructed in Section 5.6 (page 257) by superposition of surface mean operators.

In Section 9.8, we study isolated singularities for L-harmonic functions and weprove some Bôcher-type theorems. As an application, we prove that the kernel ap-pearing in the solid mean value formula of Theorem 5.6.1 (page 259) is constant onG \ {0} if and only if G is the Euclidean group.

9.1 L-Green Function for L-regular Domains

Definition 9.1.1 (L-Green function for an L-regular domain). Let Ω be a boundedL-regular open subset of G. We call L-Green function of Ω with pole at x ∈ Ω , thefunction GΩ(x, ·) : Ω →]−∞,∞] defined as follows

GΩ(x, y) := Γ (x−1 ◦ y) − hx(y),

where Γ is the fundamental solution for L, and hx denotes the solution to the bound-ary value problem

426 9 Representation Theorems{Lh = 0 in Ω ,

h(z) = Γ (x−1 ◦ z) for every z ∈ ∂Ω .

With the above definition, we have

GΩ(x, ·) is L-harmonic in Ω \ {x}, (9.1a)

GΩ(x, y) −→ 0 as y → z, for every z ∈ ∂Ω , (9.1b)

and

GΩ(x, y) = Γ (x−1 ◦ y) −∫

∂Ω

Γ (x−1 ◦ z) dμΩy (z), x, y ∈ Ω. (9.1c)

We recall that μΩy denotes the L-harmonic measure related to (the L-regular open

set) Ω and the point y (see Section 7.2, page 388).The following theorem states some other important properties of the L-Green

function.

Theorem 9.1.2. For every x, y ∈ Ω , x �= y, we have:

(i) GΩ(x, y) ≥ 0,(ii) GΩ(x, y) > 0 iff x and y belong to the same connected component of Ω ,

(iii) GΩ(x, y) = GΩ(y, x).

Proof. (i). Since Γ (x−1 ◦ z) → ∞ as z → x and hx is smooth in Ω , there existsr > 0 such that GΩ(x, z) > 0 for every z ∈ Bd(x, r). Moreover,

{LGΩ(x, ·) = 0 in Ω \ Bd(x, r),

limz→ζ GΩ(x, z) ≥ 0 for every ζ ∈ ∂(Ω \ Bd(x, r)),

so that, by the maximum principle, GΩ(x, z) ≥ 0 in Ω \Bd(x, r). Thus, GΩ(x, z) ≥0 for any z ∈ Ω . In particular, GΩ(x, y) ≥ 0.

(ii). Suppose x, y ∈ Ω0, with Ω0 ⊆ Ω open and connected. Assume by con-tradiction GΩ(x, y) = 0. Then, since GΩ(x, ·) is non-negative and L-harmonic inΩ0 \ {x}, by the strong maximum principle of Theorem 5.13.8 (page 296),

GΩ(x, z) = 0 for every z ∈ Ω0 \ {x}.This is impossible, because GΩ(x, z) → ∞ as z → x. Let us now suppose y ∈ Ω1,being Ω1 a connected component of Ω not containing x. Then z → Γ (x−1 ◦ z)

is L-harmonic in an open set containing Ω1, so that hx(z) = Γ (x−1 ◦ z) for everyz ∈ Ω1. It follows that GΩ(x, ·) = 0 in Ω1. In particular, GΩ(x, y) = 0.

(iii). Let y ∈ Ω be fixed. Denote by gy the Γ -potential of μΩy , i.e.

gy(z) :=∫

∂Ω

Γ (ζ−1 ◦ z) dμΩy (ζ ) =

∫

∂Ω

Γ (z−1 ◦ ζ ) dμΩy (ζ ), z ∈ G.

9.2 L-Green Function for General Domains 427

The function gy is L-harmonic in Ω since μΩy is supported on ∂Ω (see Theo-

rem 9.3.5, page 433). On the other hand, (9.1c) together with the positivity of GΩ ,gives

gy(z) ≤ Γ (z−1 ◦ y) ∀ z ∈ Ω.

It follows that lim supz→ζ gy(z) ≤ Γ (ζ−1 ◦ y) for every ζ ∈ ∂Ω . The definition ofhy and the maximum principle imply gy(z) ≤ hy(z) for every z ∈ Ω . In particular,gy(x) ≤ hy(x). Then

Γ (x−1 ◦ y) − GΩ(x, y) = gy(x) ≤ hy(x) = Γ (y−1 ◦ x) − GΩ(y, x),

so that, since Γ (x−1 ◦ y) = Γ (y−1 ◦ x),

GΩ(x, y) ≥ GΩ(y, x).

By interchanging the rôles of x and y, we also get GΩ(y, x) ≥ GΩ(x, y). Hence,GΩ(x, y) = GΩ(y, x). � Example 9.1.3. We know that the d-ball Bd(x, r) is an L-regular domain. SinceΓ (x−1 ◦ y) = Γ (r) if y ∈ ∂Bd(x, r), we have hx ≡ Γ (r). Then

GBd(x,r)(x, y) = Γ (x−1 ◦ y) − Γ (r). (9.2)

9.2 L-Green Function for General Domains

In this section, we extend the notion of L-Green function to general open sets.

Definition 9.2.1 (L-Green function for a general domain). Let Ω ⊆ G be open,and let x ∈ Ω . The function y → Γ (x−1 ◦ y) is L-superharmonic and non-negativein Ω . Then it has a greatest L-harmonic minorant in Ω . Let us denote it by hx . Thefunction

Ω × Ω � (x, y) → GΩ(x, y) := Γ (x−1 ◦ y) − hx(y) ∈ [0,∞]is the L-Green function for Ω .

We explicitly remark that hx and GΩ(x, ·) are L-harmonic, respectively, in Ω

and in Ω \ {x}. Moreover, GΩ(x, ·) is L-superharmonic in Ω and

hx = sup{v ∈ S(Ω)|v ≤ Γ (x−1 ◦ ·)}.As a consequence, 0 ≤ hx ≤ Γ (x−1 ◦ ·) and GΩ ≥ 0. For future references it isworth stating the following proposition.

Proposition 9.2.2. Let x ∈ Ω , and let v ∈ S(Ω) be such that

v ≤ GΩ(x, ·) in Ω.

Then v ≤ 0. Hence, the null function is the greatest L-harmonic minorant of thefunction Ω � x → GΩ(x, ·).

428 9 Representation Theorems

Proof. The hypothesis implies v +hx ≤ Γ (x−1 ◦ ·). Then, since v+hx ∈ S(Ω), weinfer v + hx ≤ hx , that is v ≤ 0. The second part of the proposition trivially followsfrom the first one. � Example 9.2.3. The L-Green function for G is

GG(x, y) = Γ (x−1 ◦ y), x, y ∈ G.

Indeed, since 0 ≤ hx ≤ Γ (x−1 ◦ ·) and Γ (x−1 ◦ y) → 0 as y → ∞, hx ≡ 0 andGG(x, ·) = Γ (x−1 ◦ ·).

When Ω is bounded, GΩ can be expressed in terms of the PWB operator. Indeed,the following theorem holds.

Theorem 9.2.4. Let Ω ⊆ G be open and bounded. Then, for every x ∈ Ω , thegreatest L-harmonic minorant in Ω of the map x → Γ (x−1 ◦y) is the PWB solutionto the Dirichlet problem {Lh = 0 in Ω,

h|∂Ω = Γ (x−1 ◦ ·).Remark 9.2.5. From this theorem it follows that

GΩ(x, y) = Γ (x−1 ◦ y) −∫

∂Ω

Γ(x−1 ◦ z

)dμΩ

y (z), x, y ∈ Ω, (9.3)

where, as usual, μΩy denotes the L-harmonic measure related to Ω and y. When Ω

is L-regular, this formula gives back (9.1c) of Section 9.1.

Proof (of Theorem 9.2.4). Let x ∈ Ω be fixed, and let ϕ := Γ (x−1 ◦ ·)|∂Ω . Letv ∈ S(Ω). From the maximum principle in Theorem 8.2.19 (page 409) we obtain

lim sup∂Ω

v ≤ ϕ iff v ≤ Γ (x−1 ◦ ·) in Ω.

Then, since ϕ is resolutive,

hx = sup{v ∈ S(Ω) : v ≤ Γ (x−1 ◦ ·)} = supUΩϕ = HΩ

ϕ = HΩϕ ,

as we aimed to prove. � The L-Green function GΩ is non-negative in Ω × Ω and such that, for every

fixed x ∈ Ω , GΩ(x, ·) is the sum of Γ (x−1 ◦ ·) plus an L-harmonic function on Ω .We now show that Γ (x−1 ◦ ·) does not exceed any other function sharing the sameproperty.

Proposition 9.2.6. Let x ∈ Ω , and let u ∈ S(Ω), u ≥ 0, be such that

u = Γ (x−1 ◦ ·) + v

with v ∈ S(Ω). Thenu ≥ GΩ(x, ·).


Proof. The condition u ≥ 0 implies −v ≤ Γ (x−1 ◦ ·), so that (since −v ∈ S(Ω))−v ≤ hx . Hence Γ (x−1 ◦ ·) − u ≤ hx , and the assertion follows. � Corollary 9.2.7. Let Ω1 ⊆ Ω2 ⊆ G. Then

GΩ1 ≤ GΩ2 in Ω1 × Ω1.

Proof. Let x ∈ Ω1. The function GΩ2(x, ·)|Ω1 is L-superharmonic and non-negativein Ω1 and is the sum of Γ (x−1 ◦ ·) plus an L-harmonic function. Proposition 9.2.6implies GΩ1(x, ·) ≤ GΩ2(x, ·)|Ω1 . �

The following approximation theorem also holds.

Theorem 9.2.8. Let (Ωn)n∈N be a monotone increasing sequence of open sets, andlet

Ω :=⋃

n∈N

Ωn.

Then1

limn→∞ GΩn = GΩ. (9.4)

Proof. Since Ωn ⊆ Ωn+1 ⊆ Ω , Corollary 9.2.7 gives

GΩn ≤ GΩn+1 ≤ GΩ.

Hence the limit in (9.4) exists and is ≤ GΩ . To prove the opposite inequality, we fixx ∈ Ω and consider n ∈ N such that Ωn � x. Then

Γ (x−1 ◦ ·) − GΩn(x, ·) = hn, hn := hΩn,x

andhn ≥ hn+1 ≥ 0 in Ωn.

By the Harnack convergence Theorem 5.7.10, page 268, there exists a L-harmonicfunction h ∈ H(Ω) such that hn ↓ h. It follows that

U := Γ (x−1 ◦ ·) − h

is non-negative in Ω since Γ (x−1 ◦ ·) − hn ≥ 0 in Ωn. Then, by Proposition 9.2.6,

GΩ(x, ·) ≤ U = limn→∞ GΩn(x, ·).

This completes the proof. � 1 We agree that limn→∞ GΩn

= GΩ means

x, y ∈ Ωp ⇒ limn→∞ GΩp+n

(x, y) = GΩ(x, y).


In order to prove the symmetry of the L-Green function for general domains, weneed a result which is of independent interest.

Lemma 9.2.9 (Approximation from the inside by L-regular sets). Given an openset Ω ⊆ G, there exists a monotone increasing sequence of bounded L-regular opensets (Ωn)n∈N such that ⋃

n∈N

Ωn = Ω.

Proof. We first assume that Ω is bounded. For every n ∈ N, let us cover ∂Ω by afinite family of L-gauge balls

{Bd(xn

j , rnj )

}j=1,...,pn

with 0 < rnj < 1/n.

We choose the balls in a such a way that

Bd(xn+1j , rn+1

j ) ⊆pn⋃

i=1

Bd(xni , rn

i ).

Then, defining

Ωn := Ω \pn⋃

i=1

Bd(xni , rn

i ),

we have Ωn ⊆ Ωn+1 and⋃

n Ωn = Ω . The open sets Ωn are L-regular. Indeed, ifz0 ∈ ∂Ωn, there exists j ∈ {1, . . . , pn} such that z0 ∈ ∂Bd(xn

j , rnj ). The function

w = Γ (rnj ) − Γ

((xn

j )−1 ◦ ·)

is L-harmonic and strictly positive in the complement of Bd(xnj , rn

j ), hence in Ωn.Moreover, limz→z0 w(z) = 0. Thus w is an L-barrier for Ωn at z0, i.e. z0 is L-regularfor Ωn.

Let us now suppose that Ω is unbounded and put

On := Ω ∩ Bd(0, n), n ∈ N.

Since On is bounded, we can find an increasing sequence of bounded L-regular opensets (Ok

n)k∈N such that⋃

k∈NOk

n = On. Using the first part of the proof, we canchoose Ok

n such that Okn ⊆ Ok

n+1 for every n and for every k. It follows that Okn ⊆

Omm if k, n ≤ m. Let us now put (see also Fig. 9.1)

Ωn = Onn .

Then Ωn is bounded and L-regular. Moreover,

Ω =⋃

n

On =⋃

n

( ⋃

k

Okn

)⊆

⋃

n

Onn ⊆ Ω.

Hence Ω = ⋃n Ωn, and the proof is complete. �


Fig. 9.1. Approximation of an open set by L-regular open sets

This lemma, together with Theorem 9.1.2, page 426, immediately proves thefollowing proposition.

Proposition 9.2.10 (Symmetry of the L-Green function). Let Ω ⊆ G be an openset. Then

GΩ(x, y) = GΩ(y, x) for every x, y ∈ Ω .

In particular, for every fixed y ∈ Ω , the function

x → GΩ(x, y)

is L-harmonic in Ω \ {y}.Remark 9.2.11. From (9.3) and the above Proposition 9.2.10 we immediately get thefollowing formula:

GΩ(x, y) = Γ (y−1 ◦ x) −∫

∂Ω

Γ (y−1 ◦ η) dμΩx (η) ∀ x, y ∈ Ω.

By collecting together some of the results of this section, we have the followingcharacterizations of the L-Green function, which we state for future reference andfor the reading convenience.

Proposition 9.2.12. Let Ω ⊆ G be open, and let x ∈ Ω be fixed. Let us denote byhx the greatest L-harmonic minorant of Γ (x−1 ◦ ·) in Ω . Then the following factshold:

(i) The L-Green function GΩ(x, y) = Γ (x−1 ◦y)−hx(y) is a symmetric function,i.e. GΩ(x, y) = GΩ(y, x) for all x, y ∈ Ω . Moreover, GΩ is continuous (in theextended sense) on Ω × Ω , and the greatest L-harmonic minorant of GΩ(x, ·)in Ω is the null function.

(ii) It holdshx = sup

{u ∈ S(Ω) : u ≤ Γ (x−1 ◦ ·) on Ω

}.


(iii) If Ω is a bounded domain, then hx = HΩΓ (x−1◦·), in the sense of Perron-Wiener-

Brelot, or equivalently

hx = sup{u ∈ S(Ω) : lim sup

z→ζ

u(z) ≤ Γ (x−1 ◦ ζ ) for all ζ ∈ ∂Ω}.

(iv) If Ω is a L-regular domain, then hx is the solution (in the classical sense) to

Lu = 0 in Ω and u = Γ (x−1 ◦ ·) on ∂Ω. (9.5)

(v) An equivalent definition of the L-Green function is the following one:GΩ is a non-negative function on Ω × Ω such that (for every x ∈ Ω) thefunction GΩ(x, ·) is the sum of Γ (x−1◦·) plus a L-harmonic function on Ω and,moreover, GΩ(x, ·) does not exceed any other non-negative L-superharmonicfunction on Ω which is the sum of Γ (x−1 ◦ ·) plus a L-superharmonic functionon Ω .

9.3 Potentials of Radon MeasuresDefinition 9.3.1 (GΩ -potential of a Radon measure). Let Ω ⊆ G be open and letGΩ be its L-Green function. Let μ be a Radon measure in Ω . The function

GΩ ∗ μ : Ω → [0,∞], (GΩ ∗ μ)(x) :=∫

Ω

GΩ(x, y) dμ(y)

is well defined and l.s.c. It is called the GΩ -potential of μ.

The following theorem holds.

Theorem 9.3.2 (Subharmonicity of a GΩ -potential). Suppose Ω is connected.Then GΩ ∗ μ ∈ S(Ω) if and only if there exists x0 ∈ Ω such that (GΩ ∗ μ)(x0) <

∞.

Proof. The “only if” part is trivial: actually, if GΩ ∗ μ ∈ S(Ω), then

GΩ ∗ μ < ∞ in a dense subset of Ω .

To prove the “if” part, by Corollary 8.2.3 (page 402), it is enough to check thatGΩ ∗ μ is super-mean. Given x ∈ Ω and r > 0 such that Bd(x, r) ⊆ Ω , we have

Mr (GΩ ∗ μ)(x) = md

rQ

∫

Bd(x,r)

ΨL(x−1 ◦ y) (GΩ ∗ μ)(y) dy

= md

rQ

∫

Bd(x,r)

ΨL(x−1 ◦ y)

( ∫

Ω

GΩ(y, z) dμ(z)

)dy

= md

rQ

∫

Ω

( ∫

Bd(x,r)

ΨL(x−1 ◦ y)GΩ(y, z) dy

)dμ(z)

(since y → GΩ(y, z) is L-superharmonic,

hence super-mean, see Example 7.2.7, page 390)

≤∫

Ω

GΩ(x, z) dμ(z) = (GΩ ∗ μ)(x).

Then GΩ ∗ μ is super-mean, and the proof is complete. �

9.3 Potentials of Radon Measures 433

Corollary 9.3.3. Let μ be a Radon measure in Ω such that

μ(Ω) < ∞.

Then GΩ ∗ μ ∈ S(Ω).

Proof. Let Bd(x0, r) ⊆ Ω . Since∫

Bd(x0,r)

(GΩ ∗ μ)(x) dx =∫

Ω

( ∫

Bd(x0,r)

GΩ(x, y) dx

)dμ(y)

≤ μ(Ω) supy∈Ω

∫

Bd(x0,r)

GΩ(x, y) dx < ∞,

then GΩ ∗ μ < ∞ almost everywhere in Bd(x0, r). Then the assertion follows fromthe previous theorem. �

By using the Harnack inequality, Corollary 9.3.3 can be improved as follows.

Corollary 9.3.4. Let Ω ⊆ G be open and connected, and let μ be a Radon measurein Ω such that K := supp(μ) is a compact subset of Ω . For every fixed y0 ∈ K andevery bounded and connected open set U such that

K ⊆ U, U ⊆ Ω,

there exists a positive constant C = C(U, y0) such that

(GΩ ∗ μ)(x) ≤ C GΩ(x, y0) ∀ x ∈ Ω \ U.

Proof. For every x ∈ Ω \ U , the function y → GΩ(x, y) is L-harmonic and non-negative in U . Then, by the Harnack inequality (Corollary 5.7.3, page 265), thereexists a positive constant C independent of x, such that GΩ(x, y) ≤ C GΩ(x, y0)

for every y ∈ K . As a consequence,

(GΩ ∗ μ)(x) =∫

K

GΩ(x, y) dμ(y) ≤ C μ(K)GΩ(x, y0)

for every x ∈ Ω \ U . � Theorem 9.3.5 (Sub-Laplacian in D′ of a GΩ -potential). Under the hypothesisof Theorem 9.3.2, we have

L(GΩ ∗ μ) = −μ in the weak sense of distributions.

In particular, GΩ ∗ μ is L-harmonic in Ω \ supp(μ).

Proof. By Theorem 9.3.2 and Corollary 8.2.4 (page 402), GΩ ∗ μ ∈ L1loc(Ω). More-

over, for every ϕ ∈ C∞0 (Ω),


∫

Ω

(GΩ ∗ μ)(x)Lϕ(x) dx =∫

Ω

( ∫

Ω

(Γ (y−1 ◦ x) − hy(x))Lϕ(x) dx

)dμ(y)

(since Lhy = 0) =∫

Ω

( ∫

Ω

Γ (y−1 ◦ x)Lϕ(x) dx

)dμ(y)

(by Theorem 5.3.3) = −∫

Ω

ϕ(y) dμ(y).

This proves the first part of the theorem. The second one follows from the hypoellip-ticity of L. �

From this theorem and Corollary 9.3.4 we obtain a corollary that will be usedvery soon.

Corollary 9.3.6. Let μ be a compactly supported Radon measure in Ω . Then:

(i) GΩ ∗ μ ∈ S(Ω),(ii) GΩ ∗ μ is L-harmonic in Ω \ supp(μ),

(iii) if v ∈ S(Ω) and v ≤ GΩ ∗ μ, then v ≤ 0.

Proof. (i) and (ii) directly follow from Corollary 9.3.3 and Theorem 9.3.5. Toprove (iii), we assume that Ω is connected (this is not restrictive) and use Corol-lary 9.3.4. First of all, if v ∈ S(Ω) and v ≤ GΩ ∗ μ, there exists h ∈ H(Ω) suchthat v ≤ h ≤ GΩ ∗ μ (h is the greatest L-harmonic minorant of GΩ ∗ μ). For afixed y0 ∈ K := supp(μ) and a connected bounded open set U ⊇ K , U ⊆ Ω , wehave

h(x) ≤ (GΩ ∗ μ)(x) ≤ GΩ(x, y0) ∀ x ∈ Ω \ U.

Since GΩ(·, y0) is L-harmonic, hence continuous, in Ω \ {y0}, we have

limU�x→ξ

(GΩ(x, y0) − h(x)) = GΩ(ξ, y0) − h(ξ) ≥ 0

for every ξ ∈ ∂U . Moreover,

limU\{y0}�x→y0

(GΩ(x, y0) − h(x)) = ∞.

Then, by the maximum principle, GΩ(·, y0) ≥ h in U . Summing up:

h ≤ GΩ(·, y0) in Ω

so that, by the very definition of GΩ , we get h ≤ 0. Hence v ≤ 0, and the proof iscomplete. �

If a GΩ -potential is L-superharmonic, then its L-harmonic minorants are non-positive constant functions. Indeed, the following theorem holds.

Theorem 9.3.7 (L-harmonic minorants of a GΩ -potential). Let μ be a Radonmeasure in an open and connected set Ω such that (GΩ ∗ μ)(x0) < ∞ for somex0 ∈ Ω . Let h be a L-harmonic function in Ω such that

h(x) ≤ (GΩ ∗ μ)(x) ∀ x ∈ Ω. (9.6)

Then h ≤ 0.


Proof. Let {Kn} be a sequence of compact subsets of Ω such that

Kn ⊆ Kn+1,⋃

n

Kn = Ω.

For every n ∈ N, we have

h ≤ GΩ ∗ μ = GΩ ∗ (μ|Kn) + GΩ ∗ (μ|Ω\Kn) =: vn + wn.

The functions vn and wn are non-negative and L-superharmonic in Ω (see Theo-rem 9.3.2). Moreover, the greatest L-harmonic minorant of vn is the zero function(see Corollary 9.3.6). Then, by Proposition 6.6.3 (page 358), h is less than the great-est L-harmonic minorant of wn. In particular,

h ≤ wn in Ω ∀ n ∈ N. (9.7)

On the other hand, by the monotone convergence theorem, we infer

vn ↑ GΩ ∗ μ,

so thatwn = GΩ ∗ μ − vn ↓ 0, as n → ∞.

This, together with (9.7), implies h ≤ 0 and completes the proof. �

9.3.1 The Potentials Related to the Average Operators for L

This section provides detailed computations of the L-potentials of the measures nat-urally related to the mean integral operators Mr and Mr introduced in Sections 5.5and 5.6 (pages 251 and 257, respectively, see in particular Theorems 5.5.4 and 5.6.1).For the sake of brevity, we rewrite our mean value formulas from those sections witha different notation.

Indeed, in order to normalize the mean value formulas, we make the follow-ing choice: fixed the sub-Laplacian L and its fundamental solution Γ , we taked := Γ 1/(2−Q). Then, d is an L-gauge (see Proposition 5.4.2, page 247), and byTheorem 5.5.6, page 256, the constant βd appearing in the surface mean value for-mula (see (5.43n), page 256) equals 1. Moreover, it is immediately seen that withthis choice of d , for every y ∈ ∂Bd(x, r), we have

(Q − 2) βd

rQ−1KL(x, y) = |∇LΓ |2(x−1 ◦ y)

|∇Γ (x−1 ◦ ·)∣∣(y).

As a consequence, the mean value formulas rewrite as follows:

If u ∈ C2(Ω), d := Γ 1/(2−Q) (where Γ is the fundamental solution for L) andBd(x, r) ⊂ Ω , it holds


u(x) = mr [u](x) −∫

Bd(x,r)

{Γ (x−1 ◦ y) − r2−Q

}Lu(y) dHN(y),

u(x) = Mr [u](x)

− Q

rQ

∫ r

0ρQ−1

{∫

Bd(x,ρ)

{Γ (x−1 ◦ y) − ρ2−Q

}Lu(y) dHN(y)

}dρ,

where k(x, y) := |∇LΓ |2(x−1 ◦ y)/|∇Γ (x−1 ◦ ·)∣∣(y), K := |∇Ld|2, and

mr [u](x) =∫

∂Bd(x,r)

u(y) k(x, y) dHN−1(y),

(9.8)Mr [u](x) = Q(Q − 2)

rQ

∫

Bd(x,r)

u(y) K(x−1 ◦ y) dHN(y).

We next give the notation which will be used in this section. If x ∈ G and r > 0are fixed, we introduce the measures λx,r and Λx,r defined by

dλx,r (y) := χ∂Bd(x,r)(y) k(x, y) dHN−1(y),

(9.9)

dΛx,r (y) := Q(Q − 2)

rQχBd(x,r)(y) K(x, y) dHN(y),

with k and K as above. We explicitly point out that, with this notations,

mr [u](x) =∫

u dλx,r and Mr [u](x) =∫

u dΛx,r .

We are interested in computing the Γ -potentials Γ ∗λx,r and Γ ∗Λx,r . We first needtwo lemmas.

Lemma 9.3.8. With the above notation, λx,r is equal to μBd(x,r)x (the L-harmonic

measure of Bd(x, r) at x) prolonged to zero outside ∂Bd(x, r).

Proof. This is just Theorem 7.2.9, page 391. � Lemma 9.3.9. Let Ω be L-regular and such that ∂Ω has vanishing Lebesgue mea-sure. Then, for every x ∈ Ω , we have

(Γ ∗ μΩx )(z) =

{Γ (z−1 ◦ x) if z /∈ Ω ,

HΩΓ (z−1◦·)(x) if z ∈ Ω ,

where μΩx denotes the measure obtained by prolonging μΩ

x to 0 outside ∂Ω .

The same assertion holds for a general L-regular set (without the assumptionthat ∂Ω has vanishing Lebesgue measure). A proof in this case may be obtained byarguing as in (9.19) in the proof of Theorem 9.5.1 (page 445).


Proof. Let x ∈ Ω be fixed. With the notation in the assertion, for any z ∈ G we aimto consider

(Γ ∗ μΩx )(z) =

∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y).

If z /∈ Ω , the map y → Γ (z−1 ◦ y) is L-harmonic about Ω . Hence, by the definitionof L-harmonic measure, we have

Γ (z−1 ◦ x) =∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y).

Let now z ∈ Ω . Then the map y → Γ (z−1◦y) is continuous on ∂Ω , and the equalityof ∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y) = HΩ

Γ (z−1◦·)(x)

is the very definition of the latter.We end by considering a fixed z0 ∈ ∂Ω . We set J = Γ ∗ μΩ

x . Since J ∈ S(G),by Theorem 8.2.10 (page 404) we have

Mr [J ](z0) −→ J (z0) as r → 0.

Since the function

GΩ(z, x) = Γ (z−1 ◦ x) − HΩΓ (z−1◦·)(x)

and Γ are symmetric functions, we have

HΩΓ (z−1◦·)(x) = HΩ

Γ (x−1◦·)(z), whenever x, z ∈ Ω .

Then, since Bd(z0, r) ∩ ∂Ω has zero HN -measure, we have (by applying what weproved above)

Mr [J ](z0) = Mr [JχΩ ](z0) + Mr [JχG\Ω ](z0) = Mr

[HΩ

Γ (x−1◦·) χΩ

](z0)

+ Mr [Γ (x−1 ◦ ·) χG\Ω ](z0)

= Mr [Γ (x−1 ◦ ·)](z0) − Mr [GΩ(x, ·)χΩ ](z0).

Clearly, we have

Mr [Γ (x−1 ◦ ·)](z0) −→ Γ (x−1 ◦ z0) = Γ (z−10 ◦ x) as r → 0.

On the other hand, since GΩ(x, z) vanishes with continuity as Ω � z → z0 ∈ ∂Ω ,and Mr [1] ≡ 1, we obtain

Mr [GΩ(x, ·)χΩ ](z0) −→ 0 as r → 0.



Theorem 9.3.10. For every x ∈ G and r > 0, we have (see Fig. 9.2)

(Γ ∗ λx,r )(z) = mr [Γ (z−1 ◦ ·)](x)

={

Γ (z−1 ◦ x) if z /∈ Bd(x, r),

r2−Q if z ∈ Bd(x, r).(9.10a)

(Γ ∗ Λx,r )(z) = Mr [Γ (z−1 ◦ ·)](x)

=⎧⎨

⎩

Γ (z−1 ◦ x) if z /∈ Bd(x, r),

2 − Q

2

d2(x, z)

rQ+ Q

2r2−Q if z ∈ Bd(x, r).

(9.10b)

In particular, the Γ -potentials Γ ∗ λx,r , Γ ∗ Λx,r are continuous on G. Moreover,

(Γ ∗ λx,r )(z), mr [Γ (z−1 ◦ ·)](x), (Γ ∗ Λx,r )(z), Mr [Γ (z−1 ◦ ·)](x)

are symmetric functions in x and z.

Fig. 9.2. The functions in Theorem 9.3.10

Proof. By Lemma 9.3.8, λx,r is the trivial prolongation of μBd(x,r)x outside ∂Bd(x, r).

Then, we can apply Lemma 9.3.9 for Ω = Bd(x, r) in order to compute Γ ∗ λx,r .We have to prove that

HBd(x,r)

Γ (z−1◦·)(x) = r2−Q for every z ∈ Bd(x, r).

By the symmetry of the L-Green function for Bd(x, r), we have

HBd(x,r)

Γ (z−1◦·)(x) = HBd(x,r)

Γ (x−1◦·)(z).


Now, the function of z in the above right-hand side is the L-harmonic functionon Bd(x, r) which equals Γ (x−1 ◦ ·) on ∂Bd(x, r). Since Γ (x−1 ◦ ·) ≡ r2−Q on∂Bd(x, r) (and the constants are L-harmonic), (9.10a) follows.

Finally, we have

(Γ ∗ Λx,r )(z) = Mr [Γ (z−1 ◦ ·)](x) = Q

rQ

∫ r

0ρQ−1 mρ[Γ (z−1 ◦ ·)](x) dρ.

If z /∈ Bd(x, r), then z /∈ Bd(x, ρ) also for any 0 < ρ < r , whence, by (9.10a),

(Γ ∗ Λx,r )(z) = Q

rQ

∫ r

0ρQ−1 Γ (z−1 ◦ x) dρ = Γ (z−1 ◦ x).

On the other hand, suppose z ∈ Bd(x, r). Arguing as above, we have

Q

rQ

∫ r

0ρQ−1 mρ[Γ (z−1 ◦ ·)](x) dρ

= Q

rQ

∫ d(x,z)

0ρQ−1 Γ (z−1 ◦ x) dρ + Q

rQ

∫ r

d(x,z)

ρQ−1 ρ2−Q dρ

= Γ (z−1 ◦ x)dQ(x, z)

rQ+ Q

2 rQ(r2 − d2(x, z)).

Thus, (9.10b) is proved. The last assertion of the theorem now follows from (9.10a)and (9.10b), since Γ (z−1 ◦ x) = Γ (x−1 ◦ z) and d(x, z) = d(z, x). �

For an application of Theorem 9.3.10, see Exercise 5 at the end of the chapter.The next result gives another consequence of Theorem 9.3.10. Indeed, we prove

that the mean integrals of a Γ -potential are L-superharmonic continuous functions.

Theorem 9.3.11. Let μ be a Radon measure in G and r > 0. We suppose that Γ ∗μ �≡ ∞. Then the map x → mr [Γ ∗ μ](x) belongs to S(G) ∩ C(G, R). Moreover,the map

(x, r) → mr [Γ ∗ μ](x)

is lower semicontinuous. The same assertion holds for Mr [Γ ∗ μ].Proof. (i). First we prove that mr [u] and Mr [u] are finite-valued and continuous forevery u ∈ S(G). We fix x ∈ G. Then

|Mr [u](x)| ≤ ‖K‖∞∫

Bd(x,r)

|u(y)| dHN(y) < ∞,

since u ∈ L1loc(G). Let R > r and set Ω = Bd(x, R). Set also (see Definition 9.4.1

in the next section)μ := (μ[u])|Ω.


Then, by Theorem 9.3.5, we have L(u−Γ ∗ μ) = 0 in the distributional sense in Ω ,so that2 (by the hypoellipticity of L) u = h + Γ ∗ μ on Ω for a suitable h ∈ H(Ω).Hence (being ∂Bd(x, r) ⊂ Bd(x, R))

mr [u](x) = mr [h](x) + mr [Γ ∗ μ](x) = h(x) +∫

Ω

mr [Γ (z−1 ◦ ·)](x) dμ(z)

(by (9.10a)) = h(x) +∫

Ω

min{Γ (z−1 ◦ x), r2−Q} dμ(z),

which is finite, since μ is a Radon measure and Ω is bounded. Moreover, the aboverepresentation of mr [u](x) also proves that mr [u] is continuous. Another argumentof dominated convergence then shows that Mr [u](x) = Q

rQ

∫ r

0 ρQ−1 mρ[u](x) dρ iscontinuous at x.

(ii). We now observe a simple fact (see also Theorem 8.2.20, page 410): let(A, λ) be an arbitrary measure space, and let {fx}x∈G be a family of non-negative λ-measurable functions on A such that, for every fixed z ∈ A, the function x → fx(z)

is in S(G). Then the integral function

F(x) =∫

A

fx(z) dλ(z)

is in S(G) too (unless it is ≡ ∞). Indeed, by Theorem 8.1.3, it is enough to provethat F is L-supermean. By Tonelli’s theorem we have

mr [F ](x) =∫

A

mr [y → fy(z)](x) dλ(z) ≤∫

A

fx(z) dλ(z) = F(x)

(since y → fy(z) is L-supermean), i.e. F is L-supermean.(iii). Tonelli’s theorem also ensures that

mr [Γ ∗ μ](x) =∫

G

mr [Γ (z−1 ◦ ·)](x) dμ(z)

and Mr [Γ ∗μ](x) = ∫G

Mr [Γ (z−1 ◦·)](x) dμ(z). Consequently, by (ii), the theoremis proved if we show that the maps

x → mr [Γ (z−1 ◦ ·)](x), Mr [Γ (z−1 ◦ ·)](x)

are in S(G) for any z ∈ G.(iv). Now, the L-superharmonicity of mr [Γ (z−1 ◦ ·)] follows from (9.10a). Fi-

nally, we have

Mr [Γ (z−1 ◦ ·)](x) = Q

rQ

∫ r

0ρQ−1 mρ[Γ (z−1 ◦ ·)](x) dρ.

Consequently, the L-superharmonicity of Mr [Γ (z−1 ◦ ·)] follows by an applicationof (ii) for A = [0, r] and dλ(ρ) = Q

rQ ρQ−1 dρ, and making use of what we provedat the beginning of step (iv). The semicontinuity of (x, r) → mr [Γ ∗ μ](x) is astandard consequence of Fatou’s lemma. �

2 The reader has certainly realized that we have just proved a Riesz representation type result:we investigate the topic in details in Section 9.4.

9.4 Riesz Representation Theorems for L-subharmonic Functions 441

Corollary 9.3.12. Let u ∈ S(G) and r > 0. Then mr [u], Mr [u] belong to S(G) ∩C(G, R).

Proof. See (i) in the proof of Theorem 9.3.11. �

9.4 Riesz Representation Theorems for L-subharmonicFunctions

Definition 9.4.1 (L-Riesz measure). Let u be an L-subharmonic function in an openset Ω ⊆ G. By Theorem 8.2.11 (page 405), u ∈ L1

loc(Ω) and Lu ≥ 0 in the weaksense of distributions. Then3 there exists a Radon measure μ in Ω such that

Lu = μ (9.11)

in the weak sense of distributions. The measure μ will be called the L-Riesz mea-sure of u. If u is L-superharmonic in Ω , the L-Riesz measure related to −u will bereferred to as the L-Riesz measure of u. In this case, it holds Lu = −μ, in the weaksense of distributions.

With reference to the above definition, we shall sometimes also write μ[u] or μu

instead of μ.

Example 9.4.2. If u = Γ , then μ[Γ ] = Dirac0, the Dirac mass supported at {0}.Indeed, L(−Γ ) = Dirac0, for (see (5.14), page 236)

∫

G

(−Γ )Lϕ = ϕ(0) =∫

G

Dirac0 ϕ ∀ϕ ∈ C∞0 (G).

When μ[u] is compactly supported, a representation theorem easily follows.

Theorem 9.4.3 (Riesz representation. I). Let Ω ⊆ G be open, and let u ∈ S(Ω).Let μ be the L-Riesz measure of u. Assume that supp(μ) is a compact subset of Ω .Then there exists an L-harmonic function h in Ω satisfying the identity

u = h − GΩ ∗ μ in Ω. (9.12)

Proof. By Corollary 9.3.3 and Theorem 9.3.5, v := GΩ ∗ μ is L-superharmonicin Ω , and Lv = −μ in the weak sense of distributions. It follows that L(u + v) = 0in Ω in the weak sense of distributions. Since L is hypoelliptic, there exists a func-tion h, L-harmonic in Ω , such that h(x) = u(x)+ v(x) almost everywhere in Ω . Asa consequence, for every x ∈ Ω and for every r < distd(x, ∂Ω)

3 Here we used the following result. Given a linear map L : C∞0 (Ω) → R such that L(ϕ) ≥

0 whenever ϕ ≥ 0, there exists a Radon measure μ on Ω such that L(ϕ) = ∫ϕ dμ for

every ϕ ∈ C∞0 (Ω). For a proof of this result, it suffices to rerun the proof of the classical

Riesz representation theorem of positive functionals on C0 as presented, e.g. in [Rud87].


Mr (u)(x) = −Mr (v)(x) + h(x),

where Mr is the solid average operator (5.50f) (page 259). Here we used theL-harmonicity of h to write h in place of Mr (h). Letting r tend to zero in the lastidentity and using Theorem 8.2.11-(ii) (page 405), we get

u(x) = −v(x) + h(x) ∀ x ∈ Ω.

This completes the proof. � For the future reference, we explicitly show the following theorem, which can be

proved as Theorem 9.4.3.

Theorem 9.4.4 (Riesz representation. II). Let Ω ⊆ G be open, and let u ∈ S(Ω).Let μ be the L-Riesz measure of u. Then, for every bounded open set Ω1 such thatΩ1 ⊂ Ω , there exists a function h, L-harmonic in Ω1, satisfying the identity

u(x) = −∫

Ω1

Γ (y−1 ◦ x) dμ(y) + h(x) ∀ x ∈ Ω1. (9.13)

Proof. The function

v(x) := −∫

Ω1

Γ (y−1 ◦ x) dμ(y), x ∈ G,

is L-subharmonic in G and satisfies Lv = μ|Ω1in the weak sense of distributions.

Therefore, L(u − v) = 0 in D′(Ω1). Then, just proceeding as in the proof of theprevious theorem, we show the existence of an L-harmonic function in Ω1 such thatu = v + h in Ω1. � Lemma 9.4.5. Let u ∈ S(Ω), let μ be its L-Riesz measure and let K ⊆ Ω becompact. There exists w ∈ S(Ω) such that

u = GΩ ∗ (μ|K) + w in Ω.

Proof. Let H ⊆ Ω be compact and such that K ⊆ Int(H). Let k ∈ H(Int(K)) andh ∈ H(Int(H)) be such that

u = h + GΩ ∗ (μ|H ) in Int(H).

Define

w : Ω →]−∞,∞], w :={

h + GΩ ∗ (μ|H\K) in Int(H),

u − GΩ ∗ (μ|K) in Ω \ K .

This definition is well-posed since the function GΩ ∗ (μ|K) is L-harmonic inInt(H) \ K , hence real-valued, and

h + GΩ ∗ (μ|H\K) = h + GΩ ∗ (μ|H ) − GΩ ∗ (μ|K) = u − GΩ ∗ (μ|K).

Moreover, w is L-superharmonic in Ω since it is L-superharmonic in Int(H) and inΩ \ K , and Ω = Int(H) ∪ (Ω \ K). �

9.4 Riesz Representation Theorems for L-subharmonic Functions 443

Theorem 9.4.6 (Superharmonicity of GΩ ∗ Lu). Let u ∈ S(Ω), and let μ be itsL-Riesz measure. Then

GΩ ∗ μ ∈ S(Ω) (9.14)

if and only if there exists v ∈ S(Ω) such that

v ≤ u.

Proof. We first prove the “if” part and assume u ≥ v in Ω with v ∈ S(Ω). Let(Kn)n∈N be an increasing sequence of compact sets such that ∪nKn = Ω . Let us putμn = μ|Kn . By Lemma 9.4.5, there exists wn ∈ S(Ω) such that u = wn +GΩ ∗ μn.The hypothesis implies 0 ≤ u − v = (wn − v) + GΩ ∗ μn in Ω , so that

GΩ ∗ μn ≥ v − wn.

By Corollary 9.3.6, we have v − wn ≤ 0 in Ω . Hence

u − v ≥ GΩ ∗ μn in Ω ∀ n ∈ N.

Letting n tend to infinity, we get u − v ≥ GΩ ∗ μ. Since u − v ∈ S(Ω), this implies(9.14). Vice versa, assume (9.14) is true. Then, by Theorem 9.3.5,

L(GΩ ∗ μ − u) = 0 in Ω , in the weak sense of distributions.

Since L is hypoelliptic, there exists a function h ∈ H(Ω) such that GΩ ∗ μ =u + h a.e. in Ω , hence everywhere (see Exercise 2 of Chapter 8, page 422). SinceGΩ ∗ μ ≥ 0, we have u ≥ −h, and the proof is complete. �

Theorem 9.4.7 (The Riesz representation). Let u ∈ S(Ω), and let μ be the L-Rieszmeasure of u. The following statements are equivalent:

(i) there exists h ∈ H(Ω) such that

u = GΩ ∗ μ + h in Ω, (9.15)

(ii) there exists v ∈ S(Ω) such that v ≤ u in Ω ,(iii) every connected component of Ω contains a point x0 such that

(GΩ ∗ μ)(x0) < ∞.

Moreover, (9.15) holds with h ∈ H(Ω) if and only if h is the greatest L-harmonicminorant of u in Ω .

Proof. (i) ⇒ (ii). If (9.15) holds, then h is an L-harmonic minorant of u (hence anL-subharmonic function), since GΩ ∗ μ ≥ 0.

(ii) ⇔ (iii). This follows from Theorems 9.4.6 and 9.3.2.(iii) ⇒ (i). It is not restrictive to assume that Ω is connected. Let x0 ∈ Ω be such

that (GΩ ∗ μ)(x0) < ∞, and let {Ωn}n∈N be a sequence of bounded open sets suchthat


Ωn ⊂ Ωn+1, Ωn+1 ⊂ Ω,⋃

n∈N

Ωn = Ω.

Defineμn := μ|Ωn

, n ∈ N.

Then, by the representation Theorem 9.4.3, there exists an L-harmonic function hn

such thatu(x) = (GΩ ∗ μn)(x) + hn(x) ∀ x ∈ Ωn, ∀ n ∈ N. (9.16)

Since GΩ ∗ μn ↗ GΩ ∗ μ, we have

hn(x) ≥ hn+k(x) ≥ u(x) − (GΩ ∗ μ)(x) ∀ x ∈ Ωn, ∀ n, k ∈ N.

On the other hand, by Theorem 9.3.2, GΩ ∗ μ ∈ S(Ω). Then, keeping in mind thatu(x) > −∞ for every x ∈ Ω , for any n ∈ N, we have

infk

hn+k > −∞ in a dense subset of Ωn.

By using Theorem 5.7.10 (page 268), we infer the existence of a function h : Ω →R, L-harmonic in Ω , such that

h(x) = limk→∞ hn+k(x) ∀ x ∈ Ωn, ∀ n ∈ N.

Then (9.15) follows from (9.16).We are left with the proof of the second part of the theorem. Assume (9.15) holds

with h ∈ H(Ω). Then, if k ∈ H(Ω) and k ≤ u, we have k − h ≤ GΩ ∗ μ, so that,by Theorem 9.3.7, we have k − h ≤ 0, i.e. k ≤ h. Vice versa, assume h is thegreatest L-harmonic minorant of u. Then, by (ii), there exists k ∈ H(Ω) such thatu = GΩ ∗ μ + k. This implies that k is the greatest L-harmonic minorant of u, i.e.k = h. Thus u = GΩ ∗ μ + h. The proof is complete. � Corollary 9.4.8 (Riesz representation in space). Let u ∈ S(G) be such that U :=sup u < ∞. Then

u = −Γ ∗ μ + U,

where μ is the L-Riesz measure of u.

Proof. Since U is an L-harmonic majorant of u, by the previous theorem we haveu = −Γ ∗ μ + h, where h is the least L-harmonic majorant of u. Then h ≤ U

so that, by the Liouville Theorem 5.8.1 (page 269) h ≡ U0, a real constant. Henceu = −Γ ∗ μ + U0. As a consequence U = sup u ≤ U0 ≤ U , that is, U = U0. Theproof is complete. �

We remark that a stronger version of the previous corollary will be provided inTheorem 9.6.1, page 451.

Corollary 9.4.9 (Riesz representation. III). Let u ∈ S(Ω) be such that Lu = 0outside a compact set K ⊂ Ω . Then there exists an L-harmonic function h in Ω

such thatu = −GΩ ∗ μ + h in Ω.

9.5 The Poisson–Jensen Formula 445

Proof. Since supp(μ) ⊆ K , we have GΩ ∗ μ ∈ S(Ω). Then GΩ ∗ μ < ∞ in adense subset of Ω , and the assertion follows from Theorem 9.4.7. �

For the future references, we explicitly write the following consequence of The-orems 9.3.2 and 9.3.5.

Corollary 9.4.10 (Riesz representation. IV). Let u ∈ S(Ω), and let μ be the L-Riesz measure of u. Assume (Γ ∗ μ)(x0) < ∞ at some point x0 ∈ G. Then thereexists an L-harmonic function h in Ω such that

u(x) = −(Γ ∗ μ)(x) + h(x) ∀ x ∈ Ω. (9.17)

Proof. It is sufficient to note that Γ ∗ μ ∈ SG and that L(u + Γ ∗ μ) = 0in Ω . �

9.5 The Poisson–Jensen Formula

The next theorem, when L = Δ is the classical Laplace operator, will give back theclassical Poisson–Jensen formula (see, e.g. [HK76, Theorem 3.14]). An improvedversion of Theorem 9.5.1 (removing the hypothesis of L-regularity of Ω) will begiven in Section 11 (see Theorem 11.7.6, page 518).

Theorem 9.5.1 (Poisson–Jensen’s formula). Let U, Ω be open subsets of G, Ω ⊂U and Ω be L-regular. Let u ∈ S(U) and μ = Lu be its L-Riesz measure. Then

u(x) =∫

∂Ω

u(y) dμΩx (y) −

∫

Ω

GΩ(y, x) dμ(y), x ∈ Ω. (9.18)

Here GΩ is the L-Green function of Ω and μΩx is the L-harmonic measure related

to Ω .

Proof. Let O be a bounded open set such that Ω ⊂ O ⊂ O ⊂ U . By the Rieszrepresentation Theorem 9.4.4, there exists an L-harmonic function h in O such that

u(x) = −∫

O

Γ (y−1 ◦ x) dμ(y) + h(x) =: v(x) + h(x) ∀ x ∈ O.

We have v ∈ S(G) and Lv = μ|O . Then, since

h(x) =∫

∂Ω

h(y) dμΩx (y) ∀ x ∈ Ω,

it suffices to prove (9.18) with u replaced by v. We can also suppose that v(x) >

−∞. Indeed, if v(x) = −∞, then u(x) = −∞ and

446 9 Representation Theorems∫

Ω

GΩ(y, x) dμ(y) ≥∫

Ω

Γ (y−1 ◦ x) dμ(y) = −u(x) − h(x) = ∞.

Moreover, since u ∈ S(U), the function x → ∫∂Ω

u(y) dμΩx (y) is L-harmonic,

hence real-valued, in Ω (see Exercise 18 of Chapter 6). Thus, in this case, (9.18)trivially holds.

Let us fix x ∈ Ω . We have∫

∂Ω

v(y) dμΩx (y) = −

∫

O

( ∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y)

)dμ(z).

The crucial part of the proof is to show that∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y)

={

Γ (z−1 ◦ x), z ∈ O \ Ω ,Γ (z−1 ◦ x) − GΩ(z, x), z ∈ Ω .

(9.19)

With (9.19) at hand, and keeping in mind the assumption v(x) = −∞, which impliesthat z → Γ (z−1 ◦ x) is μ-summable, we get the assertion. Indeed,

∫

∂Ω

v(y) dμΩx (y) = −

∫

O

Γ (z−1 ◦ x) dμ(z) +∫

Ω

GΩ(z, x) dμ(z)

= v(x) +∫

Ω

GΩ(z, x) dμ(z).

Then, it remains to prove (9.19). If z ∈ O \ Ω , the function Γ (z−1 ◦ ·) is harmonicin O \ {z}, so that

Γ (z−1 ◦ x) =∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y).

If z ∈ Ω , the function Γ (z−1 ◦ ·) is continuous in ∂Ω , hence the solution hz to theDirichlet problem {

Lh(x) = 0, x ∈ Ω ,h(y) = Γ (z−1 ◦ y), y ∈ ∂Ω

is given by

hz(x) :=∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y).

Then, by the definition of the L-Green function,∫

∂Ω

Γ (z−1 ◦ y) dμΩx (y) = hz(x) = Γ (z−1 ◦ x) − GΩ(z, x).

Finally, we fix z0 ∈ ∂Ω . Let us prove that∫

∂Ω

Γ (y−1 ◦ z0) dμΩx (y) = Γ (x−1 ◦ z0). (9.20)


Since Γ (x−1 ◦ z0) = Γ (z−10 ◦ x), this will give (9.19) in the case z0 ∈ ∂Ω . Let us

define

w(z) := Γ (x−1 ◦ z) −∫

∂Ω

Γ (y−1 ◦ z) dμΩx (y), z ∈ G \ {x}.

The function w is L-subharmonic in G \ {x} and

lim supz/∈∂Ω, z→ζ

w(z) = 0 ∀ ζ ∈ ∂Ω. (9.21)

Indeed, (9.19) holds in O \∂Ω and GΩ(z, x) → 0 as z → ζ from inside of Ω , sinceΩ is L-regular and GΩ is symmetric. In order to prove (9.20), we have to show thatw ≡ 0 on ∂Ω . First of all, we observe that

w(ζ ) ≥ lim supz→ζ

w(z) ≥ lim supz/∈∂Ω, z→ζ

w(z) = 0 ∀ ζ ∈ ∂Ω.

Suppose, by contradiction, that w > 0 somewhere in ∂Ω . Then we havemax∂Ω w > 0. From (9.21) it follows that there exists an open set V ⊆ O suchthat ∂Ω ⊂ V , x /∈ V and maxV w = max∂Ω w. Let ζ0 ∈ ∂Ω be such thatw(ζ0) = maxV w. Since w is L-subharmonic in V , by Theorems 8.1.2 and 8.2.1(pages 398 and 401), w ≡ w(ζ0) in the connected component of V containing ζ0.Thus,

lim supz/∈∂Ω, z→ζ0

w(z) = w(ζ0) = maxV

w > 0,

in contradiction with (9.21). This completes the proof of (9.19). � If, in the previous theorem, we take Ω = Bd(x, r), we obtain an extension of the

mean value formulas (5.45) and (5.51) (pages 256 and 259) to the L-subharmonicfunctions.

Theorem 9.5.2 (Mean value formulas for L-subharmonic functions). Let Ω ⊆ G

be open, and let u ∈ S(Ω). Then, for every x ∈ Ω and r > 0 such that

Bd(x, r) ⊂ Ω,

we have

u(x) = Mr (u)(x) −∫

Bd(x,r)

(Γ (x−1 ◦ y) − Γ (r)

)dμ(y) (9.22)

and

u(x) = Mr (u)(x) − Q

rQ

∫ r

0ρQ−1

( ∫

Bd(x,ρ)

(Γ (x−1 ◦ y) − Γ (ρ)) dμ(y)

)dρ,

(9.23)where μ := Lu is the L-Riesz measure of u and Γ (ρ) := βd ρ2−Q.


Proof. Take Ω = Bd(x, r) in Poisson–Jensen’s Theorem 9.5.1. By Theorem 7.2.9(page 391), we get

∫

∂Bd(x,r)

u(y) dμBd(x,r)x = Mr (u)(x).

Moreover, if GΩ denotes the L-Green function of Bd(x, r), then

GΩ(x, y) = Γ (x−1 ◦ y) − Γ (r). (9.24)

Then (9.22) follows from (9.18). Identity (9.23) follows from (9.22) keeping in mindthat

Mr (u)(x) = Q

rQ

∫ r

0ρQ−1 Mρ(u)(x) dρ.

This completes the proof. � Remark 9.5.3. In Chapter 11, Theorem 9.5.1 will be extended to arbitrary boundedopen sets Ω .

From Theorem 9.5.2 it is easy to derive the following left-continuity result of thesurface average.

Proposition 9.5.4 (Left-continuity of the surface operator). Let v ∈ S(Ω), andlet Bd(x, r) ⊆ Ω . Then

limρ↑r

Mρ(u)(x) = Mr (u)(x). (9.25)

Proof. We first assume that v(x) > −∞, and we write the Poisson–Jensen formula(9.22) as follows

Mρ(v)(x) = v(x) +∫

Bd(x,ρ)

(Γ (x−1 ◦ y) − Γ (ρ)

)dμ(y), 0 < ρ ≤ r. (9.26)

Since Bd(x, ρ) ↑ Bd(x, r) and, as ρ ↑ r ,(Γ (x−1 ◦ y) − Γ (ρ)

) ↑ (Γ (x−1 ◦ y) − Γ (r)

),

one obtains (9.25) from (9.26) by using the monotone convergence theorem.If v(x) = −∞, we replace v by its Perron-regularized w := vBd(x,r/2), and we

getw ∈ S(Ω), w(x) > −∞

andMρ(u)(x) = Mρ(w)(x)

for r2 < ρ ≤ r . Then, from what we have already proved,

limρ↗r

Mρ(u)(x) = limρ↗r

Mρ(w)(x) = Mr (w)(x) = Mr (u)(x). �


The previous proposition is the needed tool to prove the following result.

Proposition 9.5.5. Let v ∈ S(Ω), and let Bd(x0, r) ⊆ Ω . Denote B := Bd(x0, r).Then the function

h : B → R, h(x) =∫

∂B

v dμBx

is the least harmonic majorant of v on B.

Proof. Since v is bounded from above on B, by Theorem 6.6.1, v has a least har-monic majorant h0 on B. Then h0 ≤ h, since h is harmonic on B and h ≥ v|B . Onthe other hand, for 0 < ρ < r ,

h0(x0) = Mρ(h0)(x0) ≥ Mρ(u)(x0).

As ρ ↑ r , by using the previous proposition and Theorem 7.2.9, we get

h0(x0) ≥ Mr (v)(x0) = h(x0).

Therefore, h0 − h ≤ 0 in B and (h0 − h)(x0) = 0. The strong maximum principleof Theorem 8.2.19 implies h0 ≡ h. � We are now in the position to show that, in the present setting, Theorem 6.11.1 onpage 375 can be improved.

Theorem 9.5.6 (Envelopes in S(Ω)). Let Ω ⊆ G be open, and let F ⊆ S(Ω).Assume u0 := infF is locally bounded from below. Then u0 ∈ S(Ω) and

u0 = u0 a.e. in Ω.

Proof. We already know, from Theorem 6.11.1, that u0 ∈ S(Ω). Then, we onlyhave to prove that u0 = u0 a.e. in Ω . By Choquet’s lemma (Proposition 6.1.2),there exists a sequence (vn) in F such that v = u0, where v = infn∈N vn. Definingwn := min{v1, . . . , vn}, n ∈ N, we obtain a decreasing sequence {wn}n∈N of L-subharmonic functions in Ω such that wn ↓ v. For every L-gauge ball B with B ⊆Ω , we have ∫

∂B

u0 dμBy ≤

∫

∂B

wn dμBy ≤ wn(y) ∀ y ∈ B,

so that, letting n tend to infinity,∫

∂B

u0 dμBy ≤

∫

∂B

v dμBy ≤ v(y) ∀ y ∈ B,

and y → ∫∂B

v dμBy is L-harmonic in B (it is the limit of a decreasing sequence of

L-harmonic functions bounded from below by an L-harmonic function). Then theprevious inequalities extends to the following ones

∫

∂B

u0 dμBy ≤

∫

∂B

v dμBy ≤ v(y) = u0(y) ∀ y ∈ B. (9.27)


On the other hand, by Proposition 9.5.5, y → ∫∂B

u0 dμBy is the greatest harmonic

minorant of u0 in B. Thus, the first inequality in (9.27) actually is an equality, i.e.∫

∂B

u0 dμBy =

∫

∂B

v dμBy ∀ y ∈ B.

In particular, if B = Bd(x0, r), we have

Mr (u0)(x0) = Mr (v)(x0).

Since this identity holds for every gauge ball Bd(x0, r) with Bd(x0, r) ⊆ Ω , we alsohave

Mr (u0)(x0) = Mr (v)(x0)

for every x0 ∈ Ω and r > 0 such that Bd(x0, r) ⊆ Ω . Keeping in mind that u0 ≤u0 ≤ v, the assertion follows from the next lemma. � Lemma 9.5.7. Let Ω ⊆ G be open, and let f : Ω → [0,∞] be a Lebesgue measur-able function such that

Mr (f )(x0) = 0 (9.28)

for every x0 ∈ Ω and r > 0 such that Bd(x0, r) ⊆ Ω . Then f = 0 a.e. in Ω .

Proof. Let K be the kernel appearing in the average operator Mr (see (5.50f),page 259). Hypothesis (9.28) implies that f = 0 a.e. in

B+d (x0, r) := {

x ∈ Bd(x0, r) : K(x−10 ◦ x) > 0

}

for every x0 ∈ Ω and r > 0 such that Bd(x0, r) ⊆ Ω .Then, since B+

d (x0, r) is open, it is enough to show that Ω = ⋃B+

d (x0, r),where the union is taken over the family of the L-gauge balls with the closure con-tained in Ω . Let x ∈ Ω . Since K is smooth out of the origin and δλ-homogeneous ofdegree zero, there exists z ∈ G such that K(z) > 0 and

d(z) < r := 1

2distd(x, ∂Ω).

Then, setting x0 = x ◦ z−1, we have x−10 ◦ x = z, so that x ∈ Bd(x0, r) and

K(x−10 ◦ x) = K(z) > 0. This means that x ∈ B+

d (x0, r), and we are done, sinceBd(x0, r) ⊆ Ω . � Corollary 9.5.8. Let F and u0 be as in Theorem 9.5.6. Then

u0(x) = limr→0

Mr (u0)(x) for every x ∈ Ω.

Proof. From Theorem 8.2.11 we have

u0(x) = limr→0

Mr (u0)(x) for every x ∈ Ω.

The assertion follows, since u0 = u0 a.e. in Ω . �

9.6 Bounded-above L-subharmonic Functions in G 451

9.6 Bounded-above L-subharmonic Functions in GGG

This section is devoted to the proof of the following sharp Riesz representation the-orem for L-subharmonic functions bounded from above in G. We recall that we arealways assuming that G is equipped with a homogeneous Carnot group structure G

such that the homogeneous dimension Q of G is strictly greater than 2.

Theorem 9.6.1 (The L-Riesz measure of a bounded-above u ∈ S(GGG)). Let μ be aRadon measure in G, and let x0 ∈ G. Then μ is the L-Riesz measure of a bounded-above L-subharmonic function u in G with u(x0) > −∞ if and only if the followingcondition holds: ∫ ∞

0

μ(Bd(x0, t))

tQ−1dt < ∞. (9.29)

If this condition is satisfied, then there is a unique L-subharmonic function u inG having the L-Riesz measure μ, the least upper bound U < ∞ and such thatu(x0) > −∞. It is given by U − Γ ∗ μ, i.e.

u(x) = U −∫

G

Γ (y−1 ◦ x) dμ(y), x ∈ G. (9.30)

Proof. It is not restrictive to assume x0 = 0. For the sake of brevity, let us putn(t) := μ(Bd(0, t)). The proof is split into the “sufficiency” and the “necessity”parts.

• Sufficiency. Let μ be a Radon measure satisfying (9.29) with x0 = 0, and letU ∈ R. Consider the function

u(x) := U −∫

G

Γ (y−1 ◦ x) dμ(y), x ∈ G.

We shall show that

u(0) > −∞, (9.31a)

supG

u = U, (9.31b)

(v ∈ S(G) : Lv = μ, supG

v = U) �⇒ v = u. (9.31c)

Since (9.31a) implies u ∈ S(G) and Lu = μ (see Theorems 9.3.2 and 9.3.5), thiswill prove the sufficiency part.

Proof (of (9.31a)). We put Γ (t) := βd t2−Q. Since n(t) is monotone increasing andQ > 2, condition (9.29) implies

μ({0}) = limt↓0

n(t) = 0.

Then

u(0) − U = −∫

G\{0}Γ (y−1) dμ(y) = − lim

λ↓0

∫ 1/λ

λ

Γ (t) dn(t). (9.32)


On the other hand, we have∫ 1/λ

λ

Γ (t) dn(t) = (Γ (1/λ) n(1/λ) − Γ (λ) n(λ)

) −∫ 1/λ

λ

Γ ′(t) n(t) dt

≤ Γ (1/λ) n(1/λ) + βd (Q − 2)

∫ ∞

λ

n(t)

tQ−1dt

≤ 2 βd (Q − 2)

∫ ∞

λ

n(t)

tQ−1dt. (9.33)

Here, being n(λ) monotone increasing and Γ ′ ≤ 0, we have used the fact that

Γ (λ) n(λ) = −n(λ)

∫ ∞

λ

Γ ′(t) dt ≤ −∫ ∞

λ

Γ ′(t) n(t) dt

= βd (Q − 2)

∫ ∞

λ

n(t)

tQ−1dt ∀ λ > 0.

Identity (9.32), together with (9.33) and condition (9.29), implies (9.31a). � Proof (of (9.31b)). For a fixed R > 0, let us split u − U as follows

u − U = uR + u∞R , (9.34)

where

uR(x) := −∫

Bd(0,R)

Γ (y−1 ◦ x) dμ(y).

Thenlim

x→∞ uR(x) = 0. (9.35)

On the other hand, since u∞R is the Γ -potential of the measure

μ|G\Bd(0,R)

andu∞

R (0) ≥ u(0) − U > −∞,

by Theorem 9.3.2, u∞R is L-subharmonic in G. Hence, u∞

R is sub-mean. Thus, forevery r > 0, keeping in mind (9.33), we have

Mr (u∞R )(0) ≥ u∞

R (0) ≥ −∫ ∞

R

Γ (t) dn(t)

≥ −2 βd (Q − 2)

∫ ∞

R

n(t)

tQ−1dt.

This implies the existence of at least one point y(r, R) ∈ ∂Bd(0, r) such that

u∞R (y(r, R)) ≥ −2 βd (Q − 2)

∫ ∞

R

n(t)

tQ−1dt. (9.36)

Since d(y(r, R)) = r for every R > 0, (9.31b) follows from (9.34)–(9.36) andcondition (9.29). �

9.6 Bounded-above L-subharmonic Functions in G 453

Proof (of (9.31c)). Let v ∈ S(G) be such that Lv = μ and sup v = U . Sincev is bounded above by the L-harmonic function U , by Theorem 9.4.7 there existsan L-harmonic function h in G such that v = h − Γ ∗ μ. Since v ≤ U , we haveh − U ≤ Γ ∗ μ in G. Theorem 9.3.7 and Liouville Theorem 5.8.1 (page 269) implyh − U ≡ c with c ∈ R, c ≤ 0. As a consequence, v = U + c − Γ ∗ μ = u + c. Butv and u have the same upper bound. Hence c = 0 and v ≡ u. �

• Necessity. Let u ∈ S(G) be such that

u(0) > −∞, supG

= U < ∞.

Since u is sub-mean, we have

−∞ < u(0) ≤ Mr (u)(0) ≤ U,

so that, by the Poisson–Jensen formula (9.22),∫

Bd(0,r)

(Γ (y) − Γ (r)) dμ(y) ≤ U − u(0) ∀ r > 0. (9.37)

On the other hand, since

Γ (y) − Γ (r) ≥ (1 − (1/2)Q−2)Γ (y) if 0 < d(y) ≤ t < r/2,

we have

n(t) Γ (t) = μ(Bd(0, t)) Γ (t) ≤∫

Bd(0,t)

Γ (y) dμ(y)

≤ c∫

Bd(0,t)

(Γ (y) − Γ (r)) dμ(y)

≤ c∫

Bd(0,r)

(Γ (y) − Γ (r)) dμ(y), c = 1

1 − ( 12 )Q−2

.

Then, n(t) is bounded for 0 < t < r/2. In particular,

μ({0}) = n(0+) = limt↓0

n(t) = 0.

Therefore, for 0 < ε < r , we have

U − u(0) ≥∫

{ε<d<r}(Γ (y) − Γ (r)) dμ(y)

≥∫ r

ε

(Γ (t) − Γ (r)) dn(t)

= −(Γ (ε) − Γ (r)) n(ε) −∫ r

ε

Γ ′(t) n(t) dt

≥ −Γ (ε) n(ε) + βd (Q − 2)

∫ r

ε

n(t)

tQ−1dt.


Using the boundedness of {Γ (ε) n(ε) : 0 < ε < r/2} and letting ε tend to zeroand r tend to ∞, we get

βd (Q − 2)

∫ ∞

0

n(t)

tQ−1dt ≤ U − u(0) < ∞.

The proof is complete. � Example 9.6.2. The function u = −Γ is L-subharmonic and bounded from abovein G, for U := sup u = 0. The L-Riesz measure of u is μ := Dirac0, the Dirac masssupported at {0} (see Example 9.4.2, page 441). For every x0 ∈ G we have

μ(Bd(x0, t)) ={

1 if d(x0) < t ,0 if d(x0) ≥ t ,

so that ∫ ∞

0

μ(Bd(x0, t))

tQ−1dt =

∫ ∞

d(x0)

1

tQ−1dt.

Since Q ≥ 3, this integral is finite if and only if d(x0) > 0, i.e. if and only if x0 �= 0.The representation formula (9.30) obviously holds true for

U − Γ ∗ μ = −Γ ∗ Dirac0 = −Γ = u. � In Theorem 9.6.1, we can remove the explicit mention of the point x0. Precisely,

the following theorem holds.

Theorem 9.6.3 (L-Riesz measure of a bounded-above u ∈ S(GGG). II). A Radonmeasure μ in G is the L-Riesz measure of a bounded-above L-superharmonic func-tion in G if and only if ∫ ∞

1

μ(Bd(0, t))

tQ−1dt < ∞. (9.38)

If this condition is satisfied and U ∈ R, the function

u(x) := U −∫

RN

Γ (y−1 ◦ x) dμ(y), x ∈ G,

is the unique L-subharmonic function whose L-Riesz measure is μ and the leastupper bound is U .

Proof. Since every L-subharmonic function is finite in a dense subset of its domain,by Theorem 9.6.1 it remains only to prove the first part of the theorem.

Let u ∈ S(G). Denote by μ := Lu its L-Riesz measure. Let

v := uB, where B := uBd(0,1/2)

is the Perron-regularization of u related to the d-ball B. Then v ∈ S(G), v(0) > −∞and v = u in G \ Bd(0, 1/2). Let μv := Lv be the L-Riesz measure of v. Then, byTheorem 9.6.1,

9.7 Smoothing of L-subharmonic Functions 455

∫ ∞

0

μv(Bd(0, t))

tQ−1dt < ∞. (9.39)

Moreover, for every t > 1,

μ(Bd(0, t)) = μ(Bd(0, 1)) − μv(Bd(0, 1)) + μv(Bd(0, t)).

By using this identity in (9.39), we obtain (9.38). Vice versa, assume (9.38) is satis-fied and define

μ1 := μ|Bd(0,2) and μ2 := μ|G\Bd(0,2).

Since∫ ∞

0

μ2(Bd(0, t))

tQ−1dt =

∫ ∞

1

μ2(Bd(0, t))

tQ−1dt ≤

∫ ∞

1

μ(Bd(0, t))

tQ−1dt < ∞,

by Theorem 9.6.1 there exists a function u2 ∈ S(G) bounded from above and suchthat Lu2 = μ2. Then, letting

u = −Γ ∗ μ1 + u2,

we get u ∈ S(G), sup u ≤ sup u2 < ∞ and Lu = μ1 + μ2 = μ. �

9.7 L-subharmonic Smoothing of L-subharmonic Functions

Let � be a homogeneous norm on G (not necessarily an L-gauge for some sub-Laplacian L). Given a L1

loc-function u : Ω → [−∞,∞], Ω ⊆ G open, we knowthat the (ε, G)-mollifier of u (see Example 5.3.7, page 239)

uε(x) = (u ∗G Jε)(x) :=∫

B�(0,ε)

u(y−1 ◦ x) Jε(y) dy

is well defined and smooth in

Ωε := {x ∈ Ω : dist�(x, ∂Ω) > ε}.We recall that Jε(x) = ε−Q J(δ1/ε(x)) and J is a smooth function supported inBd(0, 1) with

∫G

J = 1.We also know (see Theorem 8.1.5, page 401) that the mapping

u → u ∗G Jε

preserves the L-sub-mean property for every sub-Laplacian L on G. As a conse-quence, by Theorem 8.2.1 (page 401),

the (ε, G)-mollifiers preserve the L-subharmonicity

for every sub-Laplacian L on G.

Note that the notion of (ε, G)-mollification depends only on the structure of (G, ◦, δλ)

and on the homogeneous norm � and does not depend on the sub-Laplacian L.The following theorem holds.


Theorem 9.7.1 (Smoothing of L-subharmonic functions). Let Ω be an open sub-set of G, let L be any sub-Laplacian on G and let u : Ω → [−∞,∞[ be anL-subharmonic function. Then:

(i) uε ∈ C∞(Ωε, R),(ii) uε −→ u in L1

loc(Ω) as ε → 0,(iii) uε is L-subharmonic in Ωε.

By using the smoothing operators constructed by superposition of the surfacemean operators (see Section 5.6, page 257), one can find monotone sequences of L-smooth and L-subharmonic functions approximating a given L-subharmonic func-tion. However, the smooth approximating functions constructed via this method areL-subharmonic only with respect to the given sub-Laplacian L.

The starting point of this construction is the following crucial lemma.

Lemma 9.7.2. Let Γ be the fundamental solution for L and, for any fixed z ∈ G, letus set

Γz(x) := Γ (z−1 ◦ x).

Then, for every r > 0,

Mr (Γz)(x) = min{Γ (z−1 ◦ x), Γ (r)}, x ∈ G. (9.40)

As a consequence, the function x → Mr (Γz)(x) is L-superharmonic in G. More-over, the map

(x, z, r) → Mr (Γz)(x)

is lower semicontinuous.

Proof. (9.40) is (9.10a), page 438. The L-superharmonicity of Mr (Γz) also followsfrom Proposition 6.5.4-(iii) (page 355). The last part of the assertion is a standardconsequence of Fatou’s lemma.

An alternative proof which makes use of the Poisson–Jensen formula is the fol-lowing one. Since Γz ∈ S(G) and LΓz = −Diracz (the Dirac mass supported at {z})the Poisson–Jensen formula (9.22) gives

Γ (z−1 ◦ x) = Γz(x) = Mr (Γz)(x) + wz(x), (9.41)

where

wz(x) ={

0 if d(z−1 ◦ x) ≥ r ,

Γ (x−1 ◦ z) − Γ (r) if d(z−1 ◦ x) < r .

Thenwz(x) = max{Γ (x−1 ◦ z) − Γ (r), 0}

and (9.40) follows from (9.41) and the symmetry of Γ . �

9.7 Smoothing of L-subharmonic Functions 457

From Theorem 9.3.11 (page 439), one can easily obtain the approximation Theo-rem 9.7.3 below. For the reading convenience, we recall that in (5.50b) on page 258,we introduced the following integral operator Φr :

Φr(u)(x) :=∫ ∞

0ϕr(ρ)Mρ(u)(x) dρ,

where ϕr(t) := 1

rϕ

(t

r

), ϕ ∈ C∞

0 (]0, 1[, R) and∫ 1

0ϕ = 1. (9.42)

Theorem 9.7.3 (Approximation I. Γ -potentials). Let u ∈ S(G) be the Γ -potentialof a Radon measure μ on G. Then, for every r > 0,

Φr(u) ∈ S(G) ∩ C∞(G), (9.43)

where Φr is as in (9.42).Moreover, the following assertions hold:

u(x) ≥ Φρ(u)(x) ≥ Φr(u)(x) ∀ x ∈ G, 0 < ρ ≤ r, (9.44a)

limr→0

Φr(u)(x) = u(x) ∀ x ∈ G. (9.44b)

Proof. Property (9.43) follows from (9.42), Theorem 9.3.11 (page 439) and The-orem 8.2.20, page 410 (in this last remark, we take Λ = (0,∞), z = ρ anddμ(ρ) = ϕr(ρ) dρ).

To prove (9.44a), let us assume 0 < ρ ≤ r . By Theorem 8.2.10 (page 404),ρ → Mρ(u) is monotone decreasing, so that

Φρ(u)(x) =∫ ∞

0ϕ(t)Mρ t (u)(x) dt

≥∫ ∞

0ϕ(t)Mr t (u)(x) dt = Φr(u)(x).

Moreover, since u is super-mean,

Φρ(u)(x) =∫ ∞

0ϕ(t)Mρ t (u)(x) dt ≤

( ∫ ∞

0ϕ(t) dt

)u(x) = u(x).

Thus, (9.44a) holds. Finally, (9.44b) follows from the inequality u(x) ≥ Φr(u)(x)

and the lower semicontinuity of u. � If we consider the integral operator Φr related to a non-smooth function ϕ, The-

orem 9.7.3 still holds with (9.43) replaced by

Φr(u) ∈ S(G). (9.43)′

In particular, by taking

ϕ(t) ={

QtQ−1 if 0 < t < 1,

0 otherwise,

we obtain the following theorem.


Theorem 9.7.4 (Approximation II. Γ -potentials). Let u ∈ S(G) be the Γ -potentialof a Radon measure μ in G. Then:

(i) x → Mr (u)(x) is L-superharmonic in G for every r > 0,(ii) Mr (u)(x) ≤ Mρ(u)(x) ≤ u(x) for every x ∈ G and 0 < ρ ≤ r < ∞,

(iii) limr↓0 Mr (u)(x) = u(x) for every x ∈ G.

Mr is the solid average operator defined in (5.50f), page 259.

By using the Riesz representation Theorem 9.4.4, Theorem 9.7.3 extends toL-subharmonic functions on arbitrary open subsets of G:

Theorem 9.7.5 (L-subharmonic-monotone-smooth approximation). Let u ∈S(Ω) and Ω ⊆ G be open. For every r > 0, let us define

ur(x) := Φr(u)(x), x ∈ Ωr := {y ∈ Ω : distd(y, ∂Ω) > r},where Φr is the smoothing operator of Theorem 9.7.3. Then:

(i) ur ∈ S(Ωr) ∩ C∞(Ωr) for every r > 0,(ii) u ≤ uρ ≤ ur for every 0 < ρ ≤ r < ∞,

(iii) limr↓0 ur(x) = u(x) for every x ∈ Ω .

Proof. Let x ∈ Ωr , and let O be a bounded open set such that

Bd(y, r) ⊂ O ⊂ O ⊂ Ω

for every y ∈ V , where V is a suitable neighborhood of x. By the Riesz-type rep-resentation Theorem 9.4.4, there exists a Radon measure μ supported in O and afunction h, L-harmonic in O, such that

u(y) = h(y) − (Γ ∗ μ)(y) ∀ y ∈ O.

As a consequence,

ur(y) = hr(y) − (Γ ∗ μ)r(y) = h(y) − (Γ ∗ μ)r(y) ∀ y ∈ V. (9.45)

Then, by (9.43),(ur)|V ∈ S(V ) ∩ C∞(V ),

and (i) is proved. Now, (ii) and (iii) directly follow from (9.45) with y = x and,respectively, (9.44a) and (9.44b). �

9.8 Isolated Singularities—Bôcher-type Theorems

Given an open set Ω ⊆ G, a point x0 ∈ Ω and an L-harmonic function u in Ω \{x0},we say that x0 is a removable singularity of u if there exists an L-harmonic function v

in Ω such thatu(x) = v(x) ∀ x ∈ Ω \ {x0}.

9.8 Isolated Singularities—Bôcher-type Theorems 459

A first result on removable singularities will easily follow from the next lemma, inwhich we shall use the notation

Bd(x0, r) := Bd(x0, r) \ {x0}.We recall that Q (the homogeneous dimension of G) is always supposed > 2.

Lemma 9.8.1. Let h : Bd(x0, r) → R be an L-harmonic function such that:

(i) limx→y h(x) = 0 for every y ∈ ∂Bd(x0, r),

(ii) lim supx→x0h(x)

(d(x−1

0 ◦ x))Q−2 = 0.

Then h ≡ 0.

Proof. Let ε > 0. Define

hε(x) := ±h(x) − ε(d(x−1

0 ◦ x))2−Q

, x ∈ Bd(x0, r).

Since d is an L-gauge and hε is L-harmonic, he is L-harmonic in Bd(x0, r). More-over, conditions (i) and (ii) imply

lim supx→y

hε(x) ≤ 0 ∀ y ∈ ∂Bd(x0, r).

Then, by the maximum principle, hε ≤ 0 in Bd(x0, r). Letting ε tend to zero, weobtain h ≤ 0. The same inequality holds for −h. Thus h = 0. � Theorem 9.8.2 (Characterization of removable singularities). Let Ω be an opensubset of G. A point x0 ∈ Ω is a removable singularity of the L-harmonic functionu : Ω \ {x0} → R if and only if

lim supx→x0

u(x)(d(x−1

0 ◦ x))Q−2 = 0. (9.46)

Proof. The “only if” part is trivial. We turn to the “if” part. Assume that (9.46) issatisfied and r > 0 is such that Bd(x0, r) ⊂ Ω . Let w be the solution to the boundaryvalue problem {Lw = 0 in Bd(x0, r),

w|∂Bd(x0,r) = u|∂Bd(x0,r).

Then h := u − w satisfies the hypothesis of Lemma 9.8.1. It follows that h = 0 inBd(x0, r), i.e. u = w in Bd(x0, r). Thus, the function

v : Ω → R, v(x) :={

u(x) in Ω \ Bd(x0, r),

w(x) in Bd(x0, r)

is L-harmonic in Ω and equals u in Ω \ {x0}. This completes the proof. � Remark 9.8.3. The function

x → Γ (x−10 ◦ x) = (

d(x−10 ◦ x)

)2−Q

is L-harmonic in G \ {x0} with a non-removable singularity at x = x0. �


Non-negative L-harmonic functions in G \ {x0} can be characterized in terms ofthe function x → Γ (x−1

0 ◦ x).We first show a local version of this statement, which extends the classical Bôcher

theorem to the sub-Laplacian setting.

Theorem 9.8.4 (Bôcher-type theorem on punctured balls). Let u be L-harmonicand non-negative in the punctured ball Bd(x0, R). Then there exist a non-negativeconstant a and an L-harmonic function h in Bd(x0, R) such that

u = a Γ + h in Bd(x0, R). (9.47)

Proof. It is not restrictive to assume x0 = 0. If u were bounded near zero, then (9.47)would follow, with a = 0, from Theorem 9.8.2. Assume that

lim supx→0

u(x) = ∞. (9.48)

For 0 < r ≤ R/2, define

S(r) := supd(x)=r

u(x) and s(r) := infd(x)=r

u(x).

The Harnack inequality on rings of Corollary 5.7.7 (page 267) implies

S(r) ≤ c s(r), 0 < r ≤ R/2, (9.49)

with a positive constant c independent of r . On the other hand, by the maximumprinciple,

max{S(r), S(R/2)} = maxr≤d(x)≤R/2

u(x), 0 < r ≤ R/2,

so that, by (9.48),limr↓0

S(r) = ∞. (9.50)

Using (9.50) and (9.49), we get lim infx→0 u(x) = ∞. Hence

limx→0

u(x) = ∞.

If we extend u at 0 letting u(0) = ∞, by Corollary 8.2.6 (page 402), u ∈S(Bd(0, R)). Let us now observe that the L-Riesz measure of u

μ := −Lu

is supported at {0}, since u is L-harmonic in Bd(0, r). Therefore,

μ = a Dirac0,

for a suitable a > 0 (here Dirac0 is the Dirac mass supported at the origin). Now, theRiesz representation result of Corollary 9.4.9 implies

u = Γ ∗ μ + h = a Γ + h,

where h is L-harmonic in Bd(0, R). The proof is complete. �

9.8 Isolated Singularities—Bôcher-type Theorems 461

Theorem 9.8.5 (Bôcher-type theorem for L). Let Ω ⊆ G be open, and let x0 ∈ Ω .Let u : Ω \ {x0} → R be L-harmonic and non-negative. Then there exist a non-negative constant a and an L-harmonic function v in Ω such that

u = a Γ + v in Ω \ {x0}. (9.51)

Proof. Let R > 0 be such that Bd(x0, R) ⊂ Ω . By Theorem 9.8.4, there exist aconstant a ≥ 0 and an L-harmonic function h in Bd(x0, R) such that

u = a Γ + h in Bd(x0, R).

Thus, the function

v : Ω \ {x0} → R, v :={

h in Bd(x0, R),

u − a Γ in Ω \ Bd(x0, R)

is an L-harmonic function in Ω satisfying (9.51). � This theorem, together with Liouville Theorem 5.8.1 (page 269), implies the

following corollary.

Corollary 9.8.6 (Non-negative L-harmonic functions in GGG \ {0}). Let u : G \{0} → R be a non-negative L-harmonic function. Then there exist two non-negativeconstants a and b such that

u = a Γ + b in G \ {0}.Proof. By Theorem 9.8.5,

u = a Γ + v in G \ {0},where a is a non-negative constant and v is L-harmonic in G. Since u ≥ 0 andΓ (x) → 0 as x → ∞, the function v is bounded from below and lim infx→∞ v(x) ≥0. Then, by Liouville Theorem 5.8.1 (page 269), v is constant, and this constant isnon-negative. �

With this corollary at hand, we can trivially characterize the δλ-homogeneousnon-negative L-harmonic functions in G \ {0}.Corollary 9.8.7 (Non-negative δλ-homogeneous functions in H(GGG)). Suppose u :G \ {0} → R is non-negative, L-harmonic and δλ-homogeneous of degree m. If u isnot identically zero, then m = 0 or m = 2 − Q. Moreover:

(i) u = constant if m = 0,(ii) u = a Γ for a suitable constant a > 0 if m = 2 − Q.

As a consequence, we immediately obtain the following improvement of Theo-rem 5.5.6 on page 256.

Corollary 9.8.8 (“Uniqueness” of the L-gauges. II). Let L be a sub-Laplacian onthe homogeneous Carnot group G. Let d be a homogeneous norm on G, smooth outof the origin and such that

L(dα) = 0 in G \ {0},for a suitable α ∈ R, α �= 0. Then α = 2 − Q and d is an L-gauge on G.


9.8.1 An Application of Bôcher’s Theorem

The aim of this section is to prove the following Proposition 9.8.9, as a conse-quence of Bôcher’s Theorem 9.8.5 above and of the Liouville-type Theorem 5.8.4(page 270). We recall that if d is an L-gauge on G, then

ΨL = |∇Ld|2

is the kernel appearing in the relevant solid mean value formula of Theorem 5.6.1(page 259).

Proposition 9.8.9. Suppose L is a sub-Laplacian on the (homogeneous) Carnotgroup G and d is an L-gauge on G such that |∇Ld|2 is constant on G \ {0}. Then G

has step 1, i.e. G = (G,+) is the Euclidean group.

Proof. Let us set, for brevity, G := G \ {0}. Let d be an L-gauge on G. Then, byDefinition 5.4.1 (page 247),

d ∈ C∞(G) ∩ C(G)

and d2−Q is L-harmonic on G (here Q is the homogeneous dimension of G). Fromthe formula

L(α(u)) = α′′(u) |∇Lu|2 + α′(u)Lu

(for suitably regular functions u : G → R, α : R → R), we have

0 = L(d2−Q) = (2 − Q)(1 − Q)d−Q|∇Ld|2 + (2 − Q)d1−Q Ld

on G, i.e.d Ld = (Q − 1) |∇Ld|2 on G. (9.52)

Analogously, we have L(d2) = 2 |∇Ld|2 + 2 d Lu. Hence, by (9.52),

L(d2) = 2 Q |∇Ld|2 on G. (9.53)

Consequently, (9.53) proves that(|∇Ld|2 = constant on G

) �⇒ (L(d2) = constant on G

). (9.54)

We claim that(|∇Ld|2 = constant on G

) �⇒ (d2 ∈ C∞(G)

). (9.55)

From this claim the proposition follows. Indeed, by continuity, the claimed (9.55)and (9.54) prove that L(d2) = constant on G. Then, thanks to the Liouville-typeTheorem 5.8.4 (page 270), d2 is a polynomial function of G-degree at most 2. De-noting, as usual, by x = (x(1), x(2), . . . , x(r)) the “stratification” of the variables ofthe (homogeneous) group G, we have

d2(x) = p2(x(1)) + p1(x

(2)),


where pi is a polynomial function of ordinary degree i (i = 1, 2). Due to the non-negativity of d2, this gives p1 ≡ 0, so that d2(x) = p2(x

(1)). But since d is positiveoutside 0 (see the definition of homogeneous norm in Section 5.1, page 229), if G

had step r ≥ 2, fixed x(2) �= 0, we would have the contradiction

0 < d2(0(1), x(2), 0(3), . . . , 0(r)) = p2(0(1)) = 0.

(Recall that p2 is a homogeneous polynomial of degree 2.)It remains to prove (9.55). Suppose |∇Ld|2 = c ∈ R and take a non-negative

cut-off function ϕ ∈ C∞0 (G) such that ϕ ≡ 1 on a neighborhood U of the origin,

say U = Bd(0, ε) with ε small. Denote by Γ the fundamental solution of L. FromTheorem 5.3.5 (page 238) the function

u(y) :=∫

G

Γ (y−1 ◦ x) ϕ(x) dx

is non-negative, smooth on G and satisfies Lu = −ϕ on G. In particular,

Lu = −1 on U. (9.56)

By (9.53) and (9.56), we have

L(d2 + 2 Qc u) = 0 on U \ {0}.Hence, d2 + 2 Qc u is L-harmonic and non-negative on U \ {0}. Furthermore, itis bounded there, for d and u are continuous on G. By Theorem 9.8.5, there existsh ∈ H(U) such that d2 + 2 Qc u = h on U \ {0}. By continuity, this holds on thewhole U , thus proving (9.55). � Bibliographical Notes. The topics developed in this chapter within the classicaltheory of Laplace’s operator can be found in all the main monographs devoted tothe potential theory; see the references in the Bibliographical Notes of Chapter 8.For Bôcher-type theorems, see [TW02b].

Some of the topics presented in this chapter also appear in [BL03,BL07].


Ex. 1) Prove that if u > 0 is L-harmonic in an open set Ω , then log(u) is L-superharmonic in Ω .


Ex. 2) Prove that any L-superharmonic function u in an open set Ω belongs toL

p

loc(Ω) for every

1 ≤ p <Q

Q − 2.

(Hint: Use Theorem 9.4.4, in order to restrict the proof to the case whereu = Γ ∗ μ with μ compactly supported. Then use the Hölder inequalityand the summability properties of Γ .)

Ex. 3) Prove that there exists an L-superharmonic function u in G such that u doesnot belong to L

p

loc(Ω) with p = QQ−2 . (Hint: u = Γ .)

Ex. 4) Prove that if u is an L-superharmonic function in an open set Ω , then ∇Lu

is well-defined in the weak sense of distributions, and it holds

|∇Lu| ∈ Lp

loc(Ω)

for every

1 ≤ p <Q

Q − 1.

(Hint: Use Theorem 9.4.4, to restrict to the case where u = Γ ∗ μ with μ

compactly supported. Then use the summability properties of ∇LΓ .)Ex. 5) Complete the sketched proof of the following proposition.

Proposition 9.9.1. Let u ∈ S(G) be non-negative. Then the following as-sertions hold:

a) u is a Γ -potential iff limR→∞ mR[u](0) = 0 or iff infG u = 0.b) u is a Γ -potential iff u is bounded from above by a Γ -potential.

Proof (Sketch). Let u = Γ ∗ μ, where μ is a Radon measure in G. Then(why?)

mR[u](0) =∫

G

( ∫

∂Bd(0,R)

k(0, y)Γ (z−1 ◦ y) dHN−1(y)

)dμ(z)

=∫

G

(Γ ∗ λ0,R)(z) dμ(z) =∫

G

min{Γ (z), R2−Q} dμ(z).

By monotone convergence derive limR→∞ mR[u](0) = 0. Vice versa(why?), u = infG u + Γ ∗ (μu). Hence

mR[u](0) = infG

u + mR[Γ ∗ (μu)](0).

Then, if limR→∞ mR[u](0) = 0, we obtain (why?) infG u = 0, i.e. u =Γ ∗ (μu). This argument also proves the second “iff” in (1). Finally, derive(2) from (1). �

Ex. 6) Prove the assertion in Example 9.1.3, page 427.


Ex. 7) Let u be an L-harmonic function in the punctured ball

Bd(0, 1) := Bd(0, 1) \ {0}.Assume that

lim inf|x|→0u(x) d(x)2−Q > −∞.

Then there exists b ∈ R and v, L-harmonic in Bd(0, 1), such that

u = bΓ + v in Bd(0, 1).

Ex. 8) Let Ω be an open set of G. Denote

bp(Ω) :={u L-harmonic in Ω :

∫

Ω

|u|p < ∞},

for 1 ≤ p < ∞. Prove that bp(Ω) is a closed subspace of Lp(Ω).Ex. 9) Let bp(Ω) be as in the previous exercise. Let Ω = Bd(0, 1) \ {0}, and let

u ∈ bp(Ω), p ≥ Q/(Q − 2). Prove that 0 is a removable singularity of u.Ex. 10) Let 1 ≤ p < Q/(Q − 2). Prove the existence of a function

u ∈ bp(Bd(0, 1) \ {0})such that 0 is not a removable singularity of u.

Ex. 11) Let u : G \ {0} → R be an L-harmonic function. Assume that, for somep ∈ [1,∞[, ∫

G\{0}|u|p < ∞.

Prove that:(i)

|u(x)|p ≤ c|x|−Q

∫

G\{0}|u|p ∀x �= 0,

where c > 0 is independent of u and x,(ii) u ≡ 0 in G provided that p ≥ Q/(Q − 2).

Ex. 12) Prove the following result, as a consequence of Corollary 9.8.6.

Corollary 9.9.2. Let � be any homogeneous norm on G. Suppose there ex-ists a non-constant function α : (0,∞) → (0,∞) such that α ◦ � is L-harmonic in G. Then � is an L-gauge function and α(t) = a t2−Q + b, forsome constants c, b ≥ 0.

(Hint: α ◦ � is a non-negative L-harmonic function on G. Then, by Corol-lary 9.8.6, we have (here d is the L-gauge function such that d2−Q is thefundamental solution for L)

α(�(x)) = a d2−Q(x) + b for every x ∈ G.


Since d and � are both δλ-homogeneous of degree 1, this gives, for everyξ ∈ G such that �(ξ) = 1,

α(t) = α(t �(ξ)) = α(�(δt ξ)) = a d2−Q((δt ξ)) + b

= a d2−Q(ξ) t2−Q + b. (9.57)

When t = 1, this proves that d is constant (say, a) on the level set of �

{ξ : �(ξ) = 1}, so that

d(x) = d(δ�(x) δ1/�(x)(x)

) = �(x) d(δ1/�(x)(x)

) = �(x) a,

whence � = a−1 d is an L-gauge function. Finally, (9.57) also shows thatα(t) = a t2−Q + b, with a = a a2−Q.)

Ex. 13) Let d be a homogeneous norm on G, smooth in G. We recall: if there existsa real constant γ �= 0 such that

dγ is L-harmonic in G, (9.58)

then we say that d is an L-gauge. As we know, a remarkable homogeneousnorm on G is given by

dC(x) := dX(x, 0), x ∈ G, (9.59)

where dX denotes the Carnot–Carathéodory distance related to the familyof vector fields X = {X1, . . . , Xm} (see Proposition 5.2.8). The norm dC

in (9.59) will be referred to as the LC-norm on G. In studying surface mea-sures in Carnot–Carathéodory spaces, R. Monti and F. Serra-Cassano provedthat dC satisfies the L-Eikonal equation, i.e.

|∇LdC | = 1 almost everywhere in G

(see [MSC01, Theorem 3.1]). Prove the following assertion.

Corollary 9.9.3. Let G be a Carnot group different from an Euclideangroup, and let L be a sub-Laplacian on G. Then the LC-norm (9.59) isnot an L-gauge. More explicitly, for every γ ∈ R \ {0},

dγ

C is not L-harmonic in G.

Moreover, suppose there exists a function α : (0,∞) → (0,∞) such thatα ◦ dC is L-harmonic in G. Then t → α(t) is a constant function.

(Hint: Use the cited result in [MSC01, Theorem 3.1] and our Theorem 9.8.9.Then use Corollary 9.9.2.)By using Proposition 9.8.9, prove also the following assertion.If L is a sub-Laplacian on the (homogeneous) Carnot group G and d is anL-gauge on G satisfying the L-Eikonal equation

|∇Ld| = 1 on G \ {0},then G has step 1, i.e. G = (G,+) is the Euclidean group.


Ex. 14) Prove the following result, by making use of the proof of Theorem 9.8.9.

Proposition 9.9.4. Let u : G → R be a (smooth) solution to

Lu = c in G,

where c is a real constant. Assume u is coercive, i.e.

u(x) −→ ∞ as x → ∞.

Then G is the Euclidean group.

Ex. 15) When d is an L-gauge function, we already know that the kernel ΨL =|∇Ld|2 appears in the (solid) average formula of Theorem 5.6.1 (see also(5.42) on page 252). For the sake of brevity, we introduce a new notation:

K := ΨL, K := ΨL|∇d| , mr(u) :=

∫

∂Bd(0,r)

uK dHN−1 (9.60)

(here, ∇ is the standard gradient vector in RN , and HN−1 denotes the

(N − 1)-Hausdorff measure in G). Compare to our previous notation (seeTheorem 5.5.4 and (5.42)) and notice that

K = KL(0, ·), mr(u) = 1

(Q − 2)βd

rQ−1 Mr (u)(0).


Theorem 9.9.5. Let d be an L-gauge, and let γ be the related exponent asin (9.58). If u is L-harmonic in the unit punctured ball B = Bd(0, 1) \ {0},then there exist real constants a, b such that

rγ−1mr(u) = a rγ + b for every r ∈ ]0, 1[. (9.61)

(Hint: We have proved in Section 5.5 (page 251) that if the sub-Laplacian Lis written in the divergence form L = div(A(x)∇), then we have∫

Ar,R

(g Lf − f Lg) =∫

∂Ar,R

(g 〈A∇f, ν〉 − f 〈A∇g, ν〉) dHN−1 (9.62)

for each f, g ∈ C2(B) and 0 < r < R < 1, where Ar,R := {x : r <

d(x) < R} and ν is the normal outer vector on ∂Ar,R . If f = u and g ≡ 1,then

∫

∂Bd(0,R)

〈A∇u, ν〉 dHN−1 =∫

∂Bd(0,r)

〈A∇u, ν〉 dHN−1. (9.63)

Hence,∫∂Bd(0,R)

〈A∇u, ν〉 dHN−1 is constant w.r.t. R, say α. We now choosef = u and g = dγ in (9.62). By noticing that on ∂Bd(0, ρ) we have


ν = ∇d/|∇d|, g = ργ and 〈A∇g, ν〉 = γ ργ−1 |∇Ld|2/|∇d|. Then (9.62)becomes (see the definition of mr in (9.60) and use (9.63))

0 = α Rγ − γ Rγ−1mR(u) − α rγ + γ rγ−1mr(u).

In other words, the function ]0, 1[ � r → α rγ − γ rγ−1mr(u) is constant,say β. This is exactly (9.61) with a = α/γ and b = −β/γ .)

Ex. 16) Re-derive Theorem 5.5.6 on page 256 as a consequence of the above Theo-rem 9.9.5. In other words, prove the following assertion.

Corollary 9.9.6. Let d be an L-gauge function, and let γ be the relatedexponent as in (9.58). Then

γ = 2 − Q, and Γ = βd d2−Q (9.64)

is the fundamental solution (with pole at the origin) for L. Here Q is thehomogeneous dimension of G and

β−1d = Q(Q − 2)

∫

Bd(0,1)

K. (9.65)

Hint: By means of the coarea formula, we have

∫ r

0

∫

d(x)=ρ

K(x)dHN−1(x)

|∇d(x)| =∫

d(x)<r

K(x) dx

=∫

d(ξ)<1K(δr(ξ)) rQ dξ = rQ

∫

Bd(0,1)

K(ξ) dξ =: rQ ωd.

By differentiating both sides w.r.t. r , we get

∫

d(x)=r

K(x)dHN−1(x)

|∇d(x)| = rQ−1 Qωd for all r > 0. (9.66)

Now, recalling (9.60), the above identity rewrites as

mr(1) = rQ−1 Qωd. (9.67)

On the other hand, by Theorem 9.9.5, there exist constants a, b such thatmr(1) = a r + b r1−γ , whence (9.67) gives Qωd rQ−2 = a + b r−γ whichis possible iff (recall that Q > 2) a = 0 and γ = 2 − Q.Finally, let us set Γ (x) := βd d2−Q(x) (βd as in (9.65)), and let us showthat Γ is the fundamental solution for L. First, show that

∫

Bd(0,1)

Γ (x) dx = βd HN(Bd(0, 1))Q

2< ∞.

Hence Γ ∈ L1loc(G). Moreover, for 2 − Q < 0, Γ (x) vanishes when

x → ∞. Finally, let us check that LΓ = −Dirac0 in the weak sense of


distributions. Let ϕ ∈ C∞0 (G) be fixed. By the divergence theorem, show

that∫

Γ Lϕ = limε→0

∫

d>ε

d2−Q Lϕ

= limε→0

(−

∫

d(x)=ε

d2−Q 〈∇Lϕ,∇Ld〉 dHN−1

|∇d|−

∫

d>ε

〈∇L(d2−Q),∇Lϕ〉)

= limε→0

(−I1(ε) +

∫

d=ε

ϕ (2 − Q)d1−Q KdHN−1

|∇d|+

∫

d>ε

L(d2−Q) ϕ

)

= limε→0

(−I1(ε) + I2(ε) + 0) = −ϕ(0) β−1d .

Indeed, prove that

|I1(ε)| ≤ ε2−Q sup{|∇Lϕ| |∇Ld|}

∫

d(x)=ε

dHN−1

|∇d| = c ε.

On the other hand, limε→0 I2(ε) = −ϕ(0) β−1d , since (recall (9.67))

I2(ε) = −β−1d ϕ(0) + I3(ε),

and (exploiting again (9.67))

|I3(ε)| ≤ Q(Q − 2)ωd supd(x)=ε

|ϕ(x) − ϕ(0)| → 0 as ε → 0.

Ex. 17) Let mr be as in (9.60). Consider our usual surface average operatorMr (u)(0), briefly denoted by Mr (u). Following an idea exploited byAxler, Bourdon and Ramey [ABR92] in the classical harmonic setting,given a smooth function u : B → R, we define

S(u)(x) := Md(x)(u), x ∈ B. (9.68)

If u is L-harmonic in B and d is an L-gauge, by Theorem 9.9.5 and Corol-lary 9.9.6, we have

S(u)(x) = a d2−Q(x) + b, x ∈ B,

for suitable a, b ∈ R. Thus, S(u) is L-harmonic in B and radially symmetricwith respect to d , i.e. S(u)(x) = S(u)(y) if d(x) = d(y). Moreover, sinceMr (1) = 1 and d is constant on ∂Bd(0, 1), we have

S(S(u)) = S(a d2−Q + b) = a d2−Q + b = S(u). (9.69)



Lemma 9.9.7. If u is L-harmonic on B and d is an L-gauge, there exista, b ∈ R such that

Mr (u)(0) = a r2−Q + b for every r ∈ ]0, 1[,(9.70)

S(u)(x) = a d2−Q(x) + b for every x ∈ B.

Moreover, for every n ∈ N, we have Sn(u) = S(u), where Sn denotes then-th iterate of the operator S .

(Hint: Use Theorem 9.9.5, Corollary 9.9.6 and Mr (1) = 1.)

The operator S can be used to prove the following radial-symmetry resultfor non-negative L-harmonic functions in a punctured L-gauge ball, vanish-ing on the boundary. Denote B = Bd(0, 1).

Theorem 9.9.8. Let d be an L-gauge, and let w be a non-negativeL-harmonic function in B. Assume that w is continuous up to the bound-ary of B and w(x) = 0 for every x ∈ ∂B. Then w is d-radially symmetric.More precisely, w(x) = S(w)(x) for every x ∈ B, so that, for a suitablereal constant a > 0,

w(x) = a(d2−Q(x) − 1

)for every x ∈ B.

Proof. Following an idea in [ABR92], we consider the Harnack inequalityon d-spheres in Theorem 5.16.5 (page 327). Rewrite it to derive the exis-tence of a constant c∈(0, 1) such that

c h(z) ≤ h(x) whenever 0 < d(z) = d(x) ≤ 12 and h ∈ H(D), h > 0.

(9.71)

We claim that{

h ∈ H(B) ∩ C(B),

h ≡ 0 on ∂B and h ≥ 0 on B�⇒ h − cS(h) ≥ 0 on B, (9.72)

where B = Bd(0, 1) \ {0}. Suppose that w ∈ H(B) ∩ C(B \ {0}) is non-negative on B and null on ∂B. Set

cn := 1 − (1 − c)n, n ∈ N ∪ {0}. (9.73)

Let us prove by induction that

w − cn S(w) ≥ 0 on B for all n ∈ N ∪ {0}. (9.74)

The case n = 0 is obvious. Suppose (9.74) holds, and let us prove it for n+1.The function h := w − cn S(w) satisfies the hypothesis of (9.72). Indeed,w|∂B = 0 implies S(w)|∂B = 0. Moreover, both w and S(w) belong to

H(B) ∩ C(B), since (see also Lemma 9.9.7, b = −a)


S(w)(x) = Md(x)(w)(0) = a d2−Q(x) − a. (9.75)

Consequently, by (9.72), we have on B

0 ≤ h − cS(h) = w − cn S(w) − cS(w − cn S(w)

) = w − cn+1 S(w).

(Use cn + c − c cn = cn+1 and Lemma 9.9.7.) Thus (9.74) is proved byinduction. Letting n → ∞ in it, we infer that w ≥ S(w) on B. On theother hand, the inequality w(x) > S(w)(x) for some x ∈ B is impossible,otherwise S(w)(x) > S(w)(x). Consequently, by using (9.75), w(x) =S(w)(x) = a(d2−Q(x) − 1) for x ∈ B.It remains to prove (9.72). Let h be as in (9.72). From Lemma 9.9.7 we haveS(h)(x) = a d2−Q(x) + b on B, so that

H := h − cS(h) ∈ H(B) ∩ C(B).

Furthermore, H = 0 on ∂B. If we multiply the inequality in (9.71) times(Q − 2)βd d1−Q(x)K(z) and then integrate w.r.t. z ∈ Bd(0, d(x)), we getH(x) ≥ 0 for every x ∈ Bd(0, 1/2). Then the weak maximum principleensures that H ≥ 0 on B. �

Ex. 18) Derive Theorem 9.8.5 from Theorem 9.9.8.(Hint: Let Bd(x0, ε) ⊂ Ω . Let h be the solution to the Dirichlet problemLh = 0 in Bd(x0, ε), h = u on ∂Bd(x0, ε). Consider the function w(x) :=(u−h)(x0 ∗δε(x))+Γ (x)−1. Prove, by the weak maximum principle, thatw ≥ 0 on B. Hence, by Theorem 9.9.8, there exists c ∈ R such that u = h+c Γ (x−1

0 ∗·) for x ∈ Bd(x0, ε). Then c ≥ 0, otherwise limx→x0 u(x) = −∞.Thus, v := u − Γ (x−1

0 ∗ ·) extends L-harmonically through x0.)Ex. 19) Let x0 ∈ Ω ⊆ G, where Ω is open, and let Ω = Ω \ {x0}. Suppose

w : Ω → R satisfies the following conditions:(i) w ∈ C∞(Ω) and w, ∇Lw are bounded on Ω ,

(ii) Lw = f in Ω , where f ∈ C∞(Ω).Then w extends to a C∞ function on the whole Ω .(Hint: Suppose x0 = 0. It is enough to prove that Lw = f in Ω , in the weaksense of distributions. Let ϕ ∈ C∞

0 (Ω), and let d be a homogeneous normon G, smooth on G. Argue as in the proof of Corollary 9.9.6, to show that

∫

Ω

wLϕ = limε→0

(−

∫

d(x)=ε

w 〈∇Lϕ,∇Ld〉 dHN−1

|∇d| −∫

d>ε

〈∇Lw,∇Lϕ〉)

= limε→0

(−I1(ε) +

∫

d=ε

ϕ 〈∇Lw,∇Ld〉 dHN−1

|∇d| +∫

d>ε

ϕ Lw

)

= limε→0

(−I1(ε) + I2(ε) +

∫

d>ε

f ϕ

)=

∫

Ω

f ϕ.

Indeed (see also Ex. 24, Chapter 5), |I1(ε)|, |I2(ε)| ≤ c εQ−1.)


Ex. 20) Let Ω ⊆ G be open, and let u ∈ S(Ω). Show that, for every x ∈ Ω andr > 0 such that Bd(x, r) ⊆ Ω , one has

u(x) = Mr (u)(x) − βd(Q − 2)

∫ r

0

μ(Bd(x, t))

tQ−1dt,

where μ := Lu is the L-Riesz measure of u.(Hint: Write

Γ (d) − Γ (r) =∫ r

d

Γ ′(t) dt

and use Fubini’s theorem in the Poisson–Jensen formula (9.22).)Ex. 21) Assume the same hypotheses as in the previous exercise. Show that

Mr (u)(x) − Mρ(u)(x) = βd(Q − 2)

∫ r

ρ

μ(Bd(x, t))

tQ−1dt,

for 0 < ρ < r .(Hint: First, assume u(x) > −∞ and use the Poisson–Jensen formula(9.22). Then complete the proof by replacing u with its Perron-regularizeduBd(x,ρ/2).)

Ex. 22) Let Ω ⊆ G be open, and let u ∈ S(Ω). Show that, for every x ∈ Ω ,

μ({x}) = 1

βd

limρ↘0

ρQ−2Mρ(u)(x),

where μ := −Lu is the L-Riesz measure of u.(Hint: Use the previous exercise.)

Ex. 23) Assume the same hypotheses as in the previous exercise. Show that

μ({x}) = 0 if u(x) < ∞.

(Hint: Use the previous exercise.)Ex. 24) Let f : G → R ∪ {∞} be defined as follows

f (x) =

⎧⎪⎨

⎪⎩

(d(x))−2 if 0 < d(x) < 1,

∞ if x = 0,

0 if d(x) ≥ 1.

Show that:(i) f ∈ L1(G),

(ii) x → u(x) := ∫RN Γ (y−1 ◦ x)f (y) dy is L-superharmonic in G,

(iii) u(0) = ∞,(iv) μ({0}) = 0, where μ is the L-Riesz measure of u.

10

Maximum Principle on Unbounded Domains

The aim of this chapter is to study the following version of the maximum principlefor sub-Laplacians L on Carnot groups G. An open set Ω ⊆ G is called a maximumprinciple set for L (MP set, in short) if every bounded from above L-subharmonicfunction u is ≤ 0 in Ω whenever

lim supx→y

u(x) ≤ 0 for every y ∈ ∂Ω .

We already know that bounded domains are MP sets. Here, we introduce the notionof L-thin set at infinity, and we prove that Ω is an MP set if and only if G \ Ω

is not L-thin at infinity. Then, we show some geometrical conditions for non-L-thinness at infinity, based on the notion of q-set. We prove that a sufficient conditionfor F to be not L-thin at infinity is that F is not a q-set, for some q > Q − 2.As usual, Q is the homogeneous dimension of G. To this end, a crucial tool willbe Theorem 9.6.1 (page 451), the representation theorem for bounded from aboveL-subharmonic functions on G.

It will follow, in particular, that open sets Ω satisfying an exterior G-cone condi-tion are MP sets, since G \ Ω is not a Q-set, hence not L-thin at infinity. Some otherexplicit criteria are given.

As usual, G = (RN, ◦, δλ) will denote a homogeneous Carnot group and L afixed sub-Laplacian on G.

10.1 MP Sets and L-thinness at Infinity

The maximum principle of Theorem 5.13.4 (page 295) does not hold, in general,if the open set Ω is unbounded.1 The aim of this section is to provide sufficientconditions on unbounded Ω’s for an analogue of the maximum principle to hold fora given sub-Laplacian L.

1 Consider, for example, the classical case of the ordinary Laplace operator on R2, Ω =

{(x, y) ∈ R2 : y > 0}, u(x, y) = exp(x) sin(y). Then u satisfies Δu(x, y) = 0 in Ω ,

u = 0 on ∂Ω , but u is not identically zero.

474 10 Maximum Principle on Unbounded Domains

To begin with, we introduce some notation and definitions. If Ω ⊆ G is open,Sb(Ω) will denote the (cone of the) bounded from above L-subharmonic functionsin Ω . For example, −Γ ∈ Sb(G).

Definition 10.1.1 (MP set for L). We say that Ω is a maximum principle set for L(MP set for L, in short) if the following assertion holds:

{u ∈ Sb(Ω)

lim supx→y u(x) ≤ 0 ∀ y ∈ ∂Ω�⇒ u ≤ 0 in Ω . (10.1)

By Theorem 8.2.19-(ii), page 409, it follows that

every bounded open set is an MP set for L.

We would like to remark that an u.s.c. function u : Ω → [−∞,∞[ satisfying theboundary condition in (10.1) is bounded from above if Ω is bounded.2 Then, if Ω isbounded, in (10.1), Sb(Ω) may be replaced by S(Ω).

The maximum principle is deeply related to the notion of L-thinness at infinity,which we now introduce.

Definition 10.1.2 (L-thinness at infinity). A subset F of G will be said L-thin atinfinity if there exists a bounded from above L-subharmonic function u in G suchthat3

lim supx→∞, x∈F

u(x) � lim supx→∞, x∈G

u(x). (10.2)

From this definition it follows that a set F ⊆ G is not L-thin at infinity iff forevery u ∈ Sb(G)


u(x) = lim supx→∞, x∈G

u(x).

The following theorem holds (see also Fig. 10.1).

Theorem 10.1.3 (Characterization of MP set). An open set Ω ⊆ G is an MP setfor L if and only if G \ Ω is not L-thin at infinity.

The proof of this result will easily follow from the next two lemmas.

Lemma 10.1.4. Let u : Ω → [−∞,∞[ be an L-subharmonic function in Ω satis-fying

lim supx→y

u(x) ≤ 0 ∀ y ∈ ∂Ω. (10.3)

Then the function

v : G → [−∞,∞[, v ={

max{u, 0} in Ω ,0 in G \ Ω

is L-subharmonic in G.2 See Exercise 1 at the end of the Chapter.3 We agree to let


u(x) = −∞,

if F is bounded.

10.1 MP Sets and L-thinness at Infinity 475

Fig. 10.1. A MP set Ω

Proof. Condition (10.3) implies that v is upper semicontinuous in G. Moreover,since max{u, 0} ∈ S(Ω), v is locally sub-mean in Ω . On the other hand, sincev ≥ 0,

v(x) = 0 ≤ Mr (v)(x) for every x ∈ G \ Ω .

Then, v is everywhere locally sub-mean, hence L-subharmonic in G. � Lemma 10.1.5. Let F ⊆ G and let u ∈ Sb(G). Then


u(x) = lim supx→∞, x∈G

u(x) (10.4a)

if and only ifsupF

u = supG

u. (10.4b)

Proof. We may assume that u is not constant. Then, by the maximum principle ofTheorem 8.2.19 (page 409), we have

u(x) < supG

u ∀ x ∈ G. (10.5)

As a consequence,supG

u = lim supx→∞, x∈G

u(x). (10.6)

Thenlim sup

x→∞, x∈F

u(x) ≤ supF

u ≤ supG

u = lim supx→∞, x∈G

u(x).

This shows that (10.4a) implies (10.4b).Let us now assume (10.4b). Then

supF∩Bd(0,R)

u < supF\Bd(0,R)

u ∀ R > 0. (10.7)

Indeed, if (10.7) were false, we would have


supF∩Bd(0,R)

u ≥ supF\Bd(0,R)

u,

for a certain R > 0, so that, for a suitable x0 in the closure if F ∩ Bd(0, R),

u(x0) = supF∩Bd(0,R)

u = supF

u

(by (10.4b)) = supG

u,

thus contradicting (10.5). From (10.7), we obtain


u(x) = supF

u

(by (10.4b)) = supG

u

(by (10.6)) = lim supx→∞, x∈G

u(x),

and (10.4a) follows. � From Lemma 10.1.5 and the very definition of L-thin set, we have the following

assertion.

Corollary 10.1.6 (Characterization of L-thinness at infinity). Let F ⊆ G. ThenF is L-thin at infinity iff there exists u ∈ Sb(G) such that supF u < supG u. Viceversa, F is not L-thin at infinity iff supF u = supG u for all u ∈ Sb(G).

We are now ready to prove Theorem 10.1.3.

Proof (of Theorem 10.1.3). Assuming that Ω is an MP set for L, we shall prove thatG \ Ω is not L-thin at infinity. Let u ∈ Sb(G). Put

U := supG\Ω

u. (10.8)

Thenlim supΩ�x→y

u(x) ≤ u(y) ≤ U ∀ y ∈ ∂Ω,

so that, since Ω is an MP set, u = U in Ω . Definition (10.8) now implies

supG\Ω

u = supG

u,

hence, by Lemma 10.1.5, G \ Ω is not L-thin at infinity.Vice versa, assuming that G \ Ω is not L-thin at infinity, we shall prove that Ω

is an MP set. Let u ∈ Sb(Ω) be such that

lim supx→y

u(x) ≤ 0 ∀ y ∈ ∂Ω.

10.2 q-sets and the Maximum Principle 477

By Lemma 10.1.4, the function

v : G → [−∞,∞[, v ={

max{u, 0} in Ω ,

0 in G \ Ω

is L-subharmonic in G. Then, being G \ Ω not L-thin at infinity,

supG

v = supG\Ω

v = 0.

In particular, v ≤ 0 in Ω . Hence, u ≤ 0 in Ω , and Ω is an MP set. � From Theorem 10.1.3 and Corollary 10.1.6 we obtain the following equivalent

characterization of MP sets.

Corollary 10.1.7 (Characterization of MP set. II). An open set Ω ⊆ G is an MPset for L if and only if supG\Ω u = supG u for all u ∈ Sb(G).

10.2 q-sets and the Maximum Principle

This section is devoted to some geometrical conditions for non-thinness (w.r.t. L) atinfinity. Our criteria will be based on the notion of q-set.

Throughout the sequel, L will denote a fixed sub-Laplacian on G, Γ its funda-mental solution and d any L-gauge for L. We adopt the usual notation

Γ = βd d2−Q

(see Theorem 5.5.6, page 256). We recall that Q denotes the homogeneous dimen-sion of G, and βd is a suitable positive constant.

Definition 10.2.1 (q-set for a sub-Laplacian). Let q be a real positive number.A subset F of G will be called a q-set (w.r.t. L) if there exists a finite or countablefamily of d-balls {Bd(xj , rj )}j∈J such that:

(i) F ⊆ ⋃j∈J Bd(xj , rj ),

(ii)∑

j∈J (rj

d(xj ))q < ∞.

(See also Fig. 10.2.) The next theorem will play a crucial rôle in what follows.

Theorem 10.2.2 (L-thinness at infinity and q-sets). Let F ⊆ G be L-thin at infin-ity. Then F is a q-set for every q > Q − 2.

The proof of Theorem 10.2.2 rests on the following deep lemma.

Lemma 10.2.3. Let u ∈ Sb(G). Then, for every q > Q − 2, there exists a q-setF ⊆ G such that

limx→∞, x /∈F

u = supG

u.


Fig. 10.2. A q-set F

We prove this result in the Appendix of this chapter. Note that the assertion of theabove lemma may fail to be true for q = Q − 2 (see Ex. 7 at the end of the chapter).

We are now in the position to prove Theorem 10.2.2.

Proof (of Theorem 10.2.2). Assume by contradiction that F is not a q0-set, beingq0 > Q − 2. Fixed u ∈ Sb(G), by Lemma 10.2.3, there exists a q0-set F0 ⊆ G suchthat

limx→∞, x /∈F0

u = supG

u. (10.9)

On the other hand, F \ F0 is not a q0-set, since F is not. In particular, F \ F0 isnon-empty and unbounded. Then, by (10.9),

limx→∞, x∈F\F0

u = supG

u,

so thatlim sup

x→∞, x∈F

u(x) = supG

u = lim supx→∞

u(x).

This proves that F is not L-thin at infinity, in contradiction with the hypothesis. � For the future references, it is convenient to rephrase the previous theorem as

follows.

Theorem 10.2.4. Let F ⊆ G. Assume there exists q > Q − 2 such that F is not aq-set. Then F is not L-thin at infinity.

Our more explicit geometric condition for non-L-thinness at infinity will followfrom the next proposition.

Proposition 10.2.5. Let F ⊆ G. Assume there exists a sequence of d-balls{Bd(zj , Rj )}j∈N contained in F and such that


d(zj ) → ∞, lim infj→∞

Rj

d(zj )> 0.

Then F is not a Q-set (hence, it is not L-thin at infinity).

Proof. Let c be the constant in the pseudo-triangle inequality (see Proposition 5.1.7,page 231). Let us put

M := 4c2

and choose a subsequence {zkj}j of {zj }j such that, for each j ∈ N,

d(zkj+1) ≥ M2 d(zkj), Rkj

≥ δ d(zkj), (10.10)

with a suitable δ ∈ ]0, 1/M[. Let us now define

yj := zkj, ρj := δ d(yj ), Bj := Bd(yj , ρj ), F0 :=

⋃

j∈N

Bj .

Since ρj ≤ Rkj, we have Bj ⊆ Bd(zkj

, Rkj) ⊆ F for every j ∈ N. Then, in order to

show that F is not a Q-set, it is enough to prove that F0 is not a Q-set.We argue by contradiction and assume the existence of a family of d-balls

{Bd(xk, rk)}k∈N covering F0 and such that

∞∑

k=1

(rk

d(xk)

)Q

< ∞.

Then, if we fix ε ∈ ]0, 1/M[, there exists k ∈ N such that

rk ≤ ε d(xk) ∀ k ≥ k.

Moreover, there exists j ∈ N such that the family

{Dk}k≥k, Dk := Bd(xk, rk),

is a covering of ⋃

j≥j

Bj .

For every j ∈ N, j ≥ j , we finally define

Kj := {k ∈ N | k ≥ k, Bj ∩ Dk �= ∅}.We claim that

1

M≤ d(xk)

d(yj )≤ M ∀ k ∈ Kj , ∀ j ≥ j . (10.11)

These inequalities, together with the first one in (10.10), imply


Ki ∩ Kj = ∅ if i �= j.

By means of this last fact, we infer that

∞ >∑

k≥k

(rk

d(xk)

)Q

≥∑

j≥j

∑

k∈Kj

(rk

d(xk)

)Q

(by (10.11)) ≥(

1

M

)Q ∑

j≥j

(1

d(yj )

)Q ∑

k∈Kj

(rk)Q

= 1

ωd MQ

∑

j≥j

(1

d(yj )

)Q ∑

k∈Kj

meas(Dk)

(since {Dk}k∈Kjis a covering of Bj )

≥ 1

ωd MQ

∑

j≥j

(1

d(yj )

)Q

meas(Bj )

=(

1

M

)Q ∑

j≥j

(ρj

d(yj )

)Q

= ∞,

since ρj = δ d(yj ) for every j ∈ N. This contradiction shows that F0 cannot be aQ-set.

We are then left with the proof of (10.11). If k ∈ Kj , there exists x′ ∈ Bj ∩ Dk .Hence by the pseudo-triangle inequality,

d(xk) ≤ c (d(yj ) + d(yj , xk)) ≤ c2(d(yj ) + d(yj , x′) + d(x′, xk)

)

≤ c2 (d(yj ) + ρj + rk) ≤ c2((1 + δ) d(yj ) + ε d(xk))

≤ 2 c2 d(yj ) + 1

2d(xk).

Then1

2d(xk) ≤ 2 c2 d(yj ),

and the second inequality in (10.11) follows. With the same argument, one can provethe first one too. �

Examples of subsets of G satisfying the hypothesis of Proposition 10.2.5 are theG-cones. The relevant definition is the following one.

Definition 10.2.6 (GGG-cone). A subset C of G will be said a G-cone with vertex at theorigin if (see also Fig. 10.3)

δλ(x) ∈ C ∀ x ∈ C, ∀ λ > 0.

If C is such a G-cone, we shall call

z0 ◦ C = {z0 ◦ x | x ∈ C }a G-cone with vertex at z0.


Fig. 10.3. G-cones

Proposition 10.2.7. Every open G-cone is not a Q-set.

Proof. It is enough to prove the assertion for an open G-cone C with vertex at theorigin. Let x0 ∈ C \ {0} and R0 > 0 be such that Bd(x0, R0) ⊆ C. Then, since C is aG-cone,

Bd(δλ(x0), λ R0) ⊆ C ∀ λ > 0.

The assertion follows from Proposition 10.2.5, since

zλ := δλ(x0) −→ ∞ as λ → ∞and

λ R0

d(zλ)= R0

d(x0)∀ λ > 0.

This completes the proof. � Corollary 10.2.8. Every half-space of G is not a Q-set.

Proof. Let

Π :={

x = (x1, . . . , xN) ∈ G :N∑

j=1

aj xj > α

}

be an open half-space of G (here, a1, . . . , aN , α ∈ R), and let us consider

C0 := {x = (x1, . . . , xN) ∈ Π : aj xj ≥ 0 ∀ j = 1, . . . , N

}.

C0 is a non-empty open subset of Π with the following property:

x ∈ C0, λ ≥ 1 �⇒ δλ(x) ∈ C0.

Then, if we choose a point x0 ∈ C0 and a real number R0 > 0 such that Bd(x0, R0) ⊆C0, we have Bd(δλ(x0), λ R0) ⊆ C0 for every λ ≥ 1. Since


zλ := δλ(x0) −→ ∞ as λ → ∞and

λ R0

d(zλ)= R0

d(x0)∀ λ ≥ 1,

from Proposition 10.2.5 it follows that C0, and hence Π , is not a Q-set. �

10.3 The Maximum Principle on Unbounded Domains

We collect Theorems 10.1.3, 10.2.2 and 10.2.4, Proposition 10.2.5 and Corollar-ies 10.2.7 and 10.2.8 in the following theorem.

Theorem 10.3.1. The open set Ω ⊆ G is an MP set for L if one of the following(sufficient) conditions is satisfied:

(1) G \ Ω is not L-thin at infinity (this condition is also necessary),(2) G \ Ω is not a q-set for some q > Q − 2,(3) G \ Ω contains a sequence of d-balls {Bd(zj , Rj )}j∈N such that

d(zj ) → ∞, lim infj→∞

Rj

d(zj )> 0,

(4) G \ Ω contains an open G-cone,(5) G \ Ω contains a half-space or, equivalently, Ω is contained in a half-space.

We close this section by giving an application of the previous theorem.

Corollary 10.3.2 (A maximum principle on unbounded domains). Let Ω be anopen subset of G satisfying one of the five conditions of Theorem 10.3.1. Let c :Ω → R, c ≤ 0, and let u be a bounded from above function on Ω of class C2

satisfying {Lu + c u ≥ 0 in Ω ,

lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω .(10.12)


Proof. Define v := max{u, 0} and

Ω0 := {x ∈ Ω |u(x) > 0}.We have to show that Ω0 = ∅. Assume, by contradiction, Ω0 �= ∅. Then, sincec ≤ 0 and v = u in Ω0, the function v is of class C2 in Ω0 and Lv ≥ 0 in Ω0. Itfollows that v is L-subharmonic, hence L-submean, in Ω0. On the other hand, forevery x ∈ Ω \ Ω0 and r > 0 such that Bd(x, r) ⊂ Ω ,

v(x) = 0 ≤ Mr (v)(x).

This shows that v is locally L-submean in Ω . Thus v ∈ Sb(Ω). From the boundarycondition in (10.12) we also get lim supx→y v(x) ≤ 0 for every y ∈ ∂Ω . Since Ω

is an MP set (by Theorem 10.3.1), this implies v ≤ 0 in Ω , i.e. u ≤ 0 in Ω . ThenΩ0 = ∅, and the proof is complete. �

10.4 The Proof of Lemma 10.2.3 483

10.4 Appendix to Chapter 10. The Proof of Lemma 10.2.3

The proof of Lemma 10.2.3 rests on the following deep lemma, an extension to thesub-Laplacian setting of a theorem by Cartan [Cart28] related to the classical Laplaceoperator (see also [HK76, pp. 131–134]).

In the proof of the lemma below, c denotes, as usual, the constant appearing inthe pseudo-triangle inequality for the L-gauge d (and Γ = βd d is the fundamentalsolution for L).

Lemma 10.4.1 (Cartan-type covering lemma). Let μ be a Radon measure on G

with finite total mass μ0 := μ(G) < ∞. Let q > Q − 2 (Q being the homogeneousdimension of G). Then, if h > 0, the set

{x ∈ G | (Γ ∗ μ)(x) ≥ h}can be covered by a finite or countable family of closed d-balls Bd(xn, rn) satisfyingthe following condition

∑

n

rnq < A (μ0/h)q/(Q−2).

The constant A depends only on Q, q and c.

Proof. We fix ν ∈ N and set

rν := (μ0/h)1/(Q−2) 2− 2 νq+Q−2 .

LetDν := {Dk,ν = Bd(xk,ν, rν/2) | k = 1, . . . , kν }

be a maximal family of disjoint closed balls such that

μ(Dk,ν) ≥ μ0

2ν∀ k = 1, . . . , kν.

Since μ0 < ∞ and the balls are disjoint, then kν ≤ 2ν . Define

F :=⋃

ν∈N

⋃

k≤kν

Bd(xk,ν, c rν).

We observe that if x /∈ F then Bd(x, rν/2) does not intersect any ball of themaximal family Dν . Hence

μ(Bd(x, rν/2)) <μ0

2ν∀ ν ∈ N. (10.13)

In particular, we have

μ({x}) = limν→∞ μ(Bd(x, rν/2)) = 0. (10.14)


Hence, if x /∈ F , by (10.14) we have

(Γ ∗ μ)(x) =∫

G\{x}Γ (x−1 ◦ y) dμ(y)

=( ∫

d(x−1◦y)≥ r12

+∞∑

ν=1

∫

rν+12 ≤d(x−1◦y)< rν

2

)Γ (x−1 ◦ y) dμ(y)

≤ βd (r1/2)2−Q μ0 +∞∑

ν=1

βd (rν+1/2)2−Q

∫

rν+12 ≤d(x−1◦y)< rν

2

dμ(y)

≤ (by (10.13)) βd 2Q−2 μ0

∞∑

ν=1

21−ν rν2−Q

= βd 2Q−1 h

∞∑

ν=1

2Q−2−qQ−2+q

ν =: A1 h,

where A1 depends only on Q and q. Then

{x ∈ G | (Γ ∗ μ)(x) > A1 h} ⊆⋃

ν∈N

⋃

k≤kν

Bd(xk,ν, c rν).

Moreover, since kν ≤ 2ν , we have

∑

ν∈N

∑

k≤kν

(c rν)q ≤ A2

(μ0

h

)q/(Q−2)

,

where A2 depends only on q, Q and c. This ends the proof. � We are now ready to prove Lemma 10.2.3.

Proof (of Lemma 10.2.3). Let ν ∈ N. Define

Cν := {x ∈ G | (2 c)ν < d(x) ≤ (2c)ν+1},where c denotes the constant appearing in the pseudo-triangle inequality for d (seeProposition 5.1.7, page 231). By Theorem 9.6.1 (page 451), if μ is the L-Rieszmeasure of u and U := supG u, we have

U − u(x) = (Γ ∗ μ)(x) = I1(x) + I2(x) + I3(x),

where

I1(x) :=∫

d(y)≤(2 c)ν−1βd d(x−1 ◦ y)2−Q dμ(y) ;

I2(x) :=∫

(2 c)ν−1<d(y)<(2 c)ν+2βd d(x−1 ◦ y)2−Q dμ(y) ;

I3(x) :=∫

d(y)≥(2 c)ν+2βd d(x−1 ◦ y)2−Q dμ(y).

10.4 The Proof of Lemma 10.2.3 485

From now on, we shall denote by C, C′, . . . positive constants depending only on Q,c and βd . We also let

n(t) := μ(Bd(0, t)).

If x ∈ Cν , we have

I1(x) ≤ βd (2 c)(ν−1)(2−Q)

∫

d(y)≤(2 c)ν−1dμ(y) ≤ C · (

(2 c)ν)2−Q

n((2 c)ν

)

≤ (because n(t) is increasing) C′∫ ∞

(2 c)ν

n(t)

tQ−1dt.

Analogously,

I3(x) ≤ βd

∫

d(y)≥(2 c)ν+2

(d(y)

2 c

)2−Q

dμ(y) ≤ C

∫ ∞

(2 c)νt2−Q dn(t)

= C′[

n(t)

tQ−2

]∞

(2 c)ν+ C′′

∫ ∞

(2 c)ν

n(t)

tQ−1dt ≤ C′′′

∫ ∞

(2 c)ν

n(t)

tQ−1dt.

The estimate of I2(x), x ∈ Cν , is the crucial step of the proof. Let q > Q − 2 befixed. Define

μν := μ({y ∈ G | (2 c)ν−1 < d(y) < (2 c)ν+2}),

ην := μν (2 c)(2−Q) ν,

εν := η1−(Q−2)/qν .

Then ∞∑

ν=1

ην ≤ C

∞∑

ν=1

∫ (2 c)ν+2

(2 c)ν−1

dn(t)

tQ−2≤ C′

∫ ∞

1

dn(t)

tQ−2< ∞. (10.15)

This last inequality follows from (9.29). On the other hand, by Lemma 10.4.1, thereexists a countable family of closed balls

{Bd(xk,ν, rk,ν)

}k∈Jν

such that

{x ∈ Cν | I2(x) < εν} ⊇ Cν \⋃

k∈Jν

Bd(xk,ν, rk,ν)

and∑

k∈Jν

(rk,ν)q < A

(μν

εν

)q/(Q−2)

. (10.16)

As a consequence,

(rk,ν)q ≤ A

(μν

εν

)q/(Q−2)

= Aην (2c)q ν,

so thatrk,ν ≤ (A ην)

1/q(2c)ν . (10.17)


We may also suppose Bd(xk,ν, rk,ν) ∩ Cν �= ∅ for every k ∈ Jν . This impliesd(xk,ν) ≥ (2 c)ν−2. Indeed,

d(xk,ν) ≥ (see (10.17))1

c(2c)ν − (A ην)

1/q(2c)ν

= (2c)ν(

1

c− (A ην)

1/q

),

and, since ην → 0 as ν → ∞ (see (10.15)), the claim follows. As a consequence,from (10.15), (10.16), the choice of εν and the bound for d(xk,ν) we get

∞∑

ν=1

∑

k∈Jν

(rk,ν

d(xk,ν)

)q

≤ A

∞∑

ν=1

(2 c)ν q

(2 c)(ν−2)qην = C

∞∑

ν=1

ην < ∞. (10.18)

Collecting the estimates for I1, I2 and I3, we finally obtain

U − u(x) < C

∫ ∞

(2 c)ν

n(t)

tQ−1dt + η1−(Q−2)/q

ν (10.19)

for every x ∈ Cν \ ⋃k∈Jν

Bd(xk,ν, rk,ν) and for every ν ∈ N. By (9.29) and the pos-itivity of the exponent of ην , the right-hand side of (10.19) tends to zero, as ν → ∞.Together with (10.18), this proves that the set

F =∞⋃

ν=1

⋃

k∈Jν

Bd(xk,ν, rk,ν)

is a q-set satisfyinglim

x→∞, x /∈Fu = sup

G

u.

This ends the proof. � Bibliographical Notes. In the recent literature, a special attention has been paid tothe maximum principle on unbounded domains. Such a principle plays a key rôlein looking for symmetry properties of solutions to semilinear Poisson equations (see[BN91,BCN97,BHM00]). In order to provide our analogue in the Carnot group set-ting, we have been inspired by some ideas of J. Deny published in 1947 (see Deny’stheorem [Den48, p. 142]; see also [HK76, Theorem 3.21]).

We would like to stress that maximum principles in half-spaces are crucial toolsin looking for monotonicity and symmetry properties of solutions to the semilinearequation Lu+f (u) = 0 in G. See [BHM00] and [BP99] for some recent noteworthyresults in the cases L = Δ and L = ΔHN , respectively. See also [BL02,BL03,BP01,BP02].

Furthermore, we would like to note that Theorem 10.2.2 can be used as a startingpoint in studying the asymptotic behavior of L-subharmonic functions not boundedfrom above in G.

Some of the topics presented in this chapter also appear in [BL02].



1) Suppose that Ω is a bounded open set. Let u : Ω → [−∞,∞[ be u.s.c. Supposethat

lim supx→y

u(x) ≤ 0 ∀ y ∈ ∂Ω.

Then u is bounded from above. (Hint: Let B ⊆ Ω be an open set such that∂Ω ⊂ B and u < 1 on B. Then write Ω = A ∪ B and observe that u is boundedfrom above on A which is a compact subset of Ω . Indeed, any u.s.c. functionrestricted to a compact set attains its maximum!)

2) Does there exist any u ∈ Sb(Ω), non-constant and bounded from below? In case,provide examples. (Hint: See also Theorem 9.3.10.)

3) Let F0 be a q-set. Then, F \ F0 is not a q-set, whenever F is not.4) Complete the proof of (10.11) arguing as in the last paragraph of Proposi-

tion 10.2.5.5) Prove that every subset of a q-set is a q-set and every bounded set is a q-set (for

every q > 0). Prove also that a finite union of q-sets is a q-set.6) Consider the subset of R

3

S = {(x1, x2, x3) ∈ R3 : x1 ≤ −1, x2 = 0, x3 = 0}.

Prove that S is not a q-set when q = 1 and the relevant d in Definition 10.2.1 isthe Euclidean norm. (Hint: Show that if

{Bj = B(cj , rj )}j∈N

is a sequence of ordinary balls in R3, then there exists a constant M > 0 such that

∣∣∣∣∫

H

d x

x

∣∣∣∣ ≤ M∑

j∈N

rj

|cj | , (10.20)

where

H = π1

(S ∩

⋃

j

Bj

)

andπ1 : R

3 → R, π1(x1, x2, x3) = x1.

Thus, if S were a 1-set and {Bj }j the relevant covering of S as in Defini-tion 10.2.1, we would have H = ]−∞,−1], so that the left-hand side of (10.20)would diverge, whereas the right-hand side is finite by the definition of q-set whenq = 1.)


7) In this exercise, we show that the assertion of Lemma 10.2.3 may fail to be truefor q = Q − 2.For instance, let G = (R3,+) be the usual Euclidean group on R

3, and let L = Δ

be the usual Laplace operator. Consider the function

u(x1, x2, x3) = −∫ −1

−∞((x1 − y)2 + x2

2 + x23)−1/2

1 − ydy.

Show that:a) The function u coincides with the convolution

−∫

G

Γ (y−1 ◦ x) dμ(y),

where μ is the measure on R3 supported on the set S of Ex. 6, and coinciding

there withχ(−∞,−1)(y)

1 − ydy.

Here, dy denotes the usual Hausdorff 1-dimensional measure on R1.

b) u ∈ S(R3) satisfies all the conditions in Theorem 9.6.1 (page 451). Indeed,with the notation in that theorem, we have

n(t) = μ(Bd(0, t)) ={

0 if t ∈ ]0, 1]∫ −1−t

11−y

dy = log(1 + t) − log 2 if t > 1.

Thus ∫ ∞

1

n(t)

t2dt < ∞.

c) Deduce that μ is the L-Riesz measure of u in R3, u(0) > −∞ and the least

upper bound of u is 0. Moreover, u = −∞ on S and is finite elsewhere.Note that, if Lemma 10.2.3 were true for q = Q − 2 = 1, there would exist a1-set F ⊆ R

3 such that

lim|x|→∞, x /∈Fu = sup

R3u = 0.

Hence, F would definitely cover S, so that S would be a 1-set. But this is false,as shown by Exercise 6.

11

L-capacity, L-polar Sets and Applications

The aim of this chapter is to provide an ad hoc theory of capacity and of polar sets fora sub-Laplacian L on a Carnot group G. The structure of the fundamental solutionfor L has a prominent rôle in the whole chapter.

Our starting points are a continuity principle for L-potentials and a suitable ver-sion of the Maria–Frostman domination principle. Starting from these results, wedevelop a theory of capacity following two classical approaches: one based on the no-tion of L-energy, see Section 11.4, and one based on the balayage, see Section 11.5.We compare these approaches to each other, showing that they lead to the same con-cept of capacity.

As an application of the above results we prove a Poisson–Jensen formula andthe so-called “fundamental convergence theorem”.

Notation. Throughout the chapter, we shall use the following notation and defini-tions. M denotes the set of Radon measures μ on G, i.e. of the Borel measures onG which are finite on compact sets. For μ ∈ M, we denote by supp(μ) the supportof μ, i.e. the complement of the largest open set with μ-measure zero. We denoteby M0 the subset of M of compactly supported Radon measures. Moreover, if E

is any set, we denote by M(E) (respectively, M0(E)) the set of measures μ ∈ M

(respectively, μ ∈ M0) such that supp(μ) ⊆ E. If μ ∈ M and A is any set, by μ|Awe mean the measure in M defined by μ|A(E) = μ(E ∩ A) for every Borel set E.Moreover, we write A � B whenever A is a compact subset of B. Finally, S+

(Ω)

denotes the set of the non-negative L-superharmonic functions in Ω .

11.1 The Continuity Principle for L-potentials

Theorem 11.1.1 (The continuity principle for L-potentials). Let E � G be a com-pact set, and let μ be a Radon measure in G supported in E. Let x0 ∈ E, and let thefunction (Γ ∗ μ)|E be finite and continuous at x0 (as a function in E). Then Γ ∗ μ

is also continuous at x0 (as a function in the whole G).

490 11 L-capacity, L-polar Sets and Applications

Proof. We can suppose that x0 ∈ ∂E, since otherwise there is nothing to prove. Letus set for brevity V0 := (Γ ∗ μ)(x0). We first observe that μ({x0}) = 0 since

+∞ > V0 ≥∫

{x0}Γ (y−1 ◦ x0) dμ(y) = Γ (0) · μ({x0}) = ∞ · μ({x0}).

If x0 is an isolated point of E, then there exists an open neighborhood O of x0 suchthat O ∩ E = {x0}. Since μ({x0}) = 0 and supp μ ⊆ E, we get μ(O) = 0. Then,by Theorem 9.3.5, page 433 (see also Corollary 9.3.3), Γ ∗ μ is L-harmonic in O

and hence continuous. We can then suppose that x0 ∈ ∂E is a limit point of E. Byhypothesis, we know that limx→x0(Γ ∗ μ)|E(x) = V0, i.e. for fixed ε > 0, thereexists ρ0 > 0 such that

|(Γ ∗ μ)(ξ) − V0| < ε ∀ ξ ∈ Eρ0 . (11.1)

Here and in the following we denote

Er := E ∩ Bd(x0, r).

We have to prove thatlim

x→x0(Γ ∗ μ)|(G\E)(x) = V0. (11.2)

Since μ({x0}) = 0 and∫

GΓ (y−1 ◦ x0) dμ(y) = V0 ∈ R, we have

∫

Eρ

Γ (y−1 ◦ x0) dμ(y) → 0 as ρ → 0+.

Hence there exists ρ1 > 0 (ρ1 ≤ ρ0) such that∫

Eρ

Γ (y−1 ◦ x0) dμ(y) < ε ∀ ρ ≤ ρ1. (11.3)

We now setμ1 = μ|E\Eρ1

,

and we consider Γ ∗ μ1. Such a function is L-harmonic outside the support of μ1.In particular, it is continuous in a neighborhood of x0. Therefore, the function

∫

Eρ1

Γ (y−1 ◦ ·) dμ(y) = Γ ∗ μ − Γ ∗ μ1

is continuous at x0 if restricted to E. In particular, recalling (11.3), there exists ρ2 >

0 (ρ2 < ρ1) such that∫

Eρ1

Γ (y−1 ◦ ξ) dμ(y) ≤∫

Eρ1

Γ (y−1 ◦x0) dμ(y)+ ε < 2ε ∀ ξ ∈ Eρ2 . (11.4)

By the continuity of Γ ∗ μ1 at x0, there also exists ρ3 > 0 (ρ3 ≤ ρ2) such that

|(Γ ∗ μ1)(η) − (Γ ∗ μ1)(x0)| < ε ∀ η ∈ Bd(x0, ρ3). (11.5)

11.2 L-polar Sets 491

We can now prove (11.2). Let x ∈ G \ E be such that x ∈ Bd(x0, (2c)−1ρ3), wherec≥1 is the constant in the pseudo-triangle inequality for d (see Proposition 5.1.8,page 231). Let x ∈ Eρ1 be a point that minimizes the d-distance between x and Eρ1 .We have

|(Γ ∗ μ)(x) − V0|≤ |(Γ ∗ μ)(x) − (Γ ∗ μ)(x)| + |(Γ ∗ μ)(x) − V0|≤ |(Γ ∗ μ1)(x) − (Γ ∗ μ1)(x0)| + |(Γ ∗ μ1)(x0) − (Γ ∗ μ1)(x)|

+∣∣∣∣∫

Eρ1

Γ (y−1 ◦ x) dμ(y)

∣∣∣∣+∣∣∣∣∫

Eρ1

Γ (y−1 ◦ x) dμ(y)

∣∣∣∣+ |(Γ ∗ μ)(x) − V0|.

Let us estimate the far right-hand side. The first two summands are smaller than ε bymeans of (11.5). Indeed, x ∈ Bd(x0, (2c)−1ρ3) ⊂ Bd(x0, ρ3) and x ∈ Bd(x0, ρ3)

since, recalling the choice of x,

d(x, x0) ≤ c(d(x, x) + d(x, x0)) ≤ 2c d(x, x0) < ρ3.

The third summand is smaller than 2ε by means of (11.4) (x ∈ Eρ2 , as we have justseen that d(x, x0) < ρ3 ≤ ρ2). Let us estimate the fourth summand. Again by thechoice of x, for every y ∈ Eρ1 we have

d(x, y) ≤ c(d(x, x) + d(x, y)) ≤ 2c d(x, y).

This immediately gives

Γ (y−1 ◦ x) = d2−Q(x, y) ≤ (2c)Q−2d2−Q(x, y) = (2c)Q−2Γ (y−1 ◦ x),

and we thus derive∫

Eρ1

Γ (y−1 ◦ x) dμ(y) ≤ (2c)Q−2∫

Eρ1

Γ (y−1 ◦ x) dμ(y) < (2c)Q−2 2ε

(here we have used the estimate of the third summand). Finally, the fifth summand issmaller than ε by means of (11.1). This proves (11.2) and completes the proof of thetheorem. �

11.2 L-polar Sets

Definition 11.2.1 (L-polar set). A set E ⊆ G is called L-polar if there exist an openset Ω ⊇ E and a function u ∈ S(Ω) such that u ≡ ∞ in E.

Exercise 11.2.2. (i) Any singleton {x0} is L-polar.(ii) If E1 ⊆ E2 and E2 is L-polar, then also E1 is L-polar.(iii) If E is L-polar, then E is contained in an L-polar set E of Gδ type (i.e. countableintersection of open sets).(iv) If E is L-polar, then E has Lebesgue measure zero.(v) Any countable set is L-polar.


Theorem 11.2.3 (L-polar sets and L-potentials. I). Let E ⊆ G be an L-polarset, and let y /∈ E. Then there exists μ ∈ M such that Γ ∗ μ ≡ ∞ in E and(Γ ∗ μ)(y) < ∞. In particular,1 Γ ∗ μ ∈ S(G).

Proof. By the definition of L-polar set, there exists u ∈ S(Ω) (for some open setΩ ⊇ E) such that u ≡ ∞ in E. Moreover, we can suppose that u > 0 in Ω (recallthat the lower semicontinuity of u implies that {x ∈ Ω |u(x) > 0} is open). Let Dk

be a sequence of open sets such that

Dk ⊆ Dk ⊆ Dk+1,⋃

k∈N

Dk = Ω \ {y}.

By the Riesz representation Theorem 9.4.4, page 442, there exists hk ∈ H(Dk) suchthat

u(x) = hk(x) +∫

Dk

Γ (y−1 ◦ x) dμu(y) ∀ x ∈ Dk. (11.6)

Let us set, for brevity, vk(x) := ∫Dk

Γ (y−1 ◦ x) dμu(y). We remark that vk ∈ S(G),

by means of Corollary 9.3.3, page 433. Since vk is L-harmonic outside Dk (seeTheorem 9.3.5, page 433), we have, in particular, vk(y) ∈ R for every k ∈ N. Wethen set

v(x) :=∑

k∈N

2−kvk(x)/(vk(y) + 1) ∀ x ∈ G.

Since v(y) ≤ ∑k∈N

2−k < ∞, we have v ∈ S(G) (see Corollary 8.2.8, page 403).Moreover, v ≥ 0. Hence we can use Corollary 9.4.8, page 444, and obtain

v = Γ ∗ μv + infG

v.

It is now easy to see that μ := μv satisfies the assertion of the theorem. Indeed,v(y) < ∞ immediately gives (Γ ∗ μ)(y) < ∞. Moreover, for every x0 ∈ E,there exists k0 ∈ N such that x0 ∈ Dk0 . Since u ≡ ∞ in E, (11.6) implies thatvk0(x0) = ∞. As a consequence, (Γ ∗μ)(x0) = v(x0)− infG v = ∞. This ends theproof. �Example 11.2.4. [There exist finite-valued discontinuous L-superharmonic func-tions.] Indeed, let E ⊂ G be a non-closed L-polar set (let, e.g. E = {xn}n∈N withxn → 0 and 0 /∈ E), and let x0 be a limit point for E not belonging to E. ByTheorem 11.2.3, there exists v ∈ S(G) such that v ≡ ∞ on E and v(x0) < ∞.Let now u := min{v, v(x0) + 1}. Then u clearly satisfies the requisites of theassertion. �Corollary 11.2.5 (L-polar sets and L-potentials. II). Let E ⊆ G be a boundedL-polar set, and let y /∈ E. Then there exists ν ∈ M0 such that Γ ∗ ν ≡ ∞ in E and(Γ ∗ ν)(y) < ∞.

1 By Theorem 9.3.2, page 432.

11.2 L-polar Sets 493

Proof. By Theorem 11.2.3, there exists μ ∈ M such that Γ ∗ μ ≡ ∞ in E and(Γ ∗ μ)(y) < ∞. We have

E ∪ {y} � D(0, R) =: B,

for some R > 0. Setting ν := μ|B , we have ν ∈ M0. Moreover, by the Rieszrepresentation Theorem 9.4.4, page 442 (see also Theorem 9.3.5, page 433), thereexists h ∈ H(B) such that

Γ ∗ μ = h + Γ ∗ ν

in B. It is now immediate to recognize that Γ ∗ ν has the required properties. Thiscompletes the proof. �Corollary 11.2.6. Let {Ej }j∈N be a sequence of L-polar sets. Then

⋃j∈N

Ej is L-polar.

Proof. Using Theorem 11.2.3 and observing that L-polar sets have Lebesgue mea-sure zero (since L-superharmonic functions are L1

loc), we can find y ∈ G anduj ∈ S(G), uj ≥ 0, such that uj |Ej

≡ ∞ and uj (y) < ∞ for every j ∈ N.We then set

u(x) :=∑

k∈N

2−kuk(x)

(uk(y) + 1)∀ x ∈ G.

Since u(y) ≤ ∑k∈N

2−k < ∞, we have u ∈ S(G) (see Corollary 8.2.8, page 403).Moreover, u ≡ ∞ in

⋃j∈N

Ej . �We end the section with an improvement of the maximum principle for L.

Theorem 11.2.7 (The extended maximum principle for L). Let Ω ⊂ G be abounded domain, and let u ∈ S(Ω) be bounded from above. If

lim supx→ξ

u(x) ≤ M for every ξ ∈ ∂Ω

except for an L-polar set E ⊂ ∂Ω , then u ≤ M on Ω .

Proof. Let x0 ∈ Ω . By Theorem 11.2.3, there exists ω ∈ S(G), ω < 0 and fi-nite in x0, such that ω|E ≡ −∞. Then, for ε > 0, the function uε = u + ε ω isL-subharmonic in Ω and has lim sup less than M on ∂Ω . Therefore, by the weakmaximum principle (Theorem 8.2.19, page 409), we get

u(x0) + ε ω(x0) ≤ M.

Letting ε → 0+ and recalling that ω(x0) > −∞, we finally get u(x0) ≤ M . Thisends the proof. �


11.3 The Maria–Frostman Domination Principle

Lemma 11.3.1. Let Ω ⊆ G be an open set, and let f, g ∈ S(Ω). If f ≤ g a.e. in Ω ,then f ≤ g in Ω . In particular, if f = g a.e., then f ≡ g.

Proof. By Theorem 8.2.11, page 405, we know that f (x) = limr→0+ Mr (f )(x) andg(x) = limr→0+ Mr (g)(x) for every x ∈ Ω . On the other hand, f ≤ g a.e. impliesMr (f ) ≤ Mr (g). Letting r → 0+, we get the assertion. �

The following result is a particular case of Exercise 24, Chapter 6. We providethe proof for the sake of completeness.

Lemma 11.3.2. Let Ω2 ⊆ Ω1 ⊆ G be open sets, and let s ∈ S(Ω1), u ∈ S(Ω2) besuch that lim supΩ2�x→y u(x) ≤ s(y) for every y ∈ (∂Ω2) ∩ Ω1. Then

v : Ω1 −→ [−∞,∞[, v(x) :={

max{s(x), u(x)}, x ∈ Ω2,

s(x), x ∈ Ω1 \ Ω2,

is L-subharmonic in Ω1. (See Fig. 11.1.)

Fig. 11.1. Figure of Lemma 11.3.2

Proof. Clearly we have v ∈ S(Ω1 \ Ω2) and v ∈ S(Ω2) (recall Proposition 6.5.4,page 355). Let y ∈ (∂Ω2)∩Ω1, and let us prove that lim supx→y v(x) ≤ v(y). Sinces is u.s.c. in Ω1, we have

lim supΩ1\Ω2�x→y

v(x) = lim supΩ1\Ω2�x→y

s(x) ≤ s(y) = v(y),

lim supΩ2�x→y

v(x) = lim supΩ2�x→y

(max{s(x), u(x)})

≤ max{

lim supΩ2�x→y

s(x), lim supΩ2�x→y

u(x)}≤ s(y) = v(y).

This proves that v is u.s.c. in Ω1. Moreover, v is finite in a dense subset of Ω1, sinces ≤ v has the same property. By Theorem 8.2.1, page 401, we only have to prove that

11.3 The Maria–Frostman Domination Principle 495

v is sub-mean. Let y ∈ (∂Ω2) ∩ Ω1, Bd(y, r) ⊂ Ω1. Since v ≥ s and s ∈ S(Ω1),we have

Mr (v)(y) ≥ Mr (s)(y) ≥ s(y) = v(y).

Observing that v ∈ S(Ω1\Ω2) and v ∈ S(Ω2) imply that v is sub-mean in Ω1\∂Ω2,this completes the proof. �Lemma 11.3.3 (Lusin-type theorem for potentials). Let μ ∈ M0 be such that Γ ∗μ < ∞ in K = supp μ � G.Then, for every ε > 0, there exists a compact set C ⊆ K such that:

(i) μ(K \ C) < ε,(ii) Γ ∗ (μ|C) ∈ C(G, R).

Proof. We first observe that Γ ∗μ < ∞ in G, since Γ ∗μ < ∞ in K by hypothesisand

Γ ∗ μ ∈ H(G \ supp μ)

by Theorem 9.3.5, page 433. Moreover, Γ ∗ μ is l.s.c. Hence Γ ∗ μ is measurable,and we can apply the Lusin theorem (see, e.g. [Rud87]): for every ε > 0, there existsa compact set C ⊆ K such that μ(K \ C) < ε and (Γ ∗ μ)|C is continuous in C.Since Γ ∗ μ < ∞ in G, we can write

Γ ∗ (μ|C) = Γ ∗ μ − Γ ∗ (μ|K\C).

Since Γ ∗(μ|C) is l.s.c., −Γ ∗(μ|K\C) is u.s.c. and (Γ ∗μ)|C is continuous in C, weobtain that (Γ ∗(μ|C))|C is continuous in C. We can now use the continuity principlefor potentials (Theorem 11.1.1, page 489) and get that Γ ∗ (μ|C) is continuous in C.On the other hand, Γ ∗ (μ|C) is L-harmonic and hence continuous in G\ supp (μ|C).Therefore, Γ ∗ (μ|C) is continuous in G. This ends the proof. �Theorem 11.3.4 (The Maria–Frostman domination principle). Let μ ∈ M besuch that Γ ∗μ is finite in G. Let P be an L-polar set. If u ∈ S+

(G) and u ≥ Γ ∗μ

in (supp μ) \ P , then u ≥ Γ ∗ μ in G.

Proof. Since P is L-polar, by Theorem 11.2.3 we can find ν ∈ M such that

ω := Γ ∗ ν ∈ S(G), ω ≡ ∞ in P .

Let ε > 0. Clearly, we have

u + ε ω ≥ Γ ∗ μ in suppμ. (11.7)

Let Kn � G be such that

Kn ⊆ Kn+1 and⋃

n∈NKn = G.

Let us set μn := μ|Kn . Since we obviously have Γ ∗μn ≤ Γ ∗μ < ∞ in G, we canapply Lemma 11.3.3 and find


Ln,ε � supp μn = Kn ∩ supp μ

such that μn(Kn\Ln,ε) < ε and vn,ε := Γ ∗(μn|Ln,ε ) ∈ C(G). We explicitly remarkthat

vn,ε ≤ Γ ∗ μn ≤ Γ ∗ μ < ∞ in G. (11.8)

We now set

s(x) :={

max{vn,ε(x) − u(x) − εω(x), 0}, x ∈ G \ Ln,ε,

0, x ∈ Ln,ε.

Since vn,ε is L-harmonic outside Ln,ε (see Theorem 9.3.5, page 433), we have vn,ε−u − εω ∈ S(G \ Ln,ε). Moreover, for every y ∈ ∂Ln,ε, we have

lim supG\Ln,ε�x→y

(vn,ε(x) − u(x) − εω(x))

≤ (vn,ε ∈ C(G),−u,−εω ∈ S(G))

≤ vn,ε(y) − u(y) − εω(y) ≤ 0,

by recalling thatvn,ε ≤ Γ ∗ μ ≤ u + εω in supp(μ), (11.9)

thanks to (11.8) and (11.7). We can then apply Lemma 11.3.2 with Ω1 = G andΩ2 = G \ Ln,ε and obtain that s ∈ S(G). Since

0 ≤ s ≤ vn,ε in G, s ∈ S(G), vn,ε ∈ S(G),

by Theorem 6.6.1 (page 358) there exists h ∈ H(G) such that 0 ≤ s ≤ h ≤ vn,ε

in G. We now use Theorem 9.3.7, page 434, and we obtain that h ≡ 0. This impliess ≡ 0 in G. By the definition of s, this means that vn,ε − u − εω ≤ 0 in G \ Ln,ε.Since also (11.9) holds, we obtain

vn,ε ≤ u + ε ω in G. (11.10)

For every x ∈ G \ (Kn ∩ supp μ), we have∫

(Kn∩suppμ)\Ln,ε

Γ (y−1 ◦ x) dμn(y)

≤ μn

((Kn ∩ supp μ) \ Ln,ε

) · sup{Γ (y−1 ◦ x) | y ∈ (Kn ∩ supp μ) \ Ln,ε

}

≤ ε sup{Γ (y−1 ◦ x) | y ∈ (Kn ∩ supp μ)

}< ∞,

and then, using (11.10), we obtain

(Γ ∗ μn)(x) = vn,ε(x) +∫

(Kn∩supp μ)\Ln,ε

Γ (y−1 ◦ x) dμn(y)

≤ u(x) + εω(x) +∫

(Kn∩suppμ)\Ln,ε

Γ (y−1 ◦ x) dμn(y)

≤ u(x) + εω(x) + ε sup{Γ (y−1 ◦ x) | y ∈ (Kn ∩ supp μ)

}.

11.4 L-energy and L-equilibrium Potentials 497

Since ω ∈ S(G), we have ω < ∞ a.e. (see Corollary 8.2.4, page 402). Lettingε → 0, we then obtain

Γ ∗ μn ≤ u a.e. in G \ (Kn ∩ supp μ).

On the other hand, we know by hypothesis that Γ ∗ μn ≤ Γ ∗ μ ≤ u in supp μ \ P ,hence a.e. in supp μ (P is L-polar and then has Lebesgue measure zero). Therefore,Γ ∗μn ≤ u a.e. in G. Recalling that Γ ∗μn, u ∈ S(G), by Lemma 11.3.1, we obtainΓ ∗ μn ≤ u in G. Since Γ ∗ μn → Γ ∗ μ pointwise in G, we finally get Γ ∗ μ ≤ u

in G. �

11.4 L-energy and L-equilibrium Potentials

Definition 11.4.1 (L-energy and L-equilibrium value). Let E � G be a compactset. For any Radon measure μ on G, supported in E, we set

I (μ) :=∫

G

(Γ ∗ μ)(x) dμ(x) =∫

E

∫

E

Γ (y−1 ◦ x) dμ(x) dμ(y).

I (μ) is called the L-energy of μ.We then define the L-equilibrium value V (E) of E as

V (E) := inf{I (μ) |μ Radon measure, supp μ ⊆ E, μ(E) = 1

}.

The next theorem states that the above infimum is actually a minimum.

Theorem 11.4.2 (L-equilibrium potential). Let E � G be a compact set. Thenthere exists a Radon measure μ supported in E such that μ(E) = 1 and

I (μ) = V (E) = min{I (μ) |μ Radon measure, supp μ ⊆ E, μ(E) = 1

}.

Such a measure μ will be called an L-equilibrium distribution for E. The relatedpotential Γ ∗ μ will be called an L-equilibrium potential for E.

In Section 11.5, we shall prove that the L-equilibrium potential is unique ifV (E) < ∞ (see Corollary 11.5.13).

Proof. We can assume that V (E) < ∞, otherwise there is nothing to prove. Let ustake a minimizing sequence {μn}n∈N of Radon measures such that supp μn ⊆ E,μn(E) = 1 and V (E) = limn→∞ I (μn). Then2 there exists a subsequence {μnk

}k∈N

and a Radon measure μ, supp μ ⊆ E, such that μnk

w∗−→ μ as k → ∞, i.e.

limk→∞

∫

E

f dμnk=∫

E

f dμ for every f ∈ C(E).

2 See, e.g. [Rud87].


In particular, taking f ≡ 1, we have μ(E) = 1. Then V (E) ≤ I (μ), by definition

of V (E). On the other hand, μnk

w∗−→ μ also implies that3

μnk⊗ μnk

w∗−→ μ ⊗ μ in E × E.

Observing that (x, y) �→ Γ (y−1 ◦ x) is l.s.c. in E × E, we obtain

V (E) = limk→∞ I (μnk

) = lim infk→∞

(∫

E

∫

E

Γ (y−1 ◦ x) dμnk(x) dμnk

(y)

)

≥∫

E

∫

E

Γ (y−1 ◦ x) dμ(x) dμ(y) = I (μ).

We have here used a general property of weakly convergent measures.4 ThereforeV (E) = I (μ) and the proof is complete. �

The main goal of this section is to prove Theorem 11.4.5 below as a consequenceof the Maria–Frostman domination principle proved in Section 11.3. To this end weneed the following definition.

Definition 11.4.3 (L-polar∗ set). We shall say that a set E ⊂ G is L-polar∗ ifthere exists a countable family of compact sets Fn such that V (Fn) = ∞ andE ⊆⋃

n∈NFn.

In Section 11.5 (Corollary 11.5.12) we shall prove that any L-polar∗ set is anL-polar set according to our previous Definition 11.2.1, page 491.

Lemma 11.4.4. Let E � G be a compact set and let ν be a Radon measure supportedin E such that ν(E) > 0 and I (ν) < ∞. Then, for any L-polar∗ set E1, we haveν(E1) = 0.

Proof. Let us prove the lemma in the case that E1 is a compact set such thatV (E1) = ∞. The general case will follow immediately from the definition of L-polar∗ set. If by contradiction ν(E1) > 0, we can consider μ := ν|E1/ν(E1). Obvi-ously, supp μ ⊆ E1 and μ(E1) = 1. Moreover,

I (μ) = ν−2(E1)

∫

E1

∫

E1

Γ (y−1 ◦ x) dν(x) dν(y) ≤ ν−2(E1)I (ν) < ∞.

This contradicts V (E1) = ∞ and completes the proof. �Theorem 11.4.5 (The fundamental theorem on L-equilibrium potentials). LetE � G be a compact set such that V (E) < ∞. Let μ be an L-equilibrium dis-tribution and UE = Γ ∗ μ an L-equilibrium potential for E. Then we have

{UE(x) ≤ V (E) for all x ∈ G,

UE(x) = V (E) for all x ∈ E \ P for a suitable L-polar∗ set P .(11.11)

3 Ibidem.4 Ibidem.

11.4 L-energy and L-equilibrium Potentials 499

Proof. We first observe that we only need to prove

UE ≤ V (E) in supp(μ), (11.12)

UE ≥ V (E) in E \ P for a suitable L-polar∗ set P . (11.13)

Then the assertion of the theorem will follow from the Maria–Frostman dominationprinciple (Theorem 11.3.4), observing that UE = Γ ∗ μ is L-harmonic outside thesupport of μ and then it is finite in G if (11.12) also holds.

We now want to prove (11.13) by showing that

P := {x ∈ E |UE(x) < V (E)}is L-polar∗. We have P =⋃

n∈NAn, where

An := {x ∈ E |UE(x) ≤ V (E) − 1/n}are compact sets, since UE is l.s.c. We have to show that V (An) = ∞. Let us as-sume by contradiction that V (An0) < ∞ for some n0 ∈ N, i.e. that there exists aRadon measure σ such that supp σ ⊆ An0 , σ(An0) = 1 and I (σ ) < ∞. We set

ε := (2n0)−1, and we observe that there exists a

(ε)0 ∈ supp μ such that UE(a

(ε)0 ) >

V (E) − ε. Indeed, UE ≤ V (E) − ε in supp μ would give the contradiction

V (E) =∫

UE dμ ≤ (V (E) − ε) μ(supp μ) = V (E) − ε.

Recalling again that UE is l.s.c., we can find a radius r(ε) > 0 such that

UE > V (E) − ε in Bε := Bd(a(ε)0 , r(ε))

and Bε is at a positive distance from An0 . Moreover, a(ε)0 ∈ supp μ implies that

m(ε) := μ(Bε) > 0. We also have

m(ε) ≤ μ(supp μ) = 1.

We now set σ (ε) := m(ε)σ , and we define a (signed) measure σ(ε)1 in E by

σ(ε)1 =

⎧⎨

⎩

σ (ε) in An0 ,

−μ in Bε,

0 elsewhere.

Then, for 0 < η < 1, μ(ε)1 := μ + η σ

(ε)1 is a (positive) Radon measure supported in

E and

μ(ε)1 (E) = μ(E) + η (σ (ε)(An0) − μ(Bε)) = 1.

Moreover, it is easy to see that the integral


I (σ(ε)1 ) :=

∫

E

∫

E

Γ (y−1 ◦ x) dσ(ε)1 (x) dσ

(ε)1 (y)

is convergent, since I (σ ) < ∞, I (μ) < ∞ and Bε is at a positive distance from An0 .Now, recalling that Γ (x−1 ◦ y) = Γ (y−1 ◦ x), we have

I (μ(ε)1 ) − I (μ) =

∫

E

∫

E

Γ (y−1 ◦ x) dμ(x) η dσ(ε)1 (y)

+∫

E

∫

E

Γ (y−1 ◦ x)η dσ(ε)1 (x) dμ(y) + η2I (σ

(ε)1 )

= 2η

∫

E

∫

E

Γ (y−1 ◦ x) dμ(y) dσ(ε)1 (x) + η2I (σ

(ε)1 )

= 2η

∫

E

UE dσ(ε)1 + η2I (σ

(ε)1 )

= 2η

(∫

An0

UE dσ (ε) −∫

Bε

UE dμ

)+ η2I (σ

(ε)1 )

≤ 2η{(V (E) − 2ε)σ (ε)(An0) − (V (E) − ε)μ(Bε)} + η2I (σ(ε)1 )

= 2η (I (σ(ε)1 )η/2 − εm(ε)) < 0

if η is small enough. This contradicts the minimality property of I (μ) andproves (11.13).

We now turn to the proof of (11.12). We argue by contradiction assuming thatthere exists x0 ∈ supp μ such that UE(x0) > V (E). Since UE is l.s.c., there existε > 0 and ρ(ε) > 0 such that UE > V (E) + ε in Bd(x0, ρ

(ε)). Moreover, m(ε)0 :=

μ(Bd(x0, ρ(ε))) > 0 since x0 ∈ supp μ. We have already proved that the set

E1 := {x ∈ supp μ |UE(x) < V (E)}is L-polar∗. Then μ(E1) = 0 by Lemma 11.4.4. Therefore, setting

E2 := (supp μ) ∩ Bd(x0, ρ(ε)), E3 := (supp μ) \ (E1 ∪ E2),

we have

V (E) =∫

UE dμ =∫

E2

UE dμ +∫

E3

UE dμ

≥ (V (E) + ε) μ(E2) + V (E)μ(E3)

= (V (E) + ε)m(ε)0 + V (E) (1 − m

(ε)0 ) = V (E) + ε m

(ε)0 > V (E).

This gives a contradiction and completes the proof of the theorem. �

11.5 L-balayage and L-capacity

In this section, we show some important properties of the reduced function and thebalayage in the setting of the L-harmonic spaces. We will start from the general

11.5 L-balayage and L-capacity 501

abstract results proved in Section 6.11, page 375. Our main ingredients will be thecharacterizations of the L-superharmonicity in terms of the average operators Mr

and Mr , proved in Chapter 8.To begin with, for reading convenience, we rewrite the definitions of reduced

function and balayage in the L-harmonic space.

Definition 11.5.1 (L-reduced function, L-balayage). Given A ⊆ G and u ∈S+

(G), the L-reduced function (or L-réduite) of u relative to A is

RuA := inf{f | f ∈ Φu

A},where

ΦuA := {f ∈ S+

(G) | f ≥ u in A }. (11.14)

The L-balayage of u relative to A is the lower semicontinuous regularization RuA of

RuA, i.e. for any x ∈ G,

RuA(x) := lim inf

y→xRu

A(y).

Since all the results in Chapter 6 apply to the present setting, from Theo-rems 6.11.6 and 6.11.7 (page 377) we get a series of properties of the L-reducedfunction and the L-balayage that we list here for reading convenience.

(I) RfA ≤ Rf

A in G, RfA = Rf

A in G \ ∂A, RfA = f in Int(A),

(II) RfA is L-subharmonic in G and L-harmonic in G \ A,

(III) if 0 ≤ f ≤ g then RfA ≤ Rg

A, RfA ≤ Rg

A,

(IV) if A ⊆ B ⊆ G then RfA ≤ Rf

B , RfA ≤ Rf

B .

From Theorem 9.5.6 (page 449) we also have the following crucial result

(V) RfA = Rf

A almost everywhere in G.

Moreover, by Corollary 9.5.8 (page 450),

(VI) RfA(x) = limr→0 Mr (R

fA)(x) for every x ∈ G,

where Mr is the mean value operator in (5.50f), page 259. Properties (V) and (VI)allow to give an easy proof of the following theorem.

Theorem 11.5.2 (Further properties of L-reduction and L-balayage). Let A ⊆ G,and let u, v ∈ S+

(G). One has:

(i) Ru+vA ≤ Ru

A + RvA,

(ii) Ru+vA ≤ Ru

A + RvA.

Proof. (i). If f ∈ ΦuA and g ∈ Φv

A then f + g ∈ Φu+vA . As a consequence,

Ru+vA ≤ f + g.

By taking the infimum with respect to f ∈ ΦuA and g ∈ Φv

A, we obtain (i).


(ii). By using property (VI), from (i) we immediately obtain

Ru+vA (x) = lim

r→0Mr (R

u+vA )(x) ≤ lim

r→0Mr (Ru

A)(x) + limr→0

Mr (RvA)(x)

= RuA(x) + Rv

A(x)

at any point x ∈ G. �When A ⊆ G is compact, the balayage takes the following form.

Theorem 11.5.3. Let K � G be a compact set, and let u ∈ S+(G). Then there exists

a Radon measure μ in G such that RuK = Γ ∗ μ.

Proof. Since RuK ∈ S+

(G) (see Theorem 6.11.6, page 377), by Corollary 9.4.8,page 444, we only need to prove that

infG

RuK = 0.

We first assume that u is bounded in K . Then there exists λ > 0 such that u ≤ λ Γ

in K , so that λ Γ ∈ ΦuK . As a consequence,

0 ≤ RuK ≤ Ru

K ≤ λ Γ in G,

which immediately gives infG RuK = 0.

We now consider the general case. By the cited Corollary 9.4.8, we have u =v + h in G, where v = Γ ∗ μ is an L-potential and h ≡ infG u. Since infG v = 0(see Theorem 9.3.7, page 434) and

0 ≤ RvK ≤ Rv

K ≤ v

we have infG RvK = 0. Hence Rv

K is an L-potential (again by Corollary 9.4.8). Onthe other hand, since h is bounded in K , from the first part of the proof it followsthat also Rh

K is an L-potential. Therefore, RvK + Rh

K is an L-potential and thus it hasinfimum = 0 (again by Theorem 9.3.7). We now use

RuK ≤ Rv

K + RhK

(see (6.31b), page 377), and we obtain infG RuK = 0. This ends the proof. �

We now give the definition of L-capacity. In Theorem 11.5.8 below, we shall seethat this L-capacity turns out to be equal to the inverse of the L-equilibrium valuedefined in Section 11.4.

Definition 11.5.4 (L-capacity for a compact set). Let K � G be a compact set. Wedefine

WK := R1K, VK := R1

K.

By Theorem 11.5.3, VK is an L-potential called L-capacitary potential of K . More-over, the L-Riesz measure μK of VK will be called the L-capacitary distributionfor K (so that VK = Γ ∗ μK ). We define the L-capacity of K as

C(K) := μK(K). (11.15)


We shall use the following properties of the L-capacitary potential:

0 ≤ VK ≤ WK ≤ 1 in G, VK = WK in G \ ∂K, WK = 1 in K, (11.16a)

supp μK ⊆ ∂K, VK ∈ H(G \ ∂K), lim|x|→∞VK(x) = 0. (11.16b)

The properties in (11.16a) and the inclusion VK ∈ H(G \ K) directly follow fromproperties (I) and (II) of the balayage listed in the previous pages. Let us completethe proof of (11.16b). Since the constant 1 is L-harmonic and VK = 1 in Int(K), weobtain that

VK ∈ H(G \ ∂K).

Hence, the inclusion supp μK ⊆ ∂K readily follows recalling that μK is the L-Rieszmeasure of VK . Finally, by the structure of the fundamental solution Γ = d2−Q

(being d a suitable L-gauge), we have

VK(x) =∫

K

Γ (y−1 ◦ x) dμK(y) ≤ d(x,K)2−QμK(K) → 0 as |x| → ∞.

Theorem 11.5.5 (On the L-capacitary potential). Let K � G be a compact setand let μ ∈ M(K) be a Radon measure such that Γ ∗μ ≤ 1 in G. Then Γ ∗μ ≤ VK

in G.

Proof. Let v ∈ Φ1K , i.e. v ∈ S+

(G), v ≥ 1 in K . Since

Γ ∗ μ ≤ 1 ≤ v in supp μ,

we have Γ ∗ μ ≤ v in G, in force of the Maria–Frostman domination principle(Theorem 11.3.4, page 495). Therefore,

Γ ∗ μ ≤ inf{v | v ∈ Φ1K } = WK.

Recalling that Γ ∗ μ is l.s.c., we finally obtain Γ ∗ μ ≤ VK in G. �Theorem 11.5.6 (On the L-capacity. I). Let K � G be a compact set. Then

C(K) = max{μ(K) |μ ∈ M(K), Γ ∗ μ ≤ 1 in G

}. (11.17)

Proof. We first observe that the L-capacitary distribution μK for K has the proper-ties μK ∈ M(K), Γ ∗ μK ≤ 1 in G, by (11.16a)-(11.16b). Let Ω be an open setsuch that K ⊂ Ω ⊂ Ω � G, and let ν = μΩ be the L-capacitary distribution for Ω .Then

VΩ = Γ ∗ ν ≡ 1 in Ω ⊃ K ,

by (11.16a). Let now μ ∈ M(K) be such that Γ ∗ μ ≤ 1 in G. By Theorem 11.5.5,we have

Γ ∗ μ ≤ VK = Γ ∗ μK in G.

Recalling that Γ (x) = Γ (x−1), we obtain


μ(K) =∫

K

1 dμ =∫

K

Γ ∗ ν dμ =∫

G

Γ ∗ ν dμ =∫

G

Γ ∗ μ dν ≤∫

G

Γ ∗ μK dν

=∫

G

Γ ∗ ν dμK =∫

K

Γ ∗ ν dμK =∫

K

1 dμK = μK(K) = C(K).

This completes the proof. �The L-capacitary potentials have the following sub-additivity property.

Theorem 11.5.7 (Sub-additivity of the L-capacitary potential). Let K1 and K2 becompact subsets of G. Then:

(i) WK1∪K2 + WK1∩K2 ≤ WK1 + WK2 ,(ii) VK1∪K2 + VK1∩K2 ≤ VK1 + VK2 .

Proof. (i). The inequality trivially holds on K1 ∪ K2. Indeed, if y ∈ K1 ∪ K2, theny ∈ K1 ∩ K2 or y ∈ Ki \ Kj for i �= j . In the first case,

WK1∪K2(y) = WK1∩K2(y) = WK1(y) = WK2(y) = 1,

while in the second case

WK1∪K2(y) = WK2(y) = 1, WK1∩K2(y) ≤ WK2(y).

To show that (i) holds in Ω := G \ (K1 ∪ K2), let us consider ui ∈ Φ1Ki

, i = 1, 2.Since WK = VK in G \ K , for every compact K , and VK(x) → 0 as |x| → ∞, wehave

lim inf|x|→∞ w(x) := lim inf|x|→∞(u1(x) + u2(x) − (WK1∪K2(x) + WK1∩K2(x)))

≥ lim inf|x|→∞(u1(x) + u2(x)) ≥ 0.

Moreover, if y ∈ ∂Ω = K1 ∪ K2, then

lim infΩ�x→y

w(x) ≥ u1(y) + u2(y) − lim supΩ�x→y

(WK1∪K2(x) + WK1∩K2(x)) ≥ 0.

Indeed, if y ∈ K1 ∩ K2, we have

u1(y) + u2(y) ≥ 2 ≥ supG

(WK1∪K2 + WK1∩K2),

while, if y ∈ Ki \ Kj , for i �= j ,

ui(y) ≥ 1 ≥ supG

WK1∪K2 and uj (y) ≥ WK1∩K2(y) = limx→y

WK1∩K2(x),

since WK1∩K2 is L-harmonic, hence continuous, in a neighborhood of y. Then, theminimum principle for L-superharmonic functions (Theorem 8.2.19-(ii)) impliesw ≥ 0 in Ω .

We have thus proved that


u1 + u2 ≥ WK1∪K2 + WK1∩K2 in G,

for every ui ∈ Φ1Ki

, i = 1, 2. From this inequality (i) immediately follows.(ii). From (i) and property (VI) we have

VK1∪K2 + VK1∩K2 ≤ VK1 + VK2 a.e. in G.

This inequality extends on G in force of Corollary 9.5.8, page 450. �Theorem 11.5.8 (On the L-capacity. II). Let K � G be a compact set, and letC(K) and V (K) be the L-capacity and the L-equilibrium value of K . Then we have

C(K) = (V (K))−1.

Moreover, if C(K) > 0 and μK is the L-capacitary distribution for K , thenC(K)−1μK is an L-equilibrium distribution for K .

Proof. We first observe that, by definition, it is always C(K) < ∞ and V (K) > 0.If C(K) > 0, setting ν = C(K)−1μK , we have ν(K) = 1. Moreover, supp ν ⊆ K

(see (11.16b)). Hence, by Definition 11.4.1 of V (K) and recalling that Γ ∗ μK =VK ≤ 1 (see (11.16a)), we have

V (K) ≤ I (ν) = C(K)−2∫

(Γ ∗ μK) dμK ≤ C(K)−2μK(K) = C(K)−1 < ∞.

On the other hand, if V (K) < ∞ and μ is an L-equilibrium distribution for K , then,setting μ = V (K)−1μ, we have

Γ ∗ μ = V (K)−1(Γ ∗ μ) ≤ 1 in G,

in force of the fundamental Theorem 11.4.5 on L-equilibrium potentials. Therefore,from Theorem 11.5.6 it follows that

C(K) ≥ μ(K) = V (K)−1μ(K) = V (K)−1 > 0.

This proves that C(K) > 0 iff V (K) < ∞ and that we have C(K) = V (K)−1.The last statement of the theorem is an immediate consequence of the abovearguments. �Lemma 11.5.9. If {Kj }j is a decreasing sequence of compact sets, then WKj

↓W⋂

j Kjas j → ∞.

Proof. Let us set K :=⋂j Kj . From (6.31a) (page 377), it immediately follows that

WKj↓. We set

g := limj→∞WKj

= infj

WKj= inf

{v

∣∣∣ v ∈⋃

j∈N

Φ1Kj

}.


Using Proposition 6.5.4-(iii) (page 355), it is easily seen that the family⋃

j Φ1Kj

isdown directed. Moreover, it is bounded from below by zero. From Theorem 6.11.1,page 375, it follows that the lower semicontinuous regularization g of g is L-superharmonic in G. By the Corollary 9.4.8 (page 444) we know that

g = Γ ∗ ν + infG

g in G,

where ν is the L-Riesz measure of g. Since 0 ≤ g ≤ WKjin G, we have

0 ≤ g(x) ≤ VKj(x) → 0 at infinity

(see (11.16b)). Hence infG g = 0 and

g = Γ ∗ ν in G.

For a fixed j0, (WKj)j≥j0 is a decreasing sequence of L-harmonic functions in

G \ Kj0 (see (11.16a)–(11.16b)).Hence, by Theorem 5.7.10 (page 268), also the limit g is L-harmonic in G \Kj0 .

Since j0 is arbitrary, g is L-harmonic in G \ K (in particular g = g in G \ K).Hence the support of ν is contained in K . Moreover, WKj

≤ 1 gives g ≤ 1 and thenΓ ∗ ν = g ≤ 1 in G. Thus, by Theorem 11.5.5, we have g ≤ VK . On the other hand,since

WKj≥ WK ≥ VK for every j,

(see (6.31a)), it is g ≥ VK , and then g ≥ VK . Therefore g = VK , and we obtain

g = g = limj→∞WKj

≥ WK = VK = g in G \ K.

As a consequence, limj→∞ WKj= WK in G \ K . Observing that WKj

= 1 = WK

in K , the proof is complete. �Proposition 11.5.10. Given a sequence {Kj }j of compact subsets of G, the followingproperties of the L-capacity hold:

K1 ⊆ K2 ⇒ C(K1) ≤ C(K2), (11.18)

Kj ↓ ⇒ C(Kj ) ↓ C(⋂

j

Kj

), (11.19)

C(K1 ∪ K2) + C(K1 ∩ K2) ≤ C(K1) + C(K2). (11.20)

Proof. Fix any Ki involved in any of the above statements. Let Ω be an open setsuch that Ki ⊂ Ω ⊂ Ω � G, and let ν be the L-capacitary distribution for Ω . Wehave VΩ = Γ ∗ν ≡ 1 in Ω and supp ν ⊆ ∂Ω by (11.16a)–(11.16b). Let VKi

and μKi

be, respectively, the L-capacitary potential and the L-capacitary distribution for Ki .Then supp μKi

⊆ Ki and VKi≡ WKi

outside Ki , hence in ∂Ω (again by (11.16a)–(11.16b)).


(i). If K1 ⊆ K2, then WK1 ≤ WK2 , see (6.31a), page 377. Recalling that Γ (x) =Γ (x−1), we obtain

C(K1) =∫

K1

1 dμK1 =∫

K1

Γ ∗ ν dμK1 =∫

G

Γ ∗ ν dμK1 =∫

G

Γ ∗ μK1 dν

=∫

∂Ω

VK1 dν =∫

∂Ω

WK1 dν ≤∫

∂Ω

WK2 dν =∫

∂Ω

VK2 dν

=∫

G

Γ ∗ μK2 dν =∫

G

Γ ∗ ν dμK2 =∫

K2

1 dμK2 = C(K2).

(ii). If Kj ↓ and K := ⋂j Kj , then WKj

↓ WK by Lemma 11.5.9. Recallingthat VK ≡ WK outside K , hence in ∂Ω , and arguing as in (i), we obtain

C(Kj ) =∫

G

Γ ∗ ν dμKj=∫

G

Γ ∗ μKjdν =

∫

∂Ω

VKjdν

=∫

∂Ω

WKjdν ↘

∫

∂Ω

WK dν =∫

∂Ω

VK dν

=∫

G

Γ ∗ μK dν =∫

G

Γ ∗ ν dμK = C(K).

(iii). By (6.31c) (page 377), we have WK1∪K2 + WK1∩K2 ≤ WK1 + WK2 . Thesame arguments as in (i), (ii) give

C(K1 ∪ K2) + C(K1 ∩ K2)

=∫

G

Γ ∗ ν dμK1∪K2 +∫

G

Γ ∗ ν dμK1∩K2

=∫

G

Γ ∗ μK1∪K2 dν +∫

G

Γ ∗ μK1∩K2 dν

=∫

∂Ω

VK1∪K2 dν +∫

∂Ω

VK1∩K2 dν

=∫

∂Ω

(WK1∪K2 + WK1∩K2) dν

≤∫

∂Ω

(WK1 + WK2) dν =∫

∂Ω

(VK1 + VK2) dν

=∫

G

(Γ ∗ μK1 + Γ ∗ μK2) dν =∫

G

Γ ∗ ν dμK1 +∫

G

Γ ∗ ν dμK2

= C(K1) + C(K2).

This ends the proof. �Theorem 11.5.11 (Characterization of L-polarity for K �� GGG). Let K � G be acompact set. Then K is L-polar if and only if C(K) = 0.

Proof. First, suppose K is L-polar. By Corollary 11.2.5, there exists μ ∈ M0 suchthat Γ ∗ μ ≡ ∞ in K . Recalling that Γ (x) = Γ (x−1) and that Γ ∗ μK = VK ≤ 1(see (11.16a)), we obtain


∞ · C(K) = ∞ · μK(K) =∫

Γ ∗ μ dμK =∫

Γ ∗ μK dμ ≤ μ(G) < ∞.

Therefore, it must be C(K) = 0.Suppose now C(K) = 0. By (11.19), we can find a sequence of bounded open

sets Ωj ⊃ K such that

Kj+1 := Ωj+1 ⊂ Ωj and C(Kj ) < 1/j2.

We now choose x0 /∈ K1, and we set m := maxy∈K1 Γ (y−1 ◦ x0). We have

VKj(x0) =

∫

Kj

Γ (y−1 ◦ x0) dμKj(y) ≤ mμKj

(Kj ) = m C(Kj ) < m/j2.

We now define ω = ∑j∈N

VKj. Then ω ∈ S(G), since ω(x0) < ∞ (see Corol-

lary 8.2.8, page 403). Moreover, ω ≡ ∞ in K , since VKj≡ 1 in Ωj ⊃ K

(see (11.16a)). Therefore, K is L-polar. �Corollary 11.5.12 (L-polar∗ ⇒ L-polar). Any L-polar∗ set is L-polar.

Proof. Let E ⊂ G be an L-polar∗ set, i.e. there exists a countable family of com-pact sets Fn such that V (Fn) = ∞ and E ⊆ ⋃

n∈NFn. From Theorem 11.5.8 and

Theorem 11.5.11 it follows that Fn is L-polar for every n ∈ N. We now only needto recall that any countable union of L-polar sets is L-polar, by means of Corol-lary 11.2.6. �Corollary 11.5.13 (On the L-equilibrium potential). Let K � G be a compact setwith L-equilibrium value V (K) < ∞ (or, equivalently, with L-capacity C(K) > 0).

Then there exists a unique L-equilibrium potential of K given by UK =C(K)−1VK , where VK is the L-capacitary potential of K .

Proof. Let U1 = Γ ∗ μ1, U2 = Γ ∗ μ2 be L-equilibrium potentials of K . From thefundamental Theorem 11.4.5 on L-equilibrium potentials it follows that

U1, U2 ≤ V (K) < ∞ in G

and U1 = V (K) = U2 in K \ P , for a suitable L-polar∗ set P . Observing thatP is also L-polar, by Corollary 11.5.12, and recalling that supp μi ⊆ K , we inferthat U1 = U2 in G, in force of the Maria–Frostman domination principle (Theo-rem 11.3.4). In order to complete the proof, we only need to observe that by Theo-rem 11.5.8, UK = C(K)−1VK is an L-equilibrium potential of K . �

We now aim to extend the notion of L-capacity to a larger class of sets.

Definition 11.5.14 (L-capacitable set and L-capacity). Given a non-empty setE ⊆ G, we define the interior L-capacity of E as

C∗(E) := sup{C(K) |K compact,K ⊆ E}.


We then define the exterior L-capacity of E as follows

C∗(E) := inf{C∗(Ω) |Ω open,Ω ⊇ E}.We also set C∗(∅), C∗(∅) := 0. A set E ⊆ G is called L-capacitable if C∗(E) =C∗(E). In this case, this common value is denoted by C(E) and it is called the L-capacity of E.

Proposition 11.5.15. Any Borel set is L-capacitable. Moreover, for compact sets theabove definition of L-capacity agrees with the one given in Definition 11.5.4.

Proof. With Proposition 11.5.10 at hand, one can follow verbatim the classicalcapacitability theory as presented, e.g. in [Helm69] or [AG01]. We omit furtherdetails. �Theorem 11.5.16 (Characterization of L-polarity. I). A set E ⊆ G is L-polar ifand only if E is capacitable and C(E) = 0.

Proof. First, suppose E is L-polar. Then E is contained in a Borel L-polar set E (seeEx. 11.2.2). Any compact subset K of E is L-polar and then it has null L-capacity byTheorem 11.5.11. Thus, we have C∗(E) = 0. Since E is a Borel set, it is capacitableby Proposition 11.5.15. Hence

0 = C∗(E) = C∗(E) ≥ C∗(E) ≥ C∗(E) ≥ 0

(we remark that from (11.18) it readily follows the monotonicity of interior and ex-terior capacity). Therefore E is capacitable and C(E) = 0.

We now want to prove the reverse implication. We first observe that we can as-sume that E is bounded. The general case follows by considering the sequence ofbounded sets En := E ∩ D(0, n). From C(E) = 0 we get

0 = C∗(E) ≥ C∗(En) ≥ C∗(En) ≥ 0,

and thus En is capacitable with C(En) = 0, hence L-polar; then, by Corollary 11.2.6,also E =⋃

n En is L-polar.Suppose now that E is a bounded capacitable set and C(E) = 0. Then, for every

j ∈ N, there exists an open set Ωj ⊇ E such that C(Ωj ) < 1/j2. Moreover, we cansuppose that all the Ωj ’s are contained in a fixed bounded set. In particular, thereexists x0 ∈ G such that Bd(x0, 1) ∩ Ωj = ∅ for every j , so that

Γ (ζ−1 ◦ x0) ≤ 1 ∀ ζ ∈⋃

j

Ωj . (11.21)

For a fixed j ∈ N, we consider a sequence {Bi}i of open sets such that

Bi ⊂ Bi+1,⋃

i

Bi = Ωj .

We set Ki := Bi . From (11.21) we obtain


VKi(x0) =

∫

Ki

Γ (ζ−1 ◦ x0) dμKi(ζ ) ≤

∫

Ki

dμKi= C(Ki) ≤ C(Ωj ) ≤ j−2.

From Ki ⊆ Ki+1 we infer 0 ≤ VKi≤ VKi+1 ≤ 1. If we set

ωj = limi→∞VKi

,

we have ωj (x0) ≤ j−2, 0 ≤ ωj ≤ 1. Moreover, ωj ∈ S(G) by Corollary 8.2.8,page 403. On the other hand, (11.16a) gives 1 ≥ ωj ≥ VKi

= 1 on Bi for every i, sothat ωj ≡ 1 on

⋃i Bi = Ωj ⊇ E.

Finally, setting ω =∑j∈N

ωj , we have ω(x0) ∈ R, ω ≡ ∞ on E and ω ∈ S(G)

(again by Corollary 8.2.8). This implies that E is L-polar. �

11.6 The Fundamental Convergence Theorem

The following theorem is sometimes referred to as the fundamental convergencetheorem of potential theory. It generalizes to our sub-Laplacian setting a theoremdue to Cartan [Cart45, Théorème 8.4] (see also [Helm69, Theorem 7.39] and [AG01,Theorem 5.7.1]).

Theorem 11.6.1 (The fundamental convergence theorem). Let F be a family ofL-superharmonic functions uniformly bounded from below ( for example, a family ofL-potentials).

Then, v = infu∈F u differs from its lower semicontinuous regularization v atmost on an L-polar set.

Proof. Since G is a countable union of d-balls, and a countable union of L-polar setsis L-polar, it is sufficient to prove that v = v in B := Bd(x0, r) except at most onan L-polar subset of B. We can also assume that u ≥ 0 for every u ∈ F , since F isuniformly bounded from below. Moreover, replacing F by the collection of infimumsof finitely many elements of F , if necessary, we can assume that F is down directed.

Let us now consider the L-balayage Ru

Bof u ∈ F . By Theorem 6.11.6, page 377,

we know that Ru

Bis L-superharmonic and it is equal to u in B. Moreover, Ru

Bis

the potential of a measure supported in B, by Theorem 11.5.3 and Theorem 6.11.6.Observing also that the family

{Ru

B: u ∈ F

}

is still down directed, it is then not restrictive to assume that any u ∈ F is thepotential of a measure supported in B. Finally, recalling that we are working with adown directed family, we can obviously assume that there exists u0 ∈ F such that

u ≤ u0 for every u ∈ F .

Let now D be a d-ball such that B ⊂ D, and let us consider the L-capacitarypotential

11.6 The Fundamental Convergence Theorem 511

VD = Γ ∗ μD ≡ 1 in D.

For every u = Γ ∗ μ ∈ F (μ ∈ M(B)), we have

μ(B) =∫

B

Γ ∗ μD dμ =∫

G

Γ ∗ μD dμ =∫

G

Γ ∗ μ dμD

≤∫

G

u0 dμD =∫

G

Γ ∗ μ0 dμD =∫

G

Γ ∗ μD dμ0 = μ0(B),

where u0 = Γ ∗ μ0, μ0 ∈ M(B). Hence, the set{μ(B) |u ∈ F , u = Γ ∗ μ

}

is bounded. The assertion of the theorem now follows from Lemma 11.6.2 below:we find ν ∈ M such that Γ ∗ ν ≤ v in B with equality except possibly on an L-polarset P , and we deduce that v = Γ ∗ ν ≤ v ≤ v in B \ P . �Lemma 11.6.2. Let {μi}i∈I be a family of Radon measures on G with the followingproperties:

(i) there exists a compact set K such that supp (μi) ⊆ K for every i ∈ I ,(ii) the family {Γ ∗ μi | i ∈ I } is down directed,

(iii) there exists m ∈ R such that μi(G) ≤ m for every i ∈ I .

Then there exists μ ∈ M such that

Γ ∗ μ ≤ infi∈I

Γ ∗ μi,

and there exists an L-polar set P such that

Γ ∗ μ = infi∈I

Γ ∗ μi in G \ P.

Proof. By Proposition 6.1.2, page 339, there exists a countable set I0 ⊆ I such thatif g is l.s.c. on G and g ≤ infi∈I0(Γ ∗ μi) then g ≤ infi∈I (Γ ∗ μi). Since the family

{Γ ∗ μi | i ∈ I }is down directed, it is not restrictive to assume that I0 = {in}n∈N and {Γ ∗ μin}n isdecreasing. Now,5 up to extracting a subsequence, μin converges in the w∗-topologyto some measure μ ∈ M(K), μ(G) ≤ m, and

(Γ ∗ μ)(x) =∫

G

Γ (y−1 ◦ x) dμ(y) ≤ lim infn→∞

∫

G

Γ (y−1 ◦ x) dμin(y)

= lim infn→∞ (Γ ∗ μin)(x)

= infi∈I0

(Γ ∗ μi)(x).

Recalling that Γ ∗ μ is l.s.c. on G, it follows that

5 See, e.g. [Rud87].


Γ ∗ μ ≤ infi∈I

(Γ ∗ μi).

Let us now consider the Borel set

E :={x ∈ G | (Γ ∗ μ)(x) < lim

n→∞(Γ ∗ μin)(x)}.

Since {x ∈ G | (Γ ∗ μ)(x) < inf

i∈I(Γ ∗ μi)(x)

}⊆ E,

it remains only to prove that E is L-polar. By means of Theorem 11.5.16 and Propo-sition 11.5.15, it suffices to show that C∗(E) = 0.

Assume by contradiction that there exists a compact set C ⊆ E such that C(C) >

0, and consider the L-capacitary distribution μC . Then

μC(C) = C(C) > 0 and Γ ∗ μC ≤ 1.

By Lemma 11.3.3, we infer that there exists a compact set C′ ⊆ C, μC(C′) > 0,such that setting ν = μC |C′ we have Γ ∗ ν ∈ C(G, R). Thus

∫

G

Γ ∗ ν dμin

n→∞−→∫

G

Γ ∗ ν dμ

and, by Fatou’s lemma,∫

G

limn→∞Γ ∗ μin dν ≤ lim inf

n→∞

∫

G

Γ ∗ μin dν

= lim infn→∞

∫

G

Γ ∗ ν dμin

=∫

G

Γ ∗ ν dμ =∫

G

Γ ∗ μ dν < ∞

(Γ ∗ ν is continuous and μ is compactly supported). Therefore∫

C

(Γ ∗ μ − limn→∞Γ ∗ μin) dν ≥ 0.

Since the integrand is strictly negative on C and ν(C) > 0, we get a contradiction.This completes the proof. �

For the reader’s convenience and for the future references, we collect some of theproperties of the L-balayage that we know so far.

Lemma 11.6.3. Let E ⊆ G. Let u ∈ S+(G). Then Ru

E is L-superharmonic on Ω .Moreover, the following properties hold:

(i) u ≥ RuE ≥ Ru

E ≥ 0 on G,(ii) u = Ru

E on E,(iii) u = Ru

E = RuE on the interior of E,

11.6 The Fundamental Convergence Theorem 513

(iv) RuE = Ru

E on G \ E, and they are L-harmonic on G \ E,(v) Ru

E differs from RuE on a (L-polar) subset of ∂E,

(vi) E ⊆ F ⇒ RuE ≤ Ru

F ,

(vii) v ∈ S+(G) and u ≤ v on E ⇒ Ru

E ≤ RvE ,

(viii) λ > 0 ⇒ Rλ uE = λ Ru

E ,(ix) Ru+v

E ≤ RuE + Rv

E .


Proposition 11.6.4 (RuE and Ru

E . I). Let E be any subset of G and u ∈ S+(G). Then

the L-reduced function RuE and the L-balayage Ru

E coincide on G \ E.

Proof. To begin with, from Lemma 11.6.3 we know that RuE and Ru

E may differ onlyon a subset of ∂E. From Theorem 11.6.1 we derive that Ru

E �= RuE on an L-polar set,

say P . Let Z := E ∩ P . Collecting the above remarks, we infer that Z is an L-polarsubset of E ∩ ∂E.

Let x0 /∈ E. In particular, x0 /∈ Z, so that, by Theorem 11.2.3, there existsw ∈ S+

(G) such that w(x0) ∈ R and w ≡ ∞ on Z. Then, for every ε > 0, thefunction

v := RuE + εw

belongs to S+(G) and is ≥ u on E (indeed, if z ∈ E∩P = Z then v(z) = ∞ ≥ u(z),

whereas if z ∈ E \ P , by definition of P it holds v(z) ≥ RuE(z) = Ru

E(z) = u(z),see Lemma 11.6.3).

Then, by the very Definition 11.5.1 of RuE , we have

RuE + ε w ≤ Ru

E on G.

In particular, RuE(x0) + ε w(x0) ≤ Ru

E(x0). Letting ε → 0+ (recall that w(x0) ∈R), we infer Ru

E(x0) ≤ RuE(x0), which together with Lemma 11.6.3-(i), proves

RuE(x0) = Ru

E(x0). Being x0 /∈ E arbitrary, the assertion of the proposition fol-lows. �

From Theorem 11.6.1 and Proposition 11.6.4, one immediately obtains the fol-lowing remarkable result.

Theorem 11.6.5 (RuE and Ru

E . II). Let E ⊆ G and u ∈ S+(G). Then the L-reduced

function RuE differs from the L-balayage Ru

E at most on an L-polar subset of E∩∂E.

In particular, we have the following assertion.

Corollary 11.6.6. Let Ω ⊆ G be open and u ∈ S+(G). Then Ru

Ω ≡ RuE .

Proof. Apply Theorem 11.6.5 and note that Ω ∩ ∂Ω = ∅ since Ω is open. �


11.7 The Extended Poisson–Jensen Formula

The aim of this section is to prove Theorem 11.7.6, which extends the Poisson–Jensen formula of Theorem 9.5.1 (page 445) in the case when the open set Ω isnot necessarily an L-regular set. The notion of extended L-Green function is alsoneeded.

Throughout this section, we say that Ω ⊆ G is a domain if it is an open andconnected set.

Lemma 11.7.1. Let Ω ⊂ G be a bounded domain, and let ζ0 ∈ ∂Ω . Suppose Ω �

Bd(x0, r) and set E = Bd(x0, r) \ Ω . If

UE(ζ0) = V (E) < ∞(being UE the L-equilibrium potential for E and V (E) its L-equilibrium value), thenthere exists an L-barrier for Ω at ζ0.

Proof. To begin with, we have

UE ∈ H(G \ E), UE ≤ V (E) in G

(see Theorem 11.4.5). Since UE → 0 at infinity, from the strong maximum principle(Theorem 5.13.8, page 296) it follows that

UE < V (E) in G \ E.

Moreover, UE l.s.c. and UE(ζ0) = V (E) imply that there exists

limG\E�x→ζ0

UE(x) = V (E).

Hence, the function w := V (E) − UE is an L-barrier for Ω at ζ0. �The following result is of unquestionable importance.

Theorem 11.7.2 (On the L-polarity of L-irregular points). Let Ω ⊂ G be abounded domain. Then the subset of ∂Ω of the L-irregular points is an L-polar∗set. In particular, it is L-polar.

Proof. Let {Bd(xj , rj )}j≤n be a cover of ∂Ω such that Ω � Bd(xj , rj ) for everyj ≤ n. We set Ej = Bd(xj , rj ) \ Ω and Fj = ∂Ω ∩ Bd(xj , rj ). We write

∂Ω =( ⋃

j : C(Fj )=0

Fj

)∪( ⋃

j : C(Fj )>0

Fj

)=: B1 ∪ B2.

Let us recall once for all that C(K) = V (K)−1 for any compact set K , by The-orem 11.5.8. In particular, we have that B1 is L-polar∗. Suppose next that j ∈{1, . . . , n} is such that C(Fj ) > 0. Since Fj ⊆ Ej , we also have C(Ej ) > 0 (see(11.18)). From Theorem 11.4.5 it follows that

11.7 The Extended Poisson–Jensen Formula 515

UEj(x) = V (Ej ) for all x ∈ Ej \ Pj ,

where Pj ⊆ Ej is L-polar∗. Hence, for every ζ0 ∈ Fj \ Pj , we have UEj(ζ0) =

V (Ej ) < ∞. We can then apply Lemma 11.7.1 and obtain that there exists an L-barrier for Ω at ζ0. Thus ζ0 is an L-regular point, by Bouligand’s theorem 6.10.4,page 371. We finally write

∂Ω = B1 ∪( ⋃

j : C(Fj )>0

(Pj ∩ Fj )

)∪( ⋃

j : C(Fj )>0

(Fj \ Pj )

)

=: B1 ∪ B ′2 ∪ B ′′

2 ,

and we remark that B1 ∪ B ′2 is L-polar∗, whereas all the points of B ′′

2 are L-regular.This ends the proof. �Theorem 11.7.3 (On the L-harmonic measure and L-polarity). Let Ω ⊂ G be abounded domain. Let E be an L-polar subset of ∂Ω . Then, for every x ∈ Ω , wehave μΩ

x (E) = 0.

Proof. Since μΩx is regular, it is enough to prove the theorem when E is compact.

By Theorem 6.9.3 (page 367), we know that the characteristic function χE of E isL-resolutive, and we have

μΩx (E) =

∫

E

dμΩx (y) =

∫

∂Ω

χE(y) dμΩx (y) = HΩ

χE(x).

In particular, HΩχE

∈ H(Ω) and HΩχE

is bounded. We have to prove that HΩχE

= 0 inΩ . We start by showing that

limΩ�x→ζ0

HΩχE

(x) = 0 ∀ ζ0 ∈ ∂Ω \ F, (11.22)

where F = E ∪ E0 and E0 is the set of the L-irregular points of ∂Ω . Consider asequence δn ∈ C(∂Ω), δn ↘ χE , such that there exists n0 with δn0(ζ0) = 0 (we cantake δn(ζ ) := max{1 − n dist(ζ, E), 0}). We have

0 ≤ HΩχE

(x) ≤ HΩδn

(x) for every x ∈ Ω

(see Proposition 6.7.4-(i), page 360). Moreover, since ζ0 is an L-regular point,

limΩ�x→ζ0

HΩδn0

(x) = δn0(ζ0) = 0.

Thus, (11.22) follows. On the other hand, E0 is L-polar∗ by means of Theo-rem 11.7.2, hence L-polar by Corollary 11.5.12. Moreover, E is L-polar by hypoth-esis. Thus also F is L-polar (see Corollary 11.2.6). We can now use the extendedmaximum principle for L in Lemma 11.2.7, and we obtain HΩ

χE= 0 in Ω . This ends

the proof. �


Definition 11.7.4 (The extended L-Green function). Let Ω ⊂ G be a boundeddomain, and let x ∈ Ω be fixed. We define the extended L-Green function GΩ(x, ·)for Ω by

GΩ(x, y) :=

⎧⎪⎨

⎪⎩

GΩ(x, y) if y ∈ Ω ,

0 if y ∈ G \ Ω ,

lim supΩ�z→y GΩ(x, z) if y ∈ ∂Ω .(11.23)

In particular, GΩ(x, y) = 0 for every L-regular point y ∈ ∂Ω . We have thefollowing remarkable result.

Theorem 11.7.5 (On the extended L-Green function). Let Ω ⊂ G be a boundeddomain, and let x ∈ Ω be fixed. Then the extended L-Green function for Ω is L-subharmonic in G \ {x}. Moreover, for all y ∈ G,

GΩ(x, y) = Γ (y−1 ◦ x) −∫

∂Ω

Γ(y−1 ◦ η

)dμΩ

x (η). (11.24)

Proof. Let x ∈ Ω be fixed. We define the function gΩ(x, ·) on G by

gΩ(x, ·) := Γ (x−1 ◦ ·) −∫

∂Ω

Γ((·)−1 ◦ η

)dμΩ

x (η). (11.25)

(It is easy to see that no indeterminate forms ∞−∞ may occur.) From Theorem 9.3.2(page 432) and the symmetry of Γ it follows that gΩ(x, ·) ∈ S(G \ {x}). We are leftto prove that gΩ = GΩ .

If y ∈ Ω , then from the symmetry of GΩ (see Proposition 9.2.10, page 431) itfollows that

gΩ(x, y) = GΩ(y, x) = GΩ(x, y).

Let now y ∈ G \ Ω . Since Γ (y−1 ◦ ·) ∈ H(Ω) ∩ C(Ω), we have∫

∂Ω

Γ (y−1 ◦ η) dμΩx (η) = HΩ

Γ (y−1◦·)(x) = Γ (y−1 ◦ x)

(see Proposition 6.7.7, page 361). In this case, gΩ(x, y) = 0.We now show that gΩ(x, ·) is non-negative on G. It suffices to prove it for y ∈

∂Ω . For η ∈ ∂Ω , we set fn(η) := min{n, Γ (y−1 ◦ η)}. Then fn ∈ C(∂Ω, R) andfn(η) ↗ Γ (y−1 ◦ η) as n → ∞. Then, by monotone convergence,

∫

∂Ω

Γ (y−1 ◦ η) dμΩx (η) = lim

n→∞HΩfn

(x).

If we prove that HΩfn

(x) ≤ Γ (y−1 ◦ x) for any x ∈ Ω , it follows gΩ(x, y) ≥ 0.By the extended maximum principle in Lemma 11.2.7, since the set ∂2Ω of the L-irregular points of ∂Ω is an L-polar set (see Theorem 11.7.2 and Corollary 11.5.12),it is enough to show that

11.7 The Extended Poisson–Jensen Formula 517

lim supΩ�x→ζ

(HΩ

fn(x) − Γ (y−1 ◦ x)

) ≤ 0 for ζ ∈ ∂1Ω,

where ∂1Ω is the set of the L-regular points of ∂Ω , and that HΩfn

(x) − Γ (y−1 ◦ x)

is bounded from above. The latter fact follows from

HΩfn

(x) − Γ (y−1 ◦ x) ≤ HΩfn

(x) ≤ max∂Ω

fn < ∞.

We then prove the former claim. Since each ζ ∈ ∂1Ω is an L-regular point,

lim supΩ�x→ζ

(HΩ

fn(x) − Γ (y−1 ◦ x)

) ≤ fn(ζ ) + lim supΩ�x→ζ

(−Γ (y−1 ◦ x))

which is ≤ 0 by definition of fn. This proves that gΩ(x, y) ≥ 0.In order to prove that gΩ coincides with GΩ defined in (11.23), it remains to

show that, for any y0 ∈ ∂Ω , it holds

gΩ(x, y0) = lim supΩ�y→y0

GΩ(x, y).

First, we suppose y0 ∈ ∂1Ω . Since ∂2Ω is L-polar, by Corollary 11.2.5 there existsh ∈ S(G), finite at y0, h < 0 such that h|∂2Ω ≡ −∞. For any ε > 0, let

gε(y) := gΩ(x, y) + ε h(y), y ∈ G \ {x}.Since gΩ(x, y) = GΩ(x, y) for every y ∈ Ω , gΩ(x, y) vanishes as y approaches∂1Ω from the inside of Ω and remains bounded as y ∈ Ω approaches ∂2Ω . Usingalso the fact that h is u.s.c., we have that, for any y1 ∈ ∂Ω ,

lim supΩ�y→y1

gε(y) < 0.

Moreover,lim sup

G\Ω�y→y1

gε(y) < 0,

since gΩ(x, y) = 0 for every y ∈ G \Ω . Then there exists a compact neighborhoodN of ∂Ω such that gε < 0 on N \ ∂Ω . Recalling that

gΩ(x, ·) ∈ S(G \ {x}),from the maximum principle for L-subharmonic functions (Theorem 8.2.19, page 409)we then obtain gε ≤ 0 in N . In particular, gε(y0) ≤ 0 and, letting ε vanish, we getgΩ(x, y0) ≤ 0, which (jointly with gΩ(x, ·) ≥ 0) gives gΩ(x, y0) = 0. We havethus proved that

limΩ�y→y0

GΩ(x, y) = 0 = gΩ(x, y0)

if y0 ∈ ∂1Ω .We now treat the case of y0 ∈ ∂2Ω . Let us set for brevity


E := ∂2Ω, A := G \ {x}, u := gΩ(x, ·).With this notation, we have E L-polar and u ∈ S(A). We now set

u(x) := lim supA\E�ξ→x

u(ξ), for x ∈ A ∩ E, u(x) := u(x), for x ∈ A \ E.

Since we know that u ≥ 0 is u.s.c., we immediately get 0 ≤ u ≤ u in A. Moreover,since E has Lebesgue measure zero being L-polar, the following equality of thesolid means holds: Mr(u) = Mr(u). As a consequence, u is sub-mean in A, beingu ∈ S(A), hence sub-mean by Theorem 8.2.1 (page 401). Furthermore, it is easy toverify that u is u.s.c. Therefore, again from Theorem 8.2.1, we obtain u ∈ S(A). Wenow use Lemma 11.3.1 and conclude that u ≡ u in A. In particular, we have

gΩ(x, y0) = u(y0) = u(y0) = lim supG\∂2Ω�y→y0

gΩ(x, y)

= lim supΩ�y→y0

GΩ(x, y).

In the last equality we used the following facts: gΩ(x, ·) ≥ 0, gΩ(x, ·) ≡ 0 inG \ (Ω ∪ ∂2Ω) and gΩ(x, ·) = GΩ(x, ·) in Ω . This completes the proof. �Theorem 11.7.6 (The extended Poisson–Jensen formula). Let Ω be a boundeddomain of G. Suppose u is L-subharmonic on a neighborhood of Ω . Then, for allx ∈ Ω , we have

u(x) =∫

∂1Ω

u(y) dμΩx (y) −

∫

Ω∪∂2Ω

GΩ(x, y) dμu(y), (11.26)

where ∂1Ω, ∂2Ω denote, respectively, the subsets of ∂Ω of the regular and the irreg-ular points for the Dirichlet problem related to L, μu is the L-Riesz measure of u,μΩ

x is the L-harmonic measure for Ω at x, and GΩ is the extended L-Green functionfor Ω .

Proof. Let O be a bounded domain such that Ω ⊂ O and such that u is L-subharmonic on a neighborhood of O. By the Riesz representation Theorem 9.4.4(page 442), there exists h ∈ H(O) such that

u(x) = h(x) −∫

O

Γ (y−1 ◦ x) dμu(y) ∀ x ∈ O.

We certainly have h(x) = ∫∂Ω

h(η) dμΩx (η). We now let

v(x) := −∫

O

Γ (y−1 ◦ x) dμu(y) for x ∈ G.

Recalling Theorem 9.3.5, page 433, v ∈ S(G) and μv coincides with μu on O andvanishes on G \O. We claim that it is enough to prove (11.26) with u replaced by v.

11.8 Further Results. A Miscellanea 519

Indeed, using the fact that μΩx (∂2Ω) = 0 for every x ∈ Ω (which is an immediate

consequence of Theorem 11.7.2 and Theorem 11.7.3), this would give

u(x) = h(x) + v(x)

=∫

∂Ω

h(y) dμΩx (y) +

∫

∂1Ω

v(y) dμΩx (y) −

∫

∂2Ω∪Ω

GΩ(x, y) dμv(y)

=∫

∂1Ω

(h(y) + v(y)) dμΩx (y) −

∫

∂2Ω∪Ω

GΩ(x, y) dμu(y)

=∫

∂1Ω

u(y) dμΩx (y) −

∫

∂2Ω∪Ω

GΩ(x, y) dμu(y).

We are then left to prove (11.26) for v. We recall that GΩ(x, ·) ≡ 0 on ∂1Ω and onG \ Ω (see the definition (11.23) of GΩ ). Using this fact, exploiting μΩ

x (∂2Ω) = 0,and using (11.24), we obtain that, for every x ∈ Ω ,

∫

∂1Ω

v(η) dμΩx (η) −

∫

∂2Ω∪Ω

GΩ(x, y) dμv(y)

= −∫

∂1Ω

(∫

O

Γ (y−1 ◦ η) dμu(y)

)dμΩ

x (η) −∫

∂2Ω∪Ω

GΩ(x, y) dμu(y)

= −∫

O

(∫

∂Ω

Γ (y−1 ◦ η) dμΩx (η)

)dμu(y) −

∫

O

GΩ(x, y) dμu(y)

=∫

O

(−∫

∂Ω

Γ (y−1 ◦ η) dμΩx (η) − GΩ(x, y)

)dμu(y)

= −∫

O

Γ (y−1 ◦ x) dμu(y) = v(x).


11.8 Further Results. A Miscellanea

The aim of this section is to collect a miscellanea of results concerning with theL-capacity. These assertions, besides completing the investigation of the previoussections, also provide useful results, some having an interest in its own.

Theorem 11.8.1 (The L-balayage as a L-potential). Let E ⊂ G be bounded. Thenthe following assertions hold:

(a) For every u ∈ S+(G), the L-balayage Ru

E is an L-potential. In particular, ifu ≡ 1, there exists a Radon measure νE such that R1

E = Γ ∗ νE (i.e. νE is theL-Riesz measure of R1

E);(b) With the above notation, we have

C∗(E) = νE(G).


Proof. (a). Let R � 1 be such that E ⊂ Bd(0, R). Let

K1 := Bd(0, R), K2 := Bd(0, R + 1), v := RuK1

.

Since v ∈ H(G\K1) (see Lemma 11.6.3-(iv)), there exists a � 1 such that a Γ ≥ v

on ∂K2. We set

w(x) :={

v(x) if x ∈ K2,

min{a Γ (x), v(x)} if x ∈ G \ K2.

By Lemma 11.3.2, w ∈ S+(G). Moreover,6 w ≥ u on E. Hence, by the definition

of L-balayage, w ≥ RuE ≥ Ru

E . Then, on G \ K2 we have

RuE ≤ w = min{a Γ, v} ≤ a Γ.

As a consequence,infG

RuE ≤ inf

G\K2≤ inf

G\K2a Γ = 0.

We are therefore in a position to apply Proposition 9.9.1 (see Ex. 5, Chapter 9,page 464) and derive that Ru

E is an L-potential. This proves assertion (a).(b). Let us now take u = 1 and apply what we have proved above: there exists a

Radon measure νE such that R1E = Γ ∗νE . Our task is to prove that C∗(E) = νE(G).

The proof is split in two parts: first, we suppose that E is also open, then we considerthe general case.

(b’). Let E = Ω be open and bounded. Let {Kn}n be a sequence of compactsubsets of G such that

Kn ⊂ Int(Kn+1) ⊂ Kn+1 ⊂ Ω and⋃

n

Kn = Ω.

By Ex. 15 at the end of the chapter, we infer (see also Corollary 11.6.6 and part (a)of the proof)

limn→∞ R1

Kn= R1

Ω = R1Ω = Γ ∗ νΩ. (11.27)


C(Ω) = limn→∞ C(Kn) = lim

n→∞μKn(G) = νΩ(G).

The first equality is a consequence of Ex. 16-(6) at the end of the chapter; the secondis the definition of L-capacity for a compact set; the third equality follows fromEx. 14 at the end of the chapter.7

6 Indeed, since E is contained in the interior of K1, we have w = v = RuK1

= RuK1

= u

on E, see Lemma 11.6.3.7 We apply Ex. 14 with ν = νΩ , σn = μKn

, K = Ω . Indeed, note that

Γ ∗ μKn= VKn

= R1Kn

↗ Γ ∗ νΩ

by (11.27) and property (vi) of Lemma 11.6.3.


(b”). To begin with, we make some remarks on the definition of R1E . We recall

thatR1

E =(

infu∈Φ1

E

u)

whereΦ1

E = {u ∈ S+(G) |u ≥ 1 on E}.

Since Φ1E is obviously down directed and 1 ∈ Φ1

E , then8

infu∈Φ1

E

u = infu∈Φ1

E, u≤1u.

Furthermore, by a very general result on the lower semicontinuous regularizationof the infimum of an arbitrary family of functions,9 there exists a sequence {un} inΦ1

E with un ≤ 1 such that

(infn

un

)=

(inf

u∈Φ1E

u)= R1

E.

We can also suppose that u1 = 1. Let

vn := min{u1, u2, . . . , un}.

We have 0 ≤ vn ≤ 1, vn ∈ S+(G), vn ≥ 1 on E and {vn}n∈N is non-increasing.

Also,lim

n→∞ vn = infn∈N

un. (11.28)

Let ε ∈ (0, 1). Let Ω be an open set such that

E ⊆ Ω and C(Ω) < C∗(E) + ε. (11.29)

We can suppose that Ω is bounded, by replacing it by Ω ∩ Bd(0, R) (with R � 1such that Bd(0, R) ⊃ E; note that C(Ω ∩ Bd(0, R)) ≤ C(Ω)). We set

wn := Rvn

Ω and Ωn := {x ∈ Ω |wn(x) > 1 − ε}.We collect some properties of wn and Ωn:

(a) wn := Rvn

Ω , for Ω is open (see Corollary 11.6.6);(b) {wn}n is decreasing (see Lemma 11.6.3-(vii));(c) Ωn is open, for wn is l.s.c.;(d) Ωn ⊇ E, for wn = Rvn

Ω = vn ≥ 1 > 1 − ε on E;

8 If F is a down directed family of functions and u0 ∈ F , then inf{u |u ∈ F} = inf{u |u ∈F , u ≤ u0}. Indeed, obviously inf{u | u ∈ F} ≤ inf{u |u ∈ F , u ≤ u0}. Moreover, forevery u ∈ F , there exists v ∈ F such that v ≤ min{u, u0} ≤ u0. Hence inf{u |u ∈ F , u ≤u0} ≤ v ≤ u. Taking the infimum over {u | u ∈ F} in the last inequality, the assertionfollows.

9 See Proposition 6.1.2, page 339.


(e) wn = Rvn

Ω ∈ H(G \ Ωn) (see Lemma 11.6.3-(iv));(f) wn ≥ R1

E (indeed, wn ∈ Φ1E by (d) above, whence wn ≥ R1

E , and take the lowersemicontinuous regularization);

(g) there exists a Radon measure σn on G such that wn = Γ ∗ σn (since wn = Rvn

Ω

and by the first part of the proof, being Ω bounded);(h) R1

Ωn= R1

Ωn= Γ ∗ νΩn (by the first part of the proof);

(i) by (e) above, it follows that σn in (g) is supported in Ωn (hence compactly sup-ported); the same is true for νΩn .

As a consequence of (f),vn ≥ Rvn

Ω = wn ≥ R1E.

Letting n → ∞ (see also (11.28)), we get

infn∈N

un ≥ W := limn→∞wn ≥ R1

E on G. (11.30)

By (b) and (e) above, it follows that, on each component of G \Ω , W is L-harmonicunless it is −∞. From (11.30) we infer that the latter is impossible. Hence, W = W

on G\Ω whence, taking lower semicontinuous regularization in (11.30), one obtains

R1E =

(infn

un

)≥ W ≥ R1

E = R1E

on G \ Ω . This proves W = R1E on G \ Ω .

We claim thatwn ≥ (1 − ε) R1

Ωn. (11.31)

Indeed, the function wn := wn/(1 − ε) belongs to S+(G) and, by the definition

of Ωn, wn > 1 on Ωn. Thus wn ∈ Φ1Ωn

, so that

R1Ωn

= infv∈Φ1

Ωn

v ≤ wn = wn/(1 − ε),

which is the claimed (11.31).By (g) and (h) above, (11.31) rewrites as follows

Γ ∗ σn ≥ (1 − ε) Γ ∗ νΩn = Γ ∗ ((1 − ε) νΩn

).

By Ex. 13 at the end of the chapter, the above inequality ensures that

σn(G) ≥ (1 − ε) νΩn(G). (11.32)

The following facts hold:

(i) supp(σn) ⊆ Ω � G for every n ∈ N,(ii) Γ ∗ σn = wn is decreasing,

(iii) on G \ Ω , it holds wn → W = R1E = Γ ∗ νE (by the definition of νE).


Hence, we are in the position to apply Ex. 14 at the end of the chapter to derivethat νE(G) = limn σn(G). By (11.32), we get

νE(G) ≥ (1 − ε) lim supn→∞

νΩn(G). (11.33)

Moreover, from part (a) of the proof, E ⊆ Ω , Lemma 11.6.3-(vi) and the definitionof νE , we get

Γ ∗ νΩ = R1Ω ≥ R1

E = Γ ∗ νE.

Hence (by another application of Ex. 13 at the end of the chapter) νΩ(G) ≥ νE(G).This inequality, together with (11.33) implies

νΩ(G) ≥ (1 − ε) lim supn→∞

νΩn(G). (11.34)

Finally, by (11.29), part (b’) of the proof (for Ω), (11.34), again part (b’) of the proof(this time for Ωn) and Ωn ⊇ E (whence C(Ωn) ≥ C∗(E)), we obtain

C∗(E) + ε > C(Ω) = νΩ(G) ≥ νE(G) ≥ (1 − ε) lim supn→∞

νΩn(G)

= (1 − ε) lim supn→∞

C(Ωn) ≥ (1 − ε) C∗(E).

Passing to the limit as ε → 0+, we get C∗(E) = νE(G), which we aimed toprove. �Proposition 11.8.2 (On the inner and outer L-capacity). Let E ⊂ G be a boundedset. Then we have:

(i) C∗(E) = sup{μ(E) |μ ∈ M(E), Γ ∗ μ ≤ 1 on G},(ii) C∗(E) = inf{μ(G) |μ ∈ M(G), Γ ∗ μ ≥ 1 on E except for a L-polar set}.Proof. (i). Let μ ∈ M(G) with K := supp(μ) ⊆ E and Γ ∗ μ ≤ 1 on G. Letalso v ∈ S+

(G) be such that v ≥ 1 on K . From Maria–Frostman Theorem 11.3.4,it follows v ≥ Γ ∗ μ on G. Hence, being v ∈ Φ1

K arbitrary, we get R1K ≥ Γ ∗ μ.

By passing to the lower semicontinuous regularization, we infer (denoting the L-capacitary distribution of K by νK )

Γ ∗ νK = VK = R1K ≥ Γ ∗ μ on G.

By Ex. 13 at the end of the chapter, this yields μ(G) ≤ νK(G), whence (beingμ(G) = μ(E), νK(G) = C(K))

μ(E) = μ(K) ≤ νK(G) ≤ sup{C(K) : K ⊆ E, K compact

} = C∗(E).

Passing to the supremum over the μ’s as above, we get the “≥” sign in assertion (i).On the other hand, for every compact subset K of E, it holds (see Theorem 11.5.6)


C(K) = max{μ(K) = μ(E) |μ ∈ M(K), Γ ∗ μ ≤ 1 in G}≤ sup{μ(E) |μ ∈ M(E), Γ ∗ μ ≤ 1 on G}.

Finally, passing to the supremum over all possible compact subsets K of E, by defi-nition of C∗(E), we get the remaining “≤” sign. Thus (i) is proved.

(ii). Let μ ∈ M be such that Γ ∗ μ ≥ 1 on E \ F , where F ⊆ E is an L-polarset. By Corollary 11.2.5 (being F ⊆ E and E bounded), there exists μ0 ∈ M0 suchthat Γ ∗ μ0 ≡ ∞ on F . Let ε > 0. Consider the measure

μ1 := ε

2 μ0(G)μ0.

Then μ1(G) = ε/2, μ1 ∈ M0 and

Γ ∗ μ1 = ε

2 μ0(G)Γ ∗ μ0 ≡ ∞ on F .

Obviously (by the minimum principle for S(G)), we have Γ ∗ μ1 > 0 on G (other-wise μ1 ≡ 0, i.e. μ0 ≡ 0 contradicting Γ ∗ μ0 ≡ ∞ on F ). Let ξ ∈ E. Then wehave

(Γ ∗ μ)(ξ) + (Γ ∗ μ1)(ξ) ≥{

0 +∞ if ξ ∈ F ,

1 + (Γ ∗ μ1)(ξ) if ξ ∈ E \ F ,> 1.

Consequently, setting μ2 := μ+μ1, we have Γ ∗μ2 = Γ ∗μ+ Γ ∗μ1 > 1 on E.Being Γ ∗μ2 l.s.c., for every ξ ∈ E there exists an open neighborhood Uξ of ξ suchthat Γ ∗ μ2 > 1 on Uξ . Then the set U := ⋃

ξ∈E Uξ is an open neighborhood of E

such that Γ ∗ μ2 > 1 on U . It is also not restrictive to suppose that U is bounded(for E is). Then Γ ∗ μ2 ∈ Φ1

U , so that

Γ ∗ μ2 ≥ R1U = R1

U = Γ ∗ νU .

Here, in the first equality we used Corollary 11.6.6, whereas in the second equalitywe used Theorem 11.8.1-(a) and the notation therein. So, we have Γ ∗μ2 ≥ Γ ∗ νU ,whence (see Ex. 13 at the end of the chapter)

μ2(G) ≥ νU (G). (11.35)

Now, from Theorem 11.8.1-(b) we have νU (G) = C∗(U) = C(U) (since U is open),whereas (by the definition of μ2 and μ1) μ2(G) < μ(G) + ε. Hence (11.35) yields

C∗(E) ≤ C(U) < μ(G) + ε.

Due to the arbitrariness of ε, we get C∗(E) ≤ μ(G). Hence, we have proved the “≤”sign in assertion (ii).

On the other hand, since E is a bounded set, it obviously holds

C∗(E) = inf{C(Ω) |Ω open and bounded,Ω ⊇ E}. (11.36)


So, let Ω be open and bounded with Ω ⊇ E. By Theorem 11.8.1, there exists νΩ ∈M such that R1

Ω = Γ ∗ νΩ and C(Ω) = νΩ(G). Hence (see Corollary 11.6.6)Γ ∗ νΩ = R1

Ω = 1 on U ⊇ E (see Lemma 11.6.3-(iii)), so that

inf{μ(G) |μ ∈ M(G), Γ ∗ μ ≥ 1 on E except for a L-polar set}≤ νΩ(G) = C(Ω).

Passing to the infimum over the above Ω’s, from (11.36) we get the “≥” sign inassertion (ii). This ends the proof. �Proposition 11.8.3. Let Z, E ⊆ G, and let Z be L-polar. Let also u ∈ S+

(G). ThenRu

E = RuE\Z .

Proof. From Lemma 11.6.3-(vi), we have RuE ≥ Ru

E\Z . We are thus left to prove the

reverse inequality. Let x0 /∈ Z. By Theorem 11.2.3, there exists v ∈ S+(G) such that

v(x0) ∈ R and v ≡ ∞ on Z. Let w ∈ ΦuE\Z (i.e. w ∈ S+

(G) and w ≥ u on E \ Z).

The function w + ε v belongs to S+(G) and is clearly ≥ u on E. Then

RuE ≤ Ru

E ≤ w + ε v on G.

Evaluating this at x0 and letting ε → 0+, one gets RuE(x0) ≤ w(x0). Being x0 ∈

G \ Z arbitrary, we infer RuE ≤ w on G \ Z, so that10 we obtain Ru

E ≤ w on G.Taking the infimum over w ∈ Φu

E\Z , this gives RuE ≤ Ru

E\Z , then taking lower

semicontinuous regularization, we get RuE ≤ Ru

E\Z . �Proposition 11.8.4 (Characterization of L-polarity. II). Let E ⊆ G be any set.Then the following statements are equivalent:

(a) E is L-polar,(b) There exists u ∈ S(G), u strictly positive on G, such that Ru

E ≡ 0,

(c) For every u ∈ S+(G), it holds Ru

E ≡ 0.

Proof. (a) ⇒ (c). See Ex. 7 at the end of the chapter.(c) ⇒ (b). Take u ≡ 1 and apply directly (c).(b) ⇒ (a). Let u be as in (b) above. Since, by Theorem 11.6.5, Ru

E differs fromRu

E at most on an L-polar set, they are equal almost everywhere on G. Hence, byhypothesis (b), Ru

E = 0 a.e. Let x0 ∈ G be such that RuE(x0) = 0. By the very

definition of RuE(x0), there exists a sequence {vn}n∈N in S+

(G) such that vn ≥ u onE and vn(x0) ≤ 2−n. Set v := ∑

n∈Nvn. Being v(x0) ≤ 1 and vn ≥ u > 0, we

have v ∈ S+(G) (see Theorem 8.2.7, page 403). Moreover, v ≡ ∞ on E, since, for

every e ∈ E, we have vn(e) ≥ u(e) > 0. Hence, by definition, E is L-polar and (a)follows. �10 Recall that L-polar sets have zero Lebesgue measure, and any L-superharmonic function

f ∈ S(Ω) satisfies f (x) = lim infΩ�y→x f (y) for every x ∈ Ω .


We have the following result, concerning with the “smallness” of L-polar setsw.r.t. the L-harmonic measures (in the classical context of Laplace’s operator, see[AG01, Theorem 5.1.9]).

Proposition 11.8.5. Let O ⊆ G be an open set, and let u ∈ S(O) be locallybounded. If μu denotes the L-Riesz measure of u on O, then μu(Z) = 0 for everyBorel L-polar set Z ⊆ O.

Proof. We only need to show that μu(Z ∩ B) = 0 for every d-ball B with B ⊂ O.First, we claim that there exists a compactly supported Radon measure ν on G

such that GB ∗ ν = ∞ on Z ∩ B. Indeed, let w ∈ S(G) be such that w|Z∩B ≡ ∞.The lower semicontinuity of w proves that w is bounded from below on B. Hence,by Riesz representation Theorem 9.4.7 (page 443), we have w = GB ∗ μw + h onB, for a suitable h ∈ H(B). Since h is finite-valued, this proves the claim.

Let α = infB u (α ∈ R since u is l.s.c.). Again from Riesz representation The-orem 9.4.7 applied to the function w = u − α (which is non-negative on B, henceendowed with an L-harmonic minorant on B), we infer the existence of h ∈ H(B)

such thatu(x) − α = (GB ∗ μu)(x) + h(x) for every x ∈ B. (11.37)

(We used μw = μu.) Since h is the greatest L-harmonic minorant of w ≥ 0 on B, itholds h ≥ 0, so that (11.37) gives

(GB ∗ μu)(x) ≤ u(x) − α for every x ∈ B. (11.38)


∞ · μu(Z ∩ B) ≤∫

B

(GB ∗ ν)(y) dμu(y) =∫

B

(GB ∗ μu)(y) dν(y)

(see (11.38)) ≤(

supB

u − α)· ν(G) < ∞ (by the local boundedness of u).

(In the equality sign, we used Fubini–Tonelli’s theorem jointly with the symmetryof GB .) This gives μu(Z ∩ B) = 0, ending the proof. �

The following proposition shows some more “smallness” properties of theL-polar sets.

Proposition 11.8.6. The following assertions hold:

(i) If A ⊆ G is an open connected set and E ⊂ A is L-polar and relatively closedin A, then A \ E is connected;

(ii) If A ⊆ G is open (and non-empty) and ∂A is L-polar, then A is connected anddense in G.

Proof. (i). With the notation of the assertion, let A0 be a connected component ofA \ E. We are done if we show that A0 = A \ E. Note that, since E is relativelyclosed in A, then A \ E is open, whence A0 is open too. Since E is L-polar, thereexists u ∈ S(G) such that u ≡ ∞ on E. Set

11.9 Further Reading and the Quasi-continuity Property 527

v : A → (−∞,∞], v(x) ={

u(x) if x ∈ A0,

∞ if x ∈ A \ A0.

Note that A := (A \ E) \ A0 is an open set since it coincides with the union of theconnected components of the open set A \ E, except for A0. It is easily seen that v

is lower semicontinuous, not identically ∞ and L-supermean on A, since u|A0 is L-superharmonic. Then v ∈ S(A). But v ≡ ∞ on the open set A, and this can happenonly if A = ∅, for L-superharmonic functions are locally integrable. This proves thatA contains no connected components other than A0, thus completing the proof of (i).

(ii). Since ∂A is closed in G (connected), by (i) we have G \ ∂A connected. Ifby contradiction G \ A �= ∅, then G \ ∂A = A ∪ (G \ A) is the union of two opendisjoint and non-empty sets, i.e. G \ ∂A is not connected (contradicting the aboveargument). This proves that A = G. Moreover, the above argument also gives

G \ ∂A = A ∪ (G \ A) = A,

so that A is connected, for G \ ∂A is. �

11.9 Further Reading and the Quasi-continuity Property

The aim of this section is to sketch a brief investigation of the so-called quasi-continuity property of L-subharmonic functions.

As we have seen in Section 8.3 following [JLMS07], a function u which is sub-harmonic w.r.t. every sub-Laplacian on G is v-convex (even under the assumption ofupper semicontinuity instead of continuity in the definition of v-convexity) and hasfine regularity properties. On the other hand, if a function is superharmonic with re-spect to only one sub-Laplacian L, it nonetheless possesses some (weaker) continuityproperty. Indeed, the following quasi-continuity property holds (see Theorem 11.9.1below for the precise statement):

(Q-C) Every L-superharmonic function is continuous, if restricted to the comple-ment of an open set with arbitrarily small L-capacity.

The quasi-continuity of superharmonic functions has important applications inthe theory of PDE’s, where it is often desirable to know the accurate pointwise be-havior of the relevant super- and sub-solutions and their fine properties.

In the classical case of the Laplace operator on G, the (Q-C) property was provedby H. Cartan [Cart28]. Proofs of the (Q-C) property for more general operators arenowadays available: see [HKM93, Theorem 10.9] for a class of non-linear ellipticoperators; see [MV97b, Theorem 14.3] for a class of subelliptic operators generaliz-ing the latter; see [TW02b, Theorem 4.1] for a wide class of quasilinear elliptic andsubelliptic operators. The last two results also cover the case of sub-Laplacians.

Nonetheless, in the significant case of Carnot groups, a direct and simpler ap-proach can be applied in proving (Q-C). See, for instance, [BC05]. In particular, thestratified context allows us to recover several fine results on the theory of the energyfor L, i.e. of the integral

528 11 L-capacity, L-polar Sets and Applications∫∫

Γ (x, y) dμ(x)dμ(y) (where μ is a Radon measure on G).

The proof of the (Q-C) property in [BC05] is in the spirit of this book and differsfrom those in [HKM93,MV97b,TW02b]. Indeed, the approach of these papers (dueto the considered wider classes of operators) requires highly non-trivial techniquessuch as deep weak-convergence results, a suitable theory of the relevant Sobolev-typefunction spaces, some results from the theory of variational inequalities, a quantita-tive use of Caccioppoli-type and Harnack inequalities (both weak and strong), Hölderestimates for weak solutions, etc.

In the remainder of this section, we sketch a line of the proof of the followingresult, as approached in the cited paper [BC05].

Theorem 11.9.1 (Quasi-continuity of L-superharmonic functions). Let u∈S(Ω). Then, for every ε > 0, there exists an open set Oε ⊆ Ω with C(Oε) < ε

and such that

the restriction of u to Ω \ Oε is continuous on Ω \ Oε.

Following Cartan’s theory of energy and the related ideas in the classical poten-tial theory, we give the following definition (we recall that M denotes the set of theRadon measures on G).

Definition 11.9.2 (Mutual L-energy). Let μ, ν ∈ M. The mutual L-energy of μ

and ν is

〈μ, ν〉 =∫

G

Γ ∗ μ dν.

The quantity ‖μ‖2 := 〈μ,μ〉 will be called the L-energy of μ ∈ M. Finally,

we set E+ := {μ ∈ M : ‖μ‖ < ∞}.We notice that ‖μ‖2 = I (μ) if μ ∈ M0, where I (μ) is as in Definition 11.4.1

on page 497.

Proposition 11.9.3. For every x ∈ G and r > 0, the measure λx,r introduced in (9.9)(page 436) belongs to E+ and ‖λx,r ‖2 = r2−Q.

Proof. Let x ∈ G and r > 0. Then, by (9.9) and (9.10a), we have

‖λx,r‖2 =∫

G\Bd(x,r)

Γ (z−1 ◦ x) dλx,r (z) + r2−Q

∫

Bd(x,r)

dλx,r (z)

= 0 + r2−Q

∫

∂Bd(x,r)

k(x, z) dHN−1(z) = r2−Q mr [1](x) = r2−Q.

This completes the proof. �The L-polarity is related to the set E+, as assertion (i) of the following theorem

states (see [BC05, Lemma 6.1 and Theorem 6.1]).


Theorem 11.9.4. The following assertions hold:

(i) Let E be a Borel set. Then E is L-polar iff μ(E) = 0 for every μ ∈ E+;(ii) (L-energy principle) 〈μ, ν〉 ≤ ‖μ‖ · ‖ν‖ for every μ, ν ∈ M.

We next give a new definition. We denote by E the set of formal11 differencesμ − ν, μ, ν ∈ E+. By a slight abuse of notation, the mutual L-energy and the L-energy are defined as follows

〈μ1 − ν1, μ2 − ν2〉 = 〈μ1, μ2〉 − 〈μ1, ν2〉 − 〈ν1, μ2〉 + 〈ν1, ν2〉,‖μ − ν‖2 = ‖μ‖2 − 2〈μ, ν〉 + ‖ν‖2.

Thanks to the L-energy principle in Theorem 11.9.4-(ii), we have∣∣〈μ, ν〉∣∣ ≤ ‖μ‖ · ‖ν‖ for every μ, ν ∈ E .

It can be proved that 〈·, ·〉 defines an inner product on E . We need many preliminaryresults.

Proposition 11.9.5. For every x ∈ G and every r > 0, τx,r := λx,r/2 − λx,r belongsto E and ‖τx,r‖2 = (r/2)2−Q − r2−Q.

Proof. We fix x ∈ G, r > 0. By Proposition 11.9.3, τx,r ∈ E . By (9.10a),

〈λx,r/2, λx,r 〉 =∫

G\Bd(x,r/2)

Γ (z−1 ◦ x) dλx,r (z)

+ (r/2)2−Q

∫

Bd(x,r/2)

dλx,r (z) = r2−Q,

so that ‖λx,r/2 − λx,r‖2 equals

‖λx,r/2‖2 + ‖λx,r‖2 − 2〈λx,r/2, λx,r 〉 = (r/2)2−Q − r2−Q.

This ends the proof. �We now introduce a set of functions which play a prominent rôle in the theory of

L-energy. We set

T = {Tx,r := Γ ∗ λx,r/2 − Γ ∗ λx,r | x ∈ G, r > 0}. (11.39)

In the following definition, we agree to denote by C+0 (G) the set of non-negative

compactly supported functions on G.

Definition 11.9.6 (Total set in C+0 (GGG)). A set T ⊆ C+

0 (G) is called total if it hasthe following property: for every C � G, for every open set W ⊃ C, for all ε > 0and for all v ∈ C+

0 (G) supported in C, there exist β1, . . . , βn > 0, f1, . . . , fn ∈ T

supported in W such that ‖v −∑ni=1 βi fi‖∞ < ε.

11 More precisely, E is the set of the couples (μ, ν) ∈ E+ × E+ modulo the equivalencerelation (μ1, ν1) ∼ (μ2, ν2) iff μ1 + ν2 = μ2 + ν1.


We have an important result, a G-version of a theorem due to Cartan [Cart28]:

Theorem 11.9.7 (Bonfiglioli and Cinti, [BC05], Theorem 4.3). Suppose T ⊆C+

0 (G) has the following properties:

(1) for every f ∈ T and every α ∈ G, it holds f (α ◦ ·) ∈ T ,(2) for every x ∈ G and every δ > 0, there exists f ∈ T , f �≡ 0, such that f vanishes

outside Bd(x, δ).

Then T is a total set (according to Definition 11.9.6).

From Theorem 11.9.7 and Ex. 10 at the end of this chapter we deduce the fol-lowing remarkable assertion.

Corollary 11.9.8. The set T in (11.39) is a total set of functions in C+0 (G).

We can now prove the theorem below.

Theorem 11.9.9. The square-root ‖ · ‖ of the L-energy is a norm on E , and E+ is acomplete subset of the normed space (E, ‖ · ‖).Proof. We only prove the first part of the assertion. The second part is proved in[BC05, Theorem 6.3] by delicate results on the weak and weak∗ convergence in E .We explicitly remark that (E, ‖ · ‖) is not complete even in the classical Euclideancase of Laplace’s operator (see [Cart28, p. 87]).

We must show that μ − ν ∼ 0 if μ − ν, μ, ν ∈ E+ satisfy ‖μ − ν‖ = 0.We have |〈λ,μ − ν〉| ≤ ‖λ‖ · ‖μ − ν‖ = 0, i.e. 〈λ,μ〉 = 〈λ, ν〉 for everyλ ∈ E . In particular, this holds for λ = λx,r/2 − λx,r (see Proposition 11.9.5). Thus,〈λx,r/2, μ〉 − 〈λx,r , μ〉 = 〈λx,r/2, ν〉 − 〈λx,r , ν〉, i.e.

∫

G

(Γ ∗ λx,r/2 − Γ ∗ λx,r ) dμ =∫

G

(Γ ∗ λx,r/2 − Γ ∗ λx,r ) dν

for every x ∈ G and every r > 0. By Corollary 11.9.8, the functions

Γ ∗ λx,r/2 − Γ ∗ λx,r

form a total set in C+0 (G), so that (invoking Ex. 11 at the end of this chapter) μ = ν,

and the theorem is proved. �In order to prove our (Q-C) property, we need the following lemma (whose proof

follows by collecting together Lemmas 7.1, 7.2, 7.3 in [BC05]).

Lemma 11.9.10. The following assertions hold:

(i) Let μ ∈ E+. Then there exists a sequence μj ∈ E+ converging to μ in the L-energy norm with the following properties: supp(μj ) is compact, Γ ∗μj ≤ Γ ∗μ

and Γ ∗ μj ∈ C(G, R) for every j ∈ N.(ii) Let μ, ν ∈ E+. Given ε > 0, we set E = {x ∈ G | (Γ ∗μ)(x) > (Γ ∗ν)(x)+ε}.

Then C∗(E) ≤ ε−2 ‖ μ − ν ‖2.


(iii) Let μ ∈ M be such that μ(G) ≤ m ∈ R. Then, for every ε > 0, the L-capacityof the set A = {x ∈ G | (Γ ∗ μ)(x) > ε} does not exceed m/ε.

The key-rôle in the proof of the (Q-C) property is played by the following lemma.

Lemma 11.9.11 (Quasi-continuity for Γ -potentials). Let μ ∈ M with Γ ∗μ �≡ ∞.Then, for every ε > 0, there exists an open set O ⊆ Ω with C(O) < ε such that therestriction of Γ ∗ μ to Ω \ O is continuous.

Proof. We split the proof in three steps. Throughout the proof, ε > 0 is fixed.(I) μ ∈ E+. For any n ∈ N, by Lemma 11.9.10-(i), there exists μn ∈ E+ with

compact support such that Γ ∗μn ≤ Γ ∗μ, Γ ∗μn ∈ C(G, R) and ‖ μn−μ ‖2< ε/8n.We consider

En := {x ∈ G | (Γ ∗ μ)(x) > (Γ ∗ μn)(x) + 1/2n}.En is an open set, hence L-capacitable. By Lemma 11.9.10-(ii), we have C(En) ≤4n ‖ μ − μn ‖2< ε/2n. As a consequence, the L-capacity of E = ∪nEn does notexceed ε. Set F = G \E. It is easily seen that Γ ∗μn converges uniformly to Γ ∗μ

on F . Since Γ ∗ μn ∈ C(G, R), this proves that (Γ ∗ μ)|F is continuous on F .(II) μ ∈ M0. Let p ∈ N be such that μ(G)/p < ε/2. Consider the function

min{Γ ∗μ,p}. By Proposition 9.9.1-(b) (page 464), we have min{Γ ∗μ,p} = Γ ∗ν

for some ν ∈ M. Moreover, by applying Lemma 11.9.10-(iii) with m = μ(G), itfollows C(A) < ε/2, where A = {x ∈ G | (Γ ∗μ)(x) > p}. Obviously, Γ ∗ν ≡ Γ ∗μ

on G \ A. Now, ν ∈ E+. Indeed,

‖ν‖2 ≤∫

Γ ∗ μ dν =∫

Γ ∗ ν dμ ≤ p μ(G) ∈ R.

Hence, by the part (I) of the proof, there exists an open set B with C(B) < ε/2such that (Γ ∗ ν)|G\B is continuous on G \ B. Finally, the open set O = A ∪ B hasL-capacity less than ε and satisfies the assertion of the lemma. Indeed, (Γ ∗ μ)|F iscontinuous on F = G\O since Γ ∗μ ≡ Γ ∗ν on G\A ⊇ F , and since (Γ ∗ν)|G\Bis continuous on G \ B ⊇ F .

(III) μ ∈ M with Γ ∗ μ �≡ ∞. For j ∈ N, we denote

Cj = {x ∈ G| j − 1 ≤ d(x) < j}and μj = μ|Cj

. It then follows that Γ ∗ μ = ∑∞j=1 Γ ∗ μj . Since supp(μj ) ⊆

Cj � G, we can apply the part (II) of the proof. Thus, let Oj be an open set withC(Oj ) < ε/2j and such that (Γ ∗μj )|Fj

is continuous on Fj = G\Oj . Let us provethat the set O = ⋃∞

j=1 Oj satisfies the assertion of the lemma. Indeed, C(O) < ε

and (for F = G \ O)

(Γ ∗ μ)|F is continuous on F .

To see this, we fix x0 ∈ F , n0 ∈ N such that x0 ∈ Cn0 and an open neighborhood W

of x0 with W � { n0 − 2 < d(x) < n0}. Then


Γ ∗ μ =n0−2∑

j=1

Γ ∗ μj + Γ ∗ μn0−1 + Γ ∗ μn0 +∞∑

j=n0+1

Γ ∗ μj .

Now, for j = 1, . . . , n0−2, it holds Γ ∗μj ∈ H({d(x) > n0−2}), since supp(μj ) ⊆{d(x) ≤ n0 − 2}. Moreover, for any j ≥ n0 + 1, we have

supp(μj ) ⊆ {d(x) ≥ n0}, whence Γ ∗ μj ∈ H({d(x) < n0}).As a consequence, the function

∑∞j=n0+1 Γ ∗ μj is L-harmonic near x0 since it is

majorized by Γ ∗μ �≡ ∞. Finally, (Γ ∗μj )|F is continuous at x0 for all j ∈ N, sinceF ⊆ Fj and Γ ∗ μj ≡ (Γ ∗ μj )|Fj

on F . This ends the proof. �With Lemma 11.9.11 at hand and a Riesz-representation argument, we can easily

prove the quasi-continuity of arbitrary L-superharmonic functions.

Proof (of Theorem 11.9.1). We consider a sequence {Ωj }j∈N of open sets exhaust-ing Ω with Ωj � Ω , and we let μj = (μu)|Ωj

. Then μj ∈ M0. Thus, byLemma 11.9.11, for given ε > 0, there exists an open set Oj such that

C(Oj ) < ε/2j and (Γ ∗ μj )|Fj

is continuous on Fj = G \ Oj . Then O = Ω ∩ ⋃j Oj is an open subset of Ω

whose L-capacity does not exceed ε. Finally, u|F is continuous on F = Ω \ O.Indeed, if x0 ∈ F , then x0 ∈ Ωj0 for a suitable j0 ∈ N. Now, by Riesz representationTheorem 9.4.4 on page 442, there exists h ∈ H(Ωj0) such that u = h + Γ ∗ μj0 onΩj0 . Since F ⊆ Fj0 , this completes the proof. �Bibliographical Notes. The topics developed in this chapter within the classicaltheory of Laplace’s operator can be found in all the main monographs devoted topotential theory; see the references in the Bibliographical Notes of Chapter 8. Wealso mention the following references.

The paper by N.S. Trudinger and X.-J. Wang [TW02b] (and references therein)where several results of potential theory are established for a class of quasilinearsubelliptic operators including sub-Laplacians; see also the series of papers by thesame authors [TW97,TW99,TW02a] where fully non-linear Hessian operators aretreated; see D. Labutin [Lab02] for the study of subharmonic functions related tothis last class of operators; see the axiomatic non-linear potential theory containedin the monograph by J. Heinonen, T. Kilpelainen and O. Martio [HKM93] and inI.G. Markina and S.K. Vodopyanov [MV97a,MV97b], in a more general subellipticcontext.

For recent results on capacity, see [AH96,CDG96,Cart45,DG98,Hei95b,Ne88,KR87].

Some of the topics presented in this chapter also appear in [BC05,BC04].



Ex. 1) Prove that C({x ∈ G : Γ (x) ≥ 1/r}) = r . (Hint: Let K := {x ∈ G :Γ (x) ≥ 1/r}. Compute Γ ∗ μ(0) = ∫

Γ=1/rΓ (y) dμ(y), where μ is the

L-capacitary distribution of K and recall that μ is supported in ∂K .)Ex. 2) Prove that for any compact set K , we have

C(δλ(K)) = λQ−2 C(K), C(α ◦ K) = C(K)

for every λ > 0 and every α ∈ G. (Hint: Use Theorem 11.5.6.)Ex. 3) Prove Ex. 11.2.2, page 491.Ex. 4) Show that

C(K) = sup

{(∫

K

∫

K

Γ (x, y) dμ(x) dμ(y)

)1/(2−Q) ∣∣∣μ ∈ M, μ(K) ≥ 1

}

if K ⊂ G is compact.Ex. 5) Re-derive Theorem 11.5.11 (page 507) as a consequence of Proposition

11.8.4.Ex. 6) Show that the hypothesis “u bounded from above” cannot be removed in

the extended maximum principle of Theorem 11.2.7. (Hint: Take u = Γ inΩ = Bd(0, 1) \ {0} and E = {0}.)

Ex. 7) Let Z ⊂ G be L-polar. Let u ∈ S+(G). Show that

RuZ =

{u(x) if x ∈ Z,

0 otherwise.

Derive RuZ ≡ 0. (Hint: Let y /∈ Z and v ∈ S+

(G) be such that v(y) ∈ R

and v|Z ≡ ∞. Then v/n ∈ ΦuZ for every n ∈ N, so that. . . .)

Ex. 8) (Reciprocity law.) Prove that: If Ω is an open subset of G with L-Greenfunction GΩ and μ and ν are (positive) measures on Ω , then

∫Ω

GΩ ∗μ dν = ∫

ΩGΩ ∗ ν dμ.

(Hint: Use Fubini–Tonelli’s theorem and the symmetry of the GΩ .)Ex. 9) Following the notation in Definition 11.9.2, prove that:

• 〈μ, ν〉 is well defined;• it holds 〈μ, ν〉 = 〈ν, μ〉 (use the reciprocity law);• if μ ∈ M0 and Γ ∗ μ is bounded then μ ∈ E+;• if μ ∈ E+ and λ ∈ M satisfies Γ ∗ λ ≤ Γ ∗ μ, then λ ∈ E+ and

‖λ‖ ≤ ‖μ‖.Ex. 10) Following (11.39) (page 529) and using Theorem 9.3.10 (page 438), prove

that:• the generic element Tx,r (z) of T in (11.39) is a non-negative and con-

tinuous function of z ∈ G;• it vanishes outside Bd(x, r), it equals (r/2)2−Q − r2−Q on Bd(x, r/2)

and it equals Γ (z−1 ◦ x) − r2−Q in Bd(x, r) \ Bd(x, r/2);


• show that, for every α ∈ G, the function z �→ Tx,r (α ◦ z) still belongsto T : precisely, it holds Tx,r (α ◦ z) = Tα−1◦x,r (z).

Ex. 11) Following Definition 11.9.6 (page 529), prove that if μ1, μ2 ∈ M satisfy∫G

f dμ1 = ∫G

f dμ2 for every f in a total set T , then μ1 = μ2.Ex. 12) Let u : Bd(0, 1) → R be an L-harmonic function. Let us denote by μ the

L-harmonic measure of Bd(0, 1) at x0 = 0. Then the function

λ �→∫

∂Bd(0,1)

|u(δλ(x))|pdμ(x), 1 ≤ p < ∞,

is monotone increasing for 0 < r < 1. (Hint: x �→ |u(x)|p is L-subharmonic. Then use the Poisson–Jensen formula.)

Ex. 13) Let μ, ν be Radon measures in G and suppose that Γ ∗ μ ≤ Γ ∗ ν. Thenμ(G) ≤ ν(G).Hint: It suffices to prove that μ(B(0, R)) ≤ ν(G) for every R > 0. LetK := B(0, R). As usual, μK denotes the L-capacitary distribution for K .Recall that VK = Γ ∗ μK = 1 on the interior of K , i.e. on B(0, R). Thenwe have

μ(B(0, R)) =∫

B(0,R)

1dμ =∫

B(0,R)

Γ ∗ μK dμ ≤∫

G

Γ ∗ μK dμ

=∫

G

Γ ∗ μ dμK ≤∫

G

Γ ∗ ν dμK =∫

G

Γ ∗ μK dν

=∫

G

VK dν ≤∫

G

1 dν = ν(G).

Indeed, recall that VK ≤ 1 on G (see (11.16a)).Ex. 14) Let ν, σn be Radon measures on G (for every n ∈ N) with the following

properties:(i) There exists a compact set K such that supp(σn) ⊆ K , ∀ n ∈ N;

(ii) Γ ∗ σn is decreasing;(iii) on G \ K , it holds Γ ∗ σn −→ Γ ∗ ν.Then, ν(G) = limn→∞ σn(G).Hint: Let A := B(0, r) with r � 1 such that K ⊂ B(0, r). SinceΓ ∗ σn ∈ H(G \ K) and {Γ ∗ σn}n is decreasing and non-negative, itholds Γ ∗ ν ∈ H(G \ K), whence supp(ν) ⊆ K . Let μA denote the L-capacitary distribution for A. Note that Γ ∗ νA = 1 on B(0, r) ⊃ K andsupp(μA) ⊆ ∂A ⊂ G \ K . Thus Fubini’s theorem and monotone (decreas-ing) convergence imply

σn(G) =∫

K

1 dσn =∫

K

Γ ∗ μA dσn =∫

G

Γ ∗ μA dσn =∫

G

Γ ∗ σn dμA

=∫

supp(μA)

Γ ∗ σn dμA →∫

supp(μA)

Γ ∗ ν dμA =∫

G

Γ ∗ ν dμA

=∫

G

Γ ∗ μA dν =∫

K

Γ ∗ μA dν =∫

K

1 dν = ν(G).


Ex. 15) Let u ∈ S+(G). Let, for n ∈ N, Kn ⊆ Kn+1 be compact subsets of G.

Suppose Ω :=⋃n∈N

Kn is open. Then

limn→∞ Ru

Kn= Ru

Ω.

(Hint: Provide details for the following arguments. {RuKn

}n is a decreasing

sequence in S+(G). Set v := limn→∞ Ru

Kn. Then v ∈ S+

(G) and v ≤ u

on G. Moreover,

(�) v ≤ RuKn

≤ RuΩ on G.

Being RuKn

= RuKn

up to an L-polar subset of ∂Kn, say Pn, we get RuKn

(x) =u(x) for every x ∈ Kn \ Pn. Let P :=⋃

n∈NPn. If x ∈ Ω \ P , then x /∈ Pn

for every n ∈ N, and there exists n0 ∈ N such that x ∈ Kn0 ⊆ Kn for everyn ≥ n0. Hence

RuKn

(x) = u(x) ∀ n ≥ n0.

Letting n → ∞, we derive v(x) = u(x) for every x ∈ Ω \P . Hence (recallthat P is L-polar) v = u on Ω . Thus v ∈ Φu

Ω , so that v ≥ RuΩ . By (�), this

gives v = RuΩ .)

Ex. 16) In this exercise, we state the main properties of the exterior L-capacity C∗defined in Definition 11.5.14. For the relevant proofs, one can follow verba-tim the classical capacitability theory as presented, e.g. in [Helm69].(1) Starting from the strong sub-additivity property in (11.20) of Proposi-

tion 11.5.10, one first shows that if {Ki}1≤i≤m and {Ci}1≤i≤m are finitefamilies of compact sets in G such that Ci ⊂ Ki (i ≤ m), then

C(

m⋃

i=1

Ki

)− C

(m⋃

i=1

Ci

)≤

m∑

i=1

(C(Ki) − C(Ci)

).

(2) The second step is to derive from the above that if {Ui}1≤i≤m and{Vi}1≤i≤m are finite families of open sets in G such that Vi ⊂ Ui

(i ≤ m), then

C∗

(m⋃

i=1

Ui

)+

m∑

i=1

C∗(Vi) ≤m∑

i=1

C∗(Ui) + C∗

(m⋃

i=1

Vi

).

(3) From (1) and (2) it is possible to derive that if {Ei}1≤i≤m and {Fi}1≤i≤m

are arbitrary finite families of subsets of G such that Fi ⊂ Ei (i ≤ m),then

C∗(

m⋃

i=1

Ei

)+

m∑

i=1

C∗(Fi) ≤ C∗(

m⋃

i=1

Fi

)+

m∑

i=1

C∗(Ei).

(4) From (3) it is possible to derive that C∗ is an exterior capacity. We saythat a non-negative real-valued function φ∗(·) defined on the subsets ofG is an exterior capacity if:


(i) E ⊂ F ⇒ φ∗(E) ≤ φ∗(F );(ii) if {Ej }j ⊂ G is such that Ej ⊆ Ej+1, then φ∗(

⋃j Ej ) =

limj→∞ φ∗(Ej );(iii) if {Kj }j � G is such that Kj+1 ⊆ Kj , then φ∗(

⋂j Kj ) =

limj→∞ φ∗(Kj ).(5) From (4) we infer that C∗ is numerably sub-additive. Indeed, if {Ej }j ⊆

G is a sequence of arbitrary sets, we apply (ii) to {Ej }j , where Ej :=⋃j

i=1 Ei , so that

C∗(⋃

j

Ej

)= C∗

(⋃

j

Ej

)= lim

jC∗(Ej ) ≤

∑

j

C∗(Ej ),

for (apply (3) with empty Fi’s)

C∗(Ej ) = C∗(

j⋃

i=1

Ei

)≤

j∑

i=1

C∗(Ei) ≤∞∑

i=1

C∗(Ei).

(6) As a consequence of (4) and the L-capacitability of open sets and com-pact sets, we infer that if Ω = ⋃

j Kj with Ω open and Kj ⊆ Kj+1compact sets, then

C(Ω) = limj→∞ C(Kj ).

12

L-thinness and L-fine Topology

In this chapter, we deal with the notion of fine topology generated by the superhar-monic functions related to a sub-Laplacian L. We mainly investigate the relationshipbetween L-thinness and L-regularity of boundary points for the Dirichlet problem.

As usual, throughout the chapter G = (RN, ◦, δλ) is a (homogeneous) Carnotgroup and L is a sub-Laplacian on G. Moreover, Γ = d2−Q is the fundamentalsolution for L.

12.1 The L-fine Topology: A More Intrinsic Tool

Since L-superharmonic functions play a central rôle in the potential theory for L,it seems more natural to introduce a new topology—finer than the Euclidean one—with respect to which any L-superharmonic function is continuous. In the sequel, ifa function u takes on a ∞-value at a point x0, we say that u is continuous (in theextended sense) at x0, provided limx→x0 u(x) = ∞. Moreover, if X is a topologicalspace, A ⊆ X is any subset of X and x ∈ X, we say that x is a limit point of A if, forevery open set U containing x, there exists a ∈ A ∩ U such that a �= x. We denoteby Der(A) (the derived set of A) the set of the limit points of A.

Definition 12.1.1 (The L-fine topology). The L-fine topology on G is the smallesttopology on R

N with respect to which all L-superharmonic functions are continuous(in the extended sense).

Throughout the chapter, all topological concepts (continuous, open, neighbor-hood, etc.) will be prefixed by “L-fine” when related to the L-fine topology. Instead,they will not be prefixed or prefixed by “Euclidean” or “metric”, when related to theusual Euclidean topology of R

N .A few simple remarks are in order.

Remark 12.1.2. (1) Since L-superharmonic functions are l.s.c., a basis for the L-finetopology is given by all finite intersections of the sets

538 12 L-thinness and L-fine Topology

{x ∈ Ω : u(x) < β } (12.1)

as Ω ranges over all (Euclidean-)open subsets of RN , u is any function in S(Ω), and

β ranges over ]−∞,∞].(2) The L-fine topology is properly larger than the Euclidean one (see Exam-

ple 11.2.4, page 492).(3) In particular, if limx→x0 u(x) = λ in the Euclidean sense, then the same is

true in the L-fine sense too, whereas the converse may be false. Vice versa, if {xn}n∈N

is a sequence in G converging to x0 ∈ G in the L-fine sense, then xn → x0 in theEuclidean sense too.

(4) Hence, the collection of L-fine limit points of a set A is contained (possiblyproperly) in the set of usual limit points of A,

DerL-fine(A) ⊆ DerEucl.A.

For example, if x0 ∈ G\{0}, the set of the L-fine limit points of A = {(1+ 1n) x0}n∈N

is empty whereas x0 is a limit point of A (see also Proposition 12.1.3 below).

Proposition 12.1.3. Let Z ⊂ G be an L-polar set. Then the set of the L-fine limitpoints of Z is empty. In particular, the points of Z are L-finely isolated points of Z.

Proof. By contradiction, let x be an L-fine limit point of Z. Then x is also a (Euclid-ean) limit point of Z. By Example 11.2.4 (page 492), there exists u ∈ S(G) suchthat

u(x) < limZy→x

u(y) =: r < ∞.

Set ε := 12 (r − u(x)). Then there exists a neighborhood U of x such that u(y) >

r − ε for every y ∈ (U ∩ Z) \ {x}. Since u is L-fine continuous at x (being L-superharmonic), there is an L-fine neighborhood Ufine of x such that u(y) < u(x)+ε

for every y ∈ Ufine. But r − ε = 12 (r + u(x)) = u(x) + ε, so that (U ∩ Z) ∩

Ufine = (U ∩ Ufine) ∩ Z is contained in {x}. This means that U ∩ Ufine is an L-fineneighborhood of x having in common with Z at most x, whence x is not an L-finelimit point of Z. This is a contradiction. ��

12.2 L-thinness at a Point

Consider a bounded open set Ω ⊂ G and a point x ∈ ∂Ω . We have seen in Propo-sition 7.1.5 (page 385) that if G \ Ω contains a suitable non-characteristic ball, thenx is regular for the Dirichlet problem related to L. Hence, for L-irregular boundarypoints of Ω , the complement of Ω must be quite thin (in a suitable sense, which alsotakes into account the “degeneracy” of the operator L). The notion of L-thinness willgive a more precise sense to this.

Definition 12.2.1 (L-thinness of a set at a point). A set E ⊆ G is called L-thin atx ∈ G if x is not an L-fine limit point of E.

12.2 L-thinness at a Point 539

In other words,

E is L-thin at x ⇔ x /∈ DerL-fine(E) ⇔ {x} ∩ DerL-fine(E) = ∅.

A few simple remarks are in order.

Remark 12.2.2. Obviously, E is not L-thin at x iff x is an L-fine limit point of E, i.e.

E is not L-thin at x ⇔ x ∈ DerL-fine(E).

Hence, if E is not L-thin at x, then x is a (Euclidean) limit point of E (indeed,DerL-fine(E) ⊆ DerEucl.E). Equivalently, a sufficient condition that a set E is L-thinat x is that x is not a metric limit point of E.

From Proposition 12.1.3 we immediately get the following result.

Corollary 12.2.3. A L-polar set is L-thin at every point.

The following result will be important in the sequel.

Theorem 12.2.4 (Characterization of L-thinness at a point. I). A set E ⊆ G isL-thin at a (metric limit) point x of E if and only if there exists an L-superharmonicfunction u on a (metric) neighborhood of x such that

u(x) < lim infEy→x

u(y). (12.2)

Proof. To avoid heavy notation, it is not restrictive to suppose that x /∈ E.(⇒): Suppose first that E is L-thin at x. Then there exists an L-fine neighborhood

Ufine of x such that E∩Ufine = ∅. By taking finite intersections of sets as in (12.1), asa basis for L-fine neighborhoods, we infer the existence of β1, . . . , βn ∈ ]−∞,∞]and u1, . . . , un ∈ S(W), W being an open neighborhood of x, such that Ufine =⋂n

i=1{y ∈ W : ui(y) < βi}. Since x ∈ Ufine (i.e. ui(x) < βi), by choosing ε =min{βi −ui(x)}, we can replace Ufine by the set

⋂ni=1{y ∈ W : ui(y) < ui(x)+ ε}.

Now let u := ∑ni=1 ui . We claim that u satisfies (12.2). By the lower semi-continuity

of the ui’s at x, there exists a neighborhood U ⊂ W of x such that

ui(y) > ui(x) − ε/n for every y ∈ U and i = 1, . . . , n. (12.3)

Now take any y ∈ U ∩ E. Then, since E ∩ Ufine = ∅, there exists i0 ∈ {1, . . . , n}such that

ui0(y) ≥ ui0(x) + ε. (12.4)

Collecting together (12.3) and (12.4), we get

u(y) =∑

i �=i0

ui(y) + ui0(x) >∑

i �=i0

ui(x) − n − 1

nε + ui0(x) + ε = u(x) + ε

n.

In short, u(y) > u(x) + εn

for every y ∈ U ∩ E which at once implies (12.2).


(⇐): Suppose now there exists an L-superharmonic function u on a neighbor-hood Ω of x such that u(x) < lim infEy→x u(y) =: r . We can assume that r < ∞by replacing, if needed, u by min{u, u(x) + 1}. Set ε = 1

2 (r − u(x)). ConsiderUfine := {y ∈ Ω : u(y) < u(x) + ε}, which is an L-fine neighborhood of x.From lim infEy→x u(y) = r we infer that there exists a neighborhood U of x suchthat infE∩V u > r − ε. Now we argue verbatim as in the last part of the proof ofProposition 12.1.3. ��

Since the value of an L-superharmonic function at a point coincide with its lim infthere, (12.2) gives another equivalent definition of L-thinness: E is L-thin at x if andonly if

there exists an L-superharmonic function u on a neighborhood

of x such that lim infy→x

u(y) < lim infEy→x

u(y). (12.5)

This last condition is the analogue of L-thinness at infinity given in (10.2), page 474.Another remarkable result (making L-thinness easier to handle with) is the followingone.

Theorem 12.2.5 (Characterization of L-thinness at a point. II). A set E ⊆ G is L-thin at a point x ∈ G (limit point of E) if and only if there exists an L-superharmonicfunction w on G, which is the L-potential of a compactly supported Radon measure,such that

w(x) < limEy→x

w(y) = ∞. (12.6)

Proof. The “if” part follows from Theorem 12.2.4. To prove the converse, we assumethat E is L-thin at x (and, without affecting the generality, we may suppose x /∈ E).It is enough1 to prove the existence of an open neighborhood Ω of x and u ∈ S(Ω)

such that (12.6) holds with w replaced by u.To this end, let u be as in Theorem 12.2.4. We can assume that the right-hand

side of (12.2) is finite (otherwise there is nothing to prove). Let B = Bd(x, ε) withε > 0 small enough. By arguing as in the (footnote of the) previous paragraph, wecan suppose that u = Γ ∗ ν, where ν is a Radon measure supported in B. For n largeenough, we have

1 Indeed, in this case we can take any open set O with x ∈ O ⊂ O � Ω and apply Rieszrepresentation Theorem 9.4.4 (page 442) to get

u(y) = (Γ ∗ ν)(y) + h(y) ∀ y ∈ O,

where ν = μu|O

and h is a suitable function in H(O). Hence, we have

limE→x

u(y) = limE→x

(Γ ∗ ν)(y) + h(x), whereas u(x) = (Γ ∗ ν)(x) + h(x).

Consequently, (12.6) holds if and only if (Γ ∗ν)(x) < limE→x(Γ ∗ν)(y), and the choicew := Γ ∗ ν fulfills the requirement of the assertion of the theorem.

12.2 L-thinness at a Point 541∫

Bd(x,1/n)

Γ (x, y) dν(y) +∫

B\Bd(x,1/n)

Γ (x, y) dν(y) = u(x) ∈ R,

and the second integral in the left-hand side increases2 to u(x) as n ↑ ∞. As aconsequence, setting νn := ν|Bd(x,1/n), we have limn→∞(Γ ∗ νn)(x) = 0. This en-sures the existence of εn ↓ 0 such that

∑n un(x) < ∞, where un := Γ ∗ νεn .

Then the function v := ∑n un is finite at x and belongs to S(B). Let δ :=

lim infEy→x(u(y) − u(x)) > 0. By applying once again the Riesz representationtheorem, we get u = un + hn, where hn ∈ H(Bd(x, εn)). This gives

lim infEy→x

(un(y) − un(x)) ≥ lim infEy→x

(un(y) + hn(y)) + lim infEy→x

(−hn(y) − un(x))

= lim infEy→x

u(y) − hn(x) − un(x) = lim infEy→x

(u(y) − u(x)) = δ

for every n ∈ N, so that (by summing up)

lim infEy→x

(p∑

n=1

un(y) −p∑

n=1

un(x)

)≥ p δ.

This gives lim infEy→x

∑p

n=1 un(y) ≥ ∑p

n=1 un(x) + p δ ≥ p δ. Since v =∑n un ≥ ∑p

n=1 un, we derive (letting p → ∞) lim infEy→x v(y) = ∞. ��From Theorem 12.2.5 one immediately obtains the following assertion.

Corollary 12.2.6. Let Z, E ⊆ G, y ∈ G. Suppose Z is L-polar. Then E\Z is L-thinat y if and only if E is.

Proof. Suppose first that E is L-thin at y. Then, by Theorem 12.2.4, there ex-ists an L-superharmonic function u on a neighborhood of y such that u(y) <

lim infEz→y u(z). This gives

u(y) < lim infEz→y

u(z) ≤ lim infE\Zz→y

u(z),

so that E \ Z is L-thin at y, again by Theorem 12.2.4.Vice versa, suppose that E \ Z is L-thin at y and (thanks to Theorem 12.2.5)

consider an L-superharmonic function u on G such that u(y) < limE\Zz→y u(z) =∞. Since Z \ {y} is L-polar, there exists w ∈ S +

(G) such that w ≡ ∞ on Z \ {y}and w(y) ∈ R. Consequently, we have

limEz→y

(u(z) + w(z)) = ∞ > u(y) + w(y).

This proves that E is L-thin at y by making use of Theorem 12.2.5. ��2 This follows from ν({x}) = 0. Otherwise, we would have

∞ > u(x) ≥∫

Bd(x,1/n)Γ (x, y) dν(y) → ∞ · ν({x}) = ∞.


12.3 L-thinness and L-regularity

The aim of this section is to prove that the L-regularity of a boundary point canbe characterized in terms of L-thinness (see Theorem 12.3.6 below). To this end,we provide some preliminary results. Some have an interest in their own and arecollected in the following subsection.

12.3.1 Functions Peaking at a Point

We need a definition and some preliminary results.

Definition 12.3.1 (Function peaking at a point). If u is an extended real-valuedfunction defined on a neighborhood of x, we say that u peaks at x if u(x) >

supy /∈V u(y) for every small neighborhood V of x.

Lemma 12.3.2. For every fixed x ∈ G, there exists w ∈ S +(G) ∩ C(G) such that w

has a strict absolute maximum at x (in particular, w peaks at x).

Proof. Let r > 0. Then the function w(z) := Mr (Γ (z−1 ◦ ·))(x) has the requiredproperties (see Theorem 9.3.10, page 438; see also Fig. 12.1). ��

Fig. 12.1. The function w of Lemma 12.3.2

The relevance of functions peaking at a point within the theory of L-thinnessbecomes apparent from the following result.

Lemma 12.3.3 (Characterization of L-thinness. III). Let w ∈ S +(G) be a func-

tion which peaks at x. Then a set E is L-thin at x if and only if

RwE(x) � w(x).

Proof. Since E is L-thin at x if and only if E \ {x} is L-thin at x (see, e.g. Corol-lary 12.2.6) and since Rw

E = RwE\{x} (see Proposition 11.8.3, page 525), we can

suppose that x /∈ E. As a consequence, from Theorem 11.6.5 (page 513) we derive

RwE(x) = Rw

E(x).

12.3 L-thinness and L-regularity 543

(⇐): Let us suppose RwE(x) < w(x) and prove that E is L-thin at x. We can

suppose that x is a limit point of E (otherwise, nothing is left to prove). By thedefinition of L-réduite function, there exists v ∈ S +

(G) such that v ≥ w on E andv(x) < w(x). This gives

lim infEy→x

v(y) ≥ lim infEy→x

w(y) ≥ lim infy→x

w(y) ≥ w(x) > v(x),

whence E is L-thin at x by Theorem 12.2.4.(⇒): Let now E be L-thin at x. Suppose first that x is not a limit point of E, and

let V be a neighborhood of x such that V ∩ E = ∅. Then

w(x) > supG\V

w ≥ supE

w ≥ RwE(x) = Rw

E(x).

The third inequality follows from the fact that the constant function supE w belongsto Φw

E (see the notation at the beginning of Section 11.5, page 500). We are left with

the case when x is a limit point of E. By Theorem 12.2.5, there exists v ∈ S +(G)

such that v(x) < limEy→x v(y) = ∞. Replacing v by v := min{v, v(x) + 1}, wecan suppose that v(x) < limEy→x v(y) < ∞ and v is bounded from above. Let α

and β be such thatv(x) < α < β < lim

Ey→xv(y). (12.7)

For every λ > 0, we set

wλ(·) := w(x) + λ (v(·) − α).

First, we claim that there exists a neighborhood U of x and 0 < λ � 1 such thatwλ > 0 on U for any 0 < λ ≤ λ. Indeed, from w(x) > 0 (w peaks at x) and thelower semi-continuity of w we get lim infy→x w(y) ≥ w(x) > 0. Hence, there existsa neighborhood U of x and ε > 0 such that infU w ≥ ε. We choose 0 < λ � 1 suchthat λ |v(y) − α| ≤ ε/2 (recall that v is bounded) for every y ∈ U . This gives, forevery y ∈ U and every 0 < λ ≤ λ,

wλ(y) = w(y) + λ(v(y) − α) ≥ ε − λ |v(y) − α| ≥ ε/2.

The claim is proved. According to (12.7), there exists a neighborhood U ⊆ U of x

such that v(y) > α for every y ∈ E ∩ U . Then we have

wλ(y) > w(x) ≥ w(y) ∀ y ∈ E ∩ U. (12.8)

Moreover, w(x) − w(y) ≥ γ > 0 for every y /∈ U (again by the peaking propertyof w). Since v is bounded, there exists a small positive λ0 < λ such that

λ0 |v(y) − α| ≤ γ ≤ w(x) − w(y) ∀ y /∈ U.

This proves that, for y /∈ U , we have

λ0 |v(y) − α| ≥ −λ0 |v(y) − α| ≥ −(w(x) − w(y)) = w(y) − w(x),


so thatwλ0(y) = w(x) + λ0 (v(y) − α) ≥ w(y) ∀ y /∈ U.

This last inequality, together with (see also (12.8)) the inequality

wλ0(y) ≥ w(y) ∀ y ∈ E ∩ U,

gives wλ0(y) ≥ w(y) for every y ∈ E. Moreover, from wλ0(y) ≥ 0 on U (see theclaim above) and wλ0 ≥ w ≥ 0 on E we derive that wλ0(y) ≥ 0. This proves that

w ∈ S +(G) is such that wλ0 ≥ w on E, whence, by the definition of L-réduite,

RwE(x) ≤ wλ0(x) = w(x) + λ0 (v(x) − α) < w(x).


Theorem 12.3.4 (L-thinness and L-polarity. I). Let E ⊆ G be any set. Then thesubset of E where E is L-thin is an L-polar set. In other words, the set of the L-finelyisolated points of E is an L-polar set.

Proof. Let E be L-thin at x0 ∈ E. Let w ∈ S +(G) ∩ C(G) be such that w peaks at

x0 (see Lemma 12.3.2). By Lemma 12.3.3, we have RwE(x0) � w(x0) = Rw

E(x0) (forx0 ∈ E). Then x0 belongs to P := {x ∈ G : Rw

E(x0) < RwE(x0)} which is a L-polar

subset of E ∩ ∂E (see Theorem 11.6.5, page 513). From the arbitrariness of x0, theassertion follows. ��Corollary 12.3.5 (L-thinness and L-polarity. II). A set Z ⊂ G is L-polar if andonly if Z is L-thin at any of its points.

Proof. If Z is L-polar, then Z is L-thin at every point of G (see Proposition 12.1.3).The converse assertion follows immediately from Theorem 12.3.4. ��

12.3.2 L-thinness and L-regularity

We are now in a position to state and prove the main results of the section.

Theorem 12.3.6 (Non-L-thinness and L-barriers). Let F ⊆ G be a closed set.Let x0 ∈ ∂F . Then F is not L-thin at x0 if and only if there exists a (L-barrier)function b with the following properties: b ∈ S(G \ F), b ≥ 0 on G \ F , b > 0 onW := Bd(x0, 1) \ F and limWy→x0 b(y) = 0.

From Theorem 12.3.6 and Bouligand’s Theorem 6.10.4 (page 371) we immedi-ately derive the following L-regularity criterion.

Theorem 12.3.7 (L-regularity criterion in terms of L-thinness). Let Ω ⊂ G bea bounded and connected open set. Let x0 ∈ ∂Ω . Then x0 is an L-regular point forΩ if and only if G \ Ω is not L-thin at x0. Equivalently, x0 is an L-irregular point ifand only if G \ Ω is L-thin at x0.

12.3 L-thinness and L-regularity 545

Fig. 12.2. The barrier function of Theorem 12.3.6

Proof (of Theorem 12.3.6). Let F ⊆ G be closed, and let x0 ∈ ∂F (⊆ F ).(⇒): Suppose F is not L-thin at x0. Let

w(z) := M1(Γ (z−1 ◦ ·))(x0).

Then (see Theorem 9.3.10, page 438) w ∈ S(G) ∩ C(G), w peaks at x0 and w > 0on G. Consider the L-balayage Rw

F of w relative to F . Then

RwF (x0) = w(x0).

Indeed, since F is not L-thin at x0, by Lemma 12.3.3 (and the general properties ofthe L-balayage) we have Rw

F (x0) ≥ w(x0) ≥ RwF (x0). Hence,

lim infG\Fy→x0

RwF (y) ≥ lim inf

y→xRw

F (y) ≥ RwF (x0) = w(x0). (12.9)

Let us now consider the function

b := w − RwF .

The function b is non-negative3 on G and b ∈ S(G \ F). This last fact followsfrom w ∈ S(G), Rw

F ∈ H(G \ F) and F = F . We now claim that b > 0 onW := Bd(x0, 1) \ F . Indeed, it is enough to prove that b > 0 on every connectedcomponent C of W . Suppose, to the contrary, that b vanishes at some point of C. Bythe minimum principle,4 this gives b ≡ 0 on C, whence w ≡ Rw

F on C, so that

Lw = 0 on C, (12.10)

for RwF is L-harmonic on G \ F ⊃ W ⊇ C. Now, a direct computation (see (9.10b)

in Theorem 9.3.10, page 438) shows that, for every z ∈ C, we have5

Lw(z) = 2 − Q

2L(d2(x0, z))

= Q

2 − QΓ (2Q−2)/(2−Q)(x−1

0 ◦ z) · |∇LΓ |2(x−10 ◦ z).

3 This follows from the general fact w ≥ RwF

if w ∈ S(G).4 Note that b ∈ S(W) since W is open, W ⊆ G \ F and b ∈ S(G \ F).5 Recall that d = Γ 1/(2−Q), the general formula L(α(u)) = α′′(u) |∇Lu|2 + α′(u)Lu

(where u : R → R is regular enough) and LΓ = 0 outside the origin.


The far right-hand term cannot vanish on the open set C (otherwise ∇LΓ ≡ 0 on C,i.e. Γ ≡ 0 on C, see Proposition 1.5.6, page 69) in contradiction with (12.10).

Moreover, we have

0 ≤ lim supWy→x0

b(y) = lim supF/y→x0

b(y) = lim supF/y→x0

(w(y) − RwF (y))

≤ w(x0) − lim infF/y→x0

RwF (y) ≤ 0.

Here, we exploited the continuity of w and (12.9). This finally proves thatlim supWy→x0

b(y) = 0.(⇐): Let b be as in the assertion of Theorem 12.3.6 (see also Fig. 12.2). Let

us suppose by contradiction that F is not L-thin at x0 ∈ ∂F . Then, by making useof Theorem 12.2.5, it is not difficult6 to prove the existence of a bounded functionv ∈ S(G) and a neighborhood U ⊆ Bd(x0, 1) of x0 such that v(x0) = 1 and v ≤ −1on U ∩ (F \ {x0}).

Let B1 := Bd(x0, ρ) with 0 < ρ � 1 such that B1 ⊂ U . Then

v ≤ −1 on F ∩ ∂B1 (⊆ U ∩ F \ {x0}).Since F ∩ ∂B1 is compact, this fact, together with the upper semi-continuity of v,implies the existence of an open neighborhood W of F ∩ ∂B1 such that v ≤ −1/2on W . Hence,7 for every λ > 0

v − λ b ≤ 0 on W ∩ (∂B1 \ F). (12.11)

Being b > 0 on ∂B1 \ F , we have

inf∂B1\W

b > 0, (12.12)

for ∂B1 \ W is a compact set which does not intersect a neighborhood of F and b

is l.s.c. outside F . Hence, the boundedness of v and (12.12) ensure that there existsλ0 � 1 such that

v − λ0 b ≤ 0 on ∂B1 \ W . (12.13)

Thus, for every y ∈ ∂B1 \ W and every ε > 0, one has (recalling that v − λ0 b isu.s.c. and using (12.13))

lim supB1\Fz→y

(v(z) − λ0 b(z) − ε Γ (x−1

0 ◦ z)) ≤ 0. (12.14)

The same holds for y = x0 (since Γ (0) = ∞) and for every y ∈ B1 ∩ (∂F \{x0}), for this last set is contained in U ∩ (F \ {x0}) where v ≤ −1. Now, beingv − λ0 b − ε Γ (x−1

0 ◦ ·) ∈ S(B1 \ F), from the maximum principle one gets (alsocollecting (12.11), (12.14))

6 See Ex. 5 at the end of the present Chapter.7 Indeed, v ≤ −1/2 on W , b > 0 on W = Bd(x0, 1) \ F and Bd(x0, 1) ⊇ U ⊃ B1 ⊃ ∂B1.

12.4 Wiener’s Criterion for Sub-Laplacians 547

v − λ0 b − ε Γ (x−10 ◦ ·) ≤ 0 on B1 \ F .

Letting ε → 0+, we derive v ≤ λ0 b on B1 \ F , so that

lim supB1\Fz→x0

v(z) ≤ lim supB1\Fz→x0

λ0 b(z) = 0

by the definition of the barrier function b. Since

lim supFz→x0

v(z) ≤ −1

(for v ≤ −1 on U ∩ (F \ {x0})), we finally get

lim supz→x0

v(z) ≤ 0.

But this is in contradiction to lim supz→x0v(z) = v(x0) (a consequence of the fact

that v is L-subharmonic in x0) and v(x0) = 1 (by construction). This contradictionproves the theorem. ��

12.4 Wiener’s Criterion for Sub-Laplacians

The aim of this section is to prove a test for L-thinness similar to the well-knownWiener criterion for the classical Laplace operator (in this case, see, e.g. [AG01,Section 7.7]). For a generalization to more general operators, see also [NS87].

As usual, throughout the section, G = (RN, ◦, δλ) is a Carnot group and L is asub-Laplacian on G. Moreover, Γ = d2−Q is the fundamental solution for L. Weshall also denote by β ≥ 1 the constant appearing in the following inequalities:

d(a ◦ b) ≤ β d(a) + d(b) ∀ a, b ∈ G, (12.15a)

d(a ◦ b) ≥ 1

β|d(a) − d(b)| ∀ a, b ∈ G. (12.15b)

These are (subtle) improvements of the pseudo-triangle inequality for d . (12.15a) isproved in Proposition 5.14.1 on page 306. Then, by choosing a = x ◦y and b = y−1

in (12.15a) (and using the symmetry of d), we derive d(x) − d(y) ≤ β d(x ◦ y);then, replacing x and y in the latter with, respectively, y−1 and x−1 (and using twicethe symmetry of d) we get d(y) − d(x) = d(y−1) − d(x−1) ≤ β d(y−1 ◦ x−1) =β d((x ◦ y)−1) = d(x ◦ y). This proves (12.15b).

12.4.1 A Technical Lemma

Throughout the section, we use the following notation: given a point y ∈ G and aconstant α > 1, for every n ∈ N we set

Cn := {x ∈ G : αn ≤ Γ (y−1 ◦ x) ≤ αn+1}. (12.16)


In other words, let

α :=(

1

α

) 1Q−2

,

we haveCn = Bd

(y, αn

) \ Bd

(y, αn+1).

We begin with a crucial lemma concerning with suitable estimates of the fundamentalsolution Γ for L. We remark that the proof of Lemma 12.4.1 below is more delicatein our context than in the classical case (see [AG01, Lemma 7.7.1]).

We shall henceforth suppose that

α > βQ−2, (12.17)

where β is the constant appearing in (12.15a) (note that in the Euclidean case, β = 1and (12.17) reduces to α > 1).

Lemma 12.4.1. Let α > 1 (be as in (12.17)). Then there exists c = c(α) > 0 such

that, for every y ∈ G, for every Radon measure μ on Bd(y, 12 ), for every n ∈ N,

n ≥ 2 and every x ∈ Cn it holds∫

G\(Cn−1∪Cn∪Cn+1)

Γ (x−1 ◦ z) dμ(z) ≤ c∫

G

Γ (y−1 ◦ z) dμ(z). (12.18)

Proof. Let y ∈ G and α > 1 be fixed throughout. Let also x ∈ Cn, i.e.

αn ≤ d2−Q(y−1 ◦ x) ≤ αn+1. (12.19)

We have

G \ (Cn−1 ∪ Cn ∪ Cn+1) = {z ∈ G : Γ (y−1 ◦ z) > αn+2} ∪∪ {

z ∈ G : Γ (y−1 ◦ z) < αn−1} =: I ∪ II.

We distinguish two cases:

(i): z ∈ I, i.e. d2−Q(y−1 ◦ z) > αn+2, (12.20a)

(ii): z ∈ II, i.e. d2−Q(y−1 ◦ z) < αn−1. (12.20b)

(i): Suppose first that z ∈ I . Then (see (12.15b), (12.19), (12.20a))

d(x−1 ◦ z) ≥ 1

β

(d(x−1 ◦ y) − d(y−1 ◦ z)

) ≥ 1

βα

n+22−Q (α

1Q−2 − 1).

Equivalently (exploiting once again (12.20a) in the last inequality),

Γ (x−1 ◦ z) ≤(

β

α1

Q−2 − 1

)Q−2

αn+2 =: c(α) αn+2 < c(α) Γ (y−1 ◦ z).


We have thus proved (indeed, we only used the second inequality in (12.19))

x ∈ Cn, z ∈ I �⇒ Γ (x−1 ◦ z)

Γ (y−1 ◦ z)≤ c(α). (12.21)

(ii): Suppose now that z ∈ II. In this case, we also need an estimate similar tothat in the far right-hand of (12.21), namely

x ∈ Cn, z ∈ II �⇒ Γ (x−1 ◦ z)

Γ (y−1 ◦ z)≤ c(α). (12.22)

Since this estimate will intervene to estimate the integral in the left-hand side

of (12.18), and the measure μ appearing there is supported in Bd(y, 12 ), we can

suppose thatd(y−1 ◦ z) ≤ 1/2. (12.23)

Now, (12.22) follows if we, equivalently, prove that

d(y−1 ◦ z)

d(x−1 ◦ z)≤ c(α) (12.24)

under the following8 hypotheses on z (see (12.20b), (12.23))

αn−12−Q < d(y−1 ◦ z) ≤ 1/2. (12.25)

Suitable bounds for d(x−1 ◦ z) are given by the following inequalities:

d(x−1 ◦ z) ≤ β d(x−1 ◦ y) + d(y−1 ◦ z) ≤ β αn

2−Q + d(y−1 ◦ z),

d(x−1 ◦ z) ≥ d(y−1 ◦ x ◦ x−1 ◦ z) − β d(y−1 ◦ x) ≥ d(y−1 ◦ z) − β αn

2−Q .

The first estimate has been obtained by applying (12.15a) with a = x−1 ◦ y, b =y−1 ◦ z, jointly with the first inequality in (12.19); the second estimate has beenobtained by applying (12.15a) with a = y−1 ◦ x, b = x−1 ◦ z, jointly with the firstinequality in (12.19). Thus,

d(y−1 ◦ z) − β αn

2−Q ≤ d(x−1 ◦ z) ≤ d(y−1 ◦ z) + β αn

2−Q . (12.26)

Set λ := d(y−1 ◦ z), μ := d(x−1 ◦ z), A := αn−12−Q and B := β α

n2−Q . According to

the hypotheses (12.25), (12.26) and the needed assertion (12.24), we have to prove

A < λ ≤ 1/2λ − B ≤ μ ≤ λ + B

�⇒ λ

μ≤ c. (12.27)

Now, (12.27) follows from a simple exercise of optimization (see Ex. 10 at the endof the present Chapter9) with c = A/(A − B), i.e.

8 We are tacitly supposing that n is large enough, so that αn−12−Q < 1/2, otherwise there is

nothing to prove.9 Note that the hypothesis (12.17) on α ensures that A ∈ ]0, 1/2[ and B ∈ ]0, A[.


c = αn−12−Q

αn−12−Q − β α

n2−Q

= 1

1 + β α1/(2−Q)= c(α).

(Note that c depends only on B/A and does not depend on n.)Finally, (12.18) follows straightforwardly from (12.21) and (12.22). The lemma

is completely proved. ��

12.4.2 Wiener’s Criterion for L

Let α > 1 be as in Lemma 12.4.1 at the beginning of the section. We also maintainthe notation Cn given by (12.16). We are ready to state and prove our Wiener’s testfor sub-Laplacians.

Theorem 12.4.2 (Wiener’s criterion for L). Let C∗ denote the exterior L-capacityrelative to G. Let also E ⊆ G and y ∈ G. Then the following statements are equiva-lent:

(i) E is L-thin at y,(ii) It holds

∑∞n=1 αn C∗(E ∩ Cn) < ∞,

(iii) It holds∑∞

n=1 R1E∩Cn

(y) < ∞,

(iv) It holds∫ ∞α

C∗({x ∈ E : Γ (y−1 ◦ x) ≥ t}) dt < ∞.

Proof. First, we remark that it is not restrictive to suppose that y is a limit pointof E, otherwise (for suitable large n and t) E ∩ Cn and E ∩ Bd(y, t1/(2−Q)) wouldbe contained in {y} so that (i) through (iv) all hold (recall that a set is L-thin in thecomplement of its limit points).

The proof is split into four steps: we prove one by one the equivalences (ii) ⇔(iii), (i) ⇔ (iii), (ii) ⇔ (iv).

(ii) ⇔ (iii): Since E ∩ Cn is bounded, R1E∩Cn

is an L-potential (see Theo-

rem 11.8.1), say Γ ∗ μn (where μn is supported in Cn for R1E∩Cn

is L-harmonic

in the complement of E ∩ Cn). Let us set

u(x) :=∑

n≥1

Γ ∗ μn(x).

From Theorem 11.8.1 (page 519) we also infer C∗(E ∩Cn) = μn(G). Let us assume(ii). Then (iii) holds too, for we have

u(y) =∑

n≥1

Γ ∗ μn(y) =∑

n≥1

∫

Cn

Γ (z−1 ◦ y) dμn(z)

≤∑

n≥1

αn+1 μn(G) =∑

n≥1

αn+1 C∗(E ∩ Cn) < ∞. (12.28)

Vice versa, let us assume (iii). Then (ii) holds too, for we have


∑

n≥1

αn+1 C∗(E ∩ Cn) =∑

n≥1

αn+1 μn(Cn)

≤∑

n≥1

∫

Cn

Γ (z−1 ◦ y) dμn(z) =∑

n≥1

Γ ∗ μn(y) < ∞. (12.29)

(Proving the inequalities “≤” in (12.28) and (12.29) we used the fact that, by defini-tion, αn ≤ Γ (z−1 ◦ y) ≤ αn+1 on Cn.)

(i) ⇔ (iii): First, suppose that (iii) holds. Then10 there exists a sequence bn

diverging to ∞ such that∑

n bn R1E∩Cn

(y) < ∞. Set

v(x) :=∑

n

bn R1E∩Cn

(x).

By Theorem 11.6.5 (page 513), there exists an L-polar set Pn ⊆ E ∩ Cn such that

R1E∩Cn

(x) = 1 ∀ x ∈ (E ∩ Cn) \ Pn. (12.30)

Set A = ⋃n An, P = ⋃

n Pn. From (12.30) we infer v(x) = ∑n bn = ∞ for every

x ∈ (E ∩ A) \ P . As a consequence (A being a neighborhood of y) E \ P is L-thinat y. This ensures that E is also L-thin at y (see Corollary 12.2.6).

Now, suppose that (i) holds. Then, by Theorem 12.2.5, there exists w ∈ S +(G)

such that

w(y) < limEx→y

w(x) = ∞. (12.31)

Set E0 := ⋃∞k=k0

E ∩ C2k where k0 is chosen so that C2k0 ⊂ Bd(y, 12 ). Since E0 is

bounded, by Theorem 11.8.1 (page 519), there exists a measure ν0 such that RwE0

=Γ ∗ ν0. By Theorem 11.6.5 (page 513), there exists an L-polar set F0 ⊆ E0 ∩ ∂E0such that Rw

E0= Rw

E0on G \ F0. This ensures that Γ ∗ ν0 = w on E0 \ F0, whence

(from (12.31) and E0 \ F0 ⊆ E) we derive lim(E0\F0)x→y Γ ∗ ν0(x) = ∞, i.e.

lim(E0\F0)x→y

∫Γ (x−1 ◦ z) dν0(z) = ∞. (12.32)

Furthermore, it holds∫

Γ (y−1 ◦ z) dν0(z) = Γ ∗ ν0(y) = RwE0

(y) ≤ w(y) < ∞. (12.33)

As a consequence, by applying Lemma 12.4.1 we get∫

G\(Cn−1∪Cn∪Cn+1)

Γ (x−1 ◦ z) dν0(z) ≤ c w(y) < ∞ ∀ x ∈ Cn, n ≥ 2. (12.34)

10 Here, we use the following simple analysis lemma: If∑

n an < ∞ with an ≥ 0, thenthere exists a sequence {bn} such that bn ↑ ∞ and

∑n an bn < ∞. Indeed, setting An =∑

j≥n aj , we have An ↓ 0. Choose {nk}k such that Ank ≤ 4−k and define bn = k

whenever n = nk, nk + 1, . . . , nk+1 − 1.


Now, (12.32) and (12.34) may agree only if (for a suitable diverging sequence {βn})∫

C2n

Γ (x−1 ◦ z) dν0(z) ≥ βn ∀ x ∈ C2n ∩ (E0 \ F0), n ≥ 1.

In particular, also setting μn := (ν0)|C2n, this gives

(Γ ∗ μn)(x) ≥ 1 ∀ x ∈ (E ∩ C2n) \ F0, n � 1.

By the definition of L-balayage (being Γ ∗ μn ∈ S +(G)), this proves

Γ ∗ μn ≥ R1(E∩C2n)\F0

,

i.e. thanks to Proposition 11.8.3 (page 525),

Γ ∗ (ν0|C2n) ≥ R1

E∩C2n, n � 1. (12.35)

By arguing analogously with

E1 :=⋃

k≥k0

E ∩ C2k+1, RwE1

= Γ ∗ ν1,

we have

Γ ∗ (ν1|C2n+1) ≥ R1E∩C2n+1

, n � 1. (12.36)

From (12.33), (12.35) and (12.36) we get (iii). Indeed, one has∑

n�1

R1Cn

(y) =∑

n even

+∑

n odd

≤∑

n�1

Γ ∗ (ν0|C2n)(y) +

∑

n�1

Γ ∗ (ν1|C2n+1)(y)

≤ (Γ ∗ ν0)(y) + (Γ ∗ ν1)(y) ≤ 2 w(y) < ∞.

(ii) ⇔ (iv): First, we show the implication “⇒”,

∫ ∞

α

C∗({x ∈ E : Γ (y−1 ◦ x) ≥ t})

dt =∞∑

n=1

∫ αn+1

αn

[· · ·]

≤∞∑

n=1

C∗({x ∈ E : Γ (y−1 ◦ x) ≥ αn}) ∫ αn+1

αn

dt

(from E ∩ { Γ (y−1 ◦ x) ≥ αn} ⊆ ⋃∞m=n E ∩ {αm ≤ Γ (y−1 ◦ x) ≤ αm+1}

and the sub-additivity of the exterior L-capacity C∗)

≤∞∑

n=1

∞∑

m=n

C∗(E ∩ Cm)(αn+1 − αn) =∞∑

m=1

C∗(E ∩ Cm)

m∑

n=1

(αn+1 − αn)

=∞∑

m=1

C∗(E ∩ Cm) (αm+1 − α) ≤∞∑

m=1

C∗(E ∩ Cm) αm+1 < ∞.


Finally, we show the implication “⇐”,

∞∑

n=2

C∗(E ∩ Cn) αn ≤∞∑

n=2

αn C∗({x ∈ E : Γ (y−1 ◦ x) ≥ αn})

(use the monotonicity of C∗ and αn = αα−1

∫ αn

αn−1 dt)

≤ α

α − 1

∞∑

n=2

∫ αn

αn−1C∗({x ∈ E : Γ (y−1 ◦ x) ≥ t

})dt

= α

α − 1

∫ ∞

α

C∗({x ∈ E : Γ (y−1 ◦ x) ≥ t})

dt < ∞.

This completes the proof. ��Collecting together the L-regularity criterion of Theorem 12.3.7 and Wiener’s

criterion for L of Theorem 12.4.2, we derive the following remarkable result (again,α > 1 is a constant as in Lemma 12.4.1):

Theorem 12.4.3 (Wiener’s regularity test for L). Let C∗ denote the exterior L-capacity relative to G. Let Ω be an open and connected subset of G and y ∈ ∂Ω .Then the following statements are equivalent:

(i) y is an L-regular point for Ω;(ii) G \ Ω is not L-thin at y;

(iii) It holds∑∞

n=1 αn C∗(Cn \ Ω) = ∞;(iv) It holds

∑∞n=1 R1

Cn\Ω(y) = ∞;

(v) It holds∫ ∞α

C∗({x /∈ Ω : Γ (y−1 ◦ x) ≥ t}) dt = ∞.

Here, Cn is as in (12.16) at the beginning of the section.

Bibliographical Notes. For the topics presented in this chapter, we were inspired bythe exposition of the same subjects in [Helm69, Chapter 10] and [AG01, Chapter 7].

The Wiener test in the context of the nilpotent Lie groups was proved by H. Hue-ber [Hu85] and then generalized for Hörmander vector fields by P. Negrini andV. Scornazzani [NS87].

For further results on the Wiener test, see, e.g. [HH87,Ne88] and on the finetopology see, e.g. [BR67,CC72].


Ex. 1) Prove that a set E is L-thin at x ∈ G if and only if E \ {x} is L-thin at x

(prove this by the very definition of L-thinness).


Ex. 2) Prove that any non-empty L-fine open set contains infinitely many distinctpoints. (Hint: If u is L-superharmonic, then u(x) = lim infy→x u(y).)

Ex. 3) Prove that the L-fine topology is not locally compact.(Hint: Let F be the L-fine closure of an L-fine neighborhood of a point.Then, by Exercise 1, F contains an infinite set Z = {xn}n∈N. By Proposi-tion 12.1.3, no subsequence of Z can converge.)

Ex. 4) A set is L-finely compact if and only if it is finite.(Hint: See the hint for Exercise 2.)

Ex. 5) Prove that E is L-thin at x if and only if there exists a bounded functionv ∈ S(G) and a neighborhood U of x such that v(x) = 1 and v ≤ −1on U ∩ (E \ {x}). (Hint: By Theorem 12.2.5, let u ∈ S +

(G) be such thatu(x) < limEy→x u(y) = ∞. Replace u by w = min{u, u(x)+ 1} and takeU := {y : u(y) > u(x) + 1/2} and v := 1 + 4(u(x) − u).)

Ex. 6) (The exterior L-cone regularity condition). In the following statement,we say that C is a (truncated) L-cone if

δλ(x) ∈ C ∀ x ∈ C, ∀ λ ∈]0, 1[,where {δλ}λ>0 is the family of dilations of the Carnot group G =(RN, ◦, δλ). As usual, L is a sub-Laplacian on L.Let Ω ⊆ R

N be a bounded open set, and let y ∈ ∂Ω . Assume Ω has at y theproperty of the exterior L-cone, i.e. there exists a bounded open (truncated)L-cone C such that

RN \ Ω ⊇ y ◦ C.

Then y is an L-regular point for Ω .(Hint: We follow the notation of Theorem 12.4.3. Use the results of Exer-cise 2 at the end of Chapter 11 (page 533) to show that

C∗(Cn \ Ω) ≥ c0 (1/α)n.

This follows from the fact that

Cn \ Ω ⊇ δrn{ξ ∈ C : r ≤ d(ξ) ≤ 1}, with r = α1/(2−Q).

Thus,∑

n αnC∗(Cn \ Ω) ≥ ∑n coα

n α−n = ∞, and the L-regularity of y

follows from Wiener’s test in Theorem 12.4.3.)Ex. 7) In the case of the classical Laplace operator, give a proof of the exterior

L-ball regularity condition in Ex. 9 at the end of Chapter 7 by making useof Wiener’s criterion and geometrical arguments (instead of Bouligand’stheorem).Do the same in the case of the Heisenberg–Weyl group on R

3.Ex. 8) Let Ω ⊆ R

N be a bounded open set, and let y ∈ ∂Ω . Assume there existsR > 0 and γ ∈ ]0, 1[ such that

|Bd(y, r) \ Ω | ≥ γ rQ ∀ r ∈ ]0, R[.Then y is an L-regular point for Ω . Here | · | stands for the Lebesgue mea-sure.


Ex. 9) Let Ω ⊆ RN be a bounded open set, and let y ∈ ∂Ω . For every r > 0, let

us setVr := R1

Ω ′r (y), Ω ′

r (y) = Bd(y, r) \ Ω.

Then y is not an L-regular point for Ω if and only if Vr(y) → 0 as r → 0.Ex. 10) Let A, B be fixed constants with 0 < B < A < 1/2. Prove that

A ≤ λ ≤ 1/2λ − B ≤ μ ≤ λ + B

�⇒ λ

μ≤ A

A − B.

This proves (12.27) in the proof of Lemma 12.4.1.Ex. 11) Prove that if A is L-thin at x then, for every B ⊆ A, B is L-thin at x.

Vice versa, if A is not L-thin at x then, for every B ⊇ A, B is not L-thinat x. Derive the following facts:Let Ω, Ω be open, connected and bounded subsets of G. Suppose also thatx0 ∈ ∂Ω ∩ ∂Ω . If x0 is L-regular for Ω and Ω ⊆ Ω , then x0 is L-regularfor Ω . Vice versa, if x0 is L-irregular for Ω and Ω ⊇ Ω , then x0 is L-irregular for Ω .Derive this both from Theorem 12.3.7 and from Bouligand’s Theorem 6.10.4(page 371).

Ex. 12) Let 0 < r1 < r2 < ∞. Let x0 ∈ G be fixed. Set Bi := Bd(x0, ri) (fori = 1, 2) and C := B2 \ B1. Prove that the annulus C is an L-regular set byconstructing explicit L-barrier functions (see Definition 6.10.3, page 371).(Hint: w1 := r2−Q − d2−Q(x−1

0 ◦ ·) is an L-barrier function for the pointsin ∂B1, whereas w2 := −r2−Q + d2−Q(x−1

0 ◦ ·) is an L-barrier function forthe points in ∂B2.)Note that, by Theorem 12.3.7, the L-regularity of a point y ∈ ∂C impliesthat (actually, is equivalent to) G \ C = B1 ∪ (G \ B2) is not L-thin at y. Inturn, this is equivalent to

y ∈ DerL-fine(B1 ∪ (G \ B2)

).

Since Der(A∪B) = Der(A)∪Der(B) in any topological space (prove this),then the above argument shows that ∂C ⊆ DerL-fine(B1) ∪ DerL-fine(G \B2). Recalling that DerL-fine(A) ⊆ DerEucl.(A), this easily proves thatDerL-fine(B1) = B1, DerL-fine(G \ B2) = G \ B2. Is it true that we alsohave DerL-fine(B1) = B1 and DerL-fine(G \ B2) = G \ B2?

13

d-Hausdorff Measure and L-capacity

The aim of this chapter is to provide some estimate of the “smallness” of L-polarsets in terms of the Hausdorff measure naturally related to the quasi-distance definedby an L-gauge d on G.

To this end, we first need the relevant definitions of d-Hausdorff measure andd-Hausdorff dimension, besides several preliminary results. One of the main resultsis contained in Theorem 13.2.5.

Finally, in Section 13.3, we recall some results contained in the remarkable paper[BRSC03], in order to emphasize the new phenomena which can occur when theHausdorff dimension is considered in the sub-elliptic context of Carnot groups (forinstance, the Heisenberg–Weyl group H

1).As usual, throughout the section G = (RN, ◦, δλ) is a homogeneous Carnot

group and L is a sub-Laplacian on G. Moreover, Γ is the fundamental solution for Land d is any L-gauge function (i.e. a positive constant times Γ 1/(2−Q)). Q denotes,as usual, the homogeneous dimension of G. We recall that d is a quasi-distance onG satisfying the pseudo-triangle inequality

d(x−1 ◦ y) ≤ c (d(x) + d(y)) for all x, y ∈ G. (13.1)

The constant c≥1 will always denote that appearing in (13.1). As usual, the d-ballsrelated to d are denoted by Bd(x, r).

13.1 d-Hausdorff Measure and Dimension

Definition 13.1.1 (The d-Hausdorff measure). An increasing functionφ : (0,∞) → (0,∞] vanishing as t → 0+ will be referred to as measure func-tion. Given any set E ⊆ G and any ρ > 0, we set

M(ρ)φ (E) := inf

{∑

k

φ(rk) : E ⊆⋃

k

Bd(xk, rk), rk < ρ ∀ k ∈ N

}, (13.2)

558 13 d-Hausdorff Measure and L-capacity

where the infimum is taken over all possible coverings E by a (finite or) countablefamily of d-balls {Bd(xk, rk)}k∈N such that the rays rk are strictly less than ρ (anysuch covering will be referred to as ρ-covering of E).

Since the map ]0,∞] ρ �→ M(ρ)φ is decreasing, it is well posed,

mφ(E) := limρ→0+ M

(ρ)φ (E) = sup

ρ>0M

(ρ)φ (E). (13.3)

In the sequel, mφ(E) will be referred to as the d-Hausdorff φ-measure. When φ(t) =tα ( for α > 0), we simply write M

(ρ)

(α) instead of M(ρ)tα and m(α) instead of mtα . In

the sequel, m(α)(E) will be referred to as the d-Hausdorff (α)-measure of the set E.

Remark 13.1.2. With the above notation, it can be proved that M(ρ)φ and mφ are outer

measures on G. Properly restricted to their related measurable sets, they are Borelregular measures. We leave the proof as an exercise.

We begin with some lemmas.

Lemma 13.1.3 (The d-Hausdorff dimension). For every bounded set E ⊂ G, thereexists a number α(E) ∈ [0,Q] (where Q is the homogeneous dimension of G) suchthat

m(α)(E) = ∞ for all α < α(E), m(α)(E) = 0 for all α > α(E). (13.4)

We thus have

α(E) = inf{α > 0 : m(α)(E) = 0}. (13.5)

We call α(E) the d-Hausdorff dimension of E.

Proof. Let α > Q be fixed and take any ε ∈ ]0, 1[. Let Ω be any bounded openset containing E. Then there exists a countable collection of disjoint d-balls Bk :=Bd(xk,

rk4 ) (with rk

2 < ε for every k) and a set N such that Ω = N ∪⋃k Bk and such

that⋃

k Bd(xk, c rk) ⊇ Ω . This gives (for a suitable structural constant cQ)

M(ε)(α)(E) ≤

∑

k

(c rk)α ≤ εα−Q

∑

k

(c rk)Q = (4c)Qεα−Q

∑

k

(rk

4

)Q

= cQεα−Q∑

k

meas(Bk) ≤ cQεα−Q meas(Ω) → 0 as ε → 0+.

This proves that m(α)(E) = 0 for every α > Q. If we define α(E) as in (13.5), wethus get α(E) ∈ [0,Q].

Let now α > α(E) and ε > 0. Hence, there exist a suitable β ∈ ]α(E), α[ andan ε-covering {Bd(xk, rk)}k of E such that

∑k r

βk < 1. Consequently,

M(ε)(α)(E) ≤

∑

k

rαk =

∑

k

rα−βk r

βk ≤ εα−β → 0 as ε → 0+.

13.1 d-Hausdorff Measure and Dimension 559

This proves the second part of (13.4). Let now α ∈ ]0, α(E)[. By the definition ofα(E), there exists β ∈ ]α, α(E)[ with m(β)(E) > 0. If {Bd(xk, rk)}k is an ε-covering

of E, then∑

k rαk = ∑

k rα−βk r

βk ≥ εα−β M

(ε)(β)(E), whence

M(ε)(α)(E) ≥ εα−β M

(ε)(β)(E) → ∞ as ε → 0+.

This proves the first part of (13.4), completing the proof of the lemma. ��Example 13.1.4. The d-Hausdorff dimension of G is Q. Indeed, there exists a posi-tive constant c such that

m(Q)(E) = c HN(E) for every E ⊆ G,

where HN stands for the usual (Euclidean) Hausdorff N -dimensional measure onR

N ≡ G. The proof is left as an exercise.

Remark 13.1.5. It is easy to see that m(α) is invariant under left-translation and ho-mogeneous of degree α with respect to the dilation of G. In other words, for everyx ∈ G and every λ > 0, we have

m(α)(x ◦ E) = m(α)(E), m(α)(δλ(E)) = λα m(α)(E),

where E is any subset of G. The proof is left as an exercise.

We next aim to prove Theorem 13.1.7 below. In order to do this, we need a lemmahaving an interest in its own.

Lemma 13.1.6. Suppose K � G is not an L-polar set. Then there exists a Radonmeasure μ supported in K with μ(K) > 0 such that Γ ∗ μ ∈ C(G, R).

Proof. By Proposition 11.5.11 (page 507), since K is not L-polar, we have

C(K) = μK(K) > 0.

Since Γ ∗ μK ≤ 1, by Lemma 11.3.3 (page 495) there exists C � K such that

μK(K \ C) < μK(K)/2 and Γ ∗ (μK |C) ∈ C(G, R).

We claim that μ := (μK)|C fulfills the assertion of the lemma. Indeed,

μK(C) = μK(K) − μK(K \ C) > μK(K)/2 > 0.

This ends the proof. ��We are now ready to give the proof of the following result.

Theorem 13.1.7 (d-Hausdorff measure and L-polarity. I). Let E ⊂ G be abounded L-capacitable set satisfying m(Q−2)(E) < ∞ (as usual, Q is the homo-geneous dimension of G). Then E is L-polar.


For a partial converse of Theorem 13.1.7, see also Theorem 13.2.5.

Proof. We argue by contradiction. Let E be a non-L-polar capacitable set, i.e.C(E) > 0. This also gives C∗(E) > 0, i.e. there exists K � E with C(K) > 0.By the sub-additivity of the L-capacity, there exists an open d-ball B1 of radius 1such that C(K ∩ B1) > 0. Then, it is not restrictive to suppose that K is containedin such a d-ball B1. From C(K) > 0 and Lemma 13.1.6 we derive the existence of aRadon measure μ supported in K such that μ(K) > 0 and u := Γ ∗ μ ∈ C(G, R).For any 0 < ρ < 1 and x ∈ G, we set

uρ(x) :=∫

G

Γ (y−1 ◦ x) max

{0, 1 − d(y−1 ◦ x)

2ρ

}dμ(y).

For any fixed x0 ∈ G, we have

uρ(x0) ≤ lim infx→x0

uρ(x) ≤ lim supx→x0

uρ(x)

≤ lim supx→x0

u(x) + lim supx→x0

(uρ(x) − u(x))

= u(x0) − lim infx→x0

(u(x) − uρ(x))

= u(x0) − lim infx→x0

∫

G

Γ (y−1 ◦ x)

(1 − max

{0, 1 − d(y−1 ◦ x)

2ρ

})dμ(y)

≤ u(x0) −∫

G

lim infx→x0

[· · ·]

= u(x0) −∫

G

Γ (y−1 ◦ x0)

(1 − max

{0, 1 − d(y−1 ◦ x0)

2ρ

})dμ(y)

= uρ(x0).

In the first and the last inequalities, we used Fatou’s lemma; in the first equality, weused the continuity of u. This proves that uρ is continuous on G.

Being Γ ∗ μ finite-valued, we have μ({x}) = 0 for every x ∈ G. As a conse-quence, it holds (by dominated convergence)

limρ→0+ uρ(x) = lim

ρ→0+

∫

G\{x}[· · ·] dμ

=∫

G\{x}Γ (y−1 ◦ x) lim

ρ→0+ max

{0, 1 − d(y−1 ◦ x)

2ρ

}dμ(y) = 0.

The convergence is also monotone, for uρ′(x) ≤ uρ(x) whenever 0 < ρ′ < ρ.By Dini’s theorem, the convergence is uniform on compact sets. Consequently, thereexists a sequence ρn ↓ 0 such that uρn < 2−n on B1 for every n ∈ N. Let us introducethe function φ : [0,∞[→ R,

φ(t) := 1

2t2−Q

∞∑

n=1

χ[0,ρn](t).

13.2 d-Hausdorff Measure and L-capacity 561

Since χ[0,ρn](d(y−1 ◦ x)) = χBd(x,ρn)(y) (and by monotone convergence), we have

∫

K

φ(d(y−1 ◦ x)) dμ(y) =∞∑

n=1

1

2

∫

G

Γ (y−1 ◦ x) χBd(x,ρn)(y) dμ(y)

≤∞∑

n=1

1

2

∫

G

Γ (y−1 ◦ x) max

{0, 2 − d(y−1 ◦ x)

ρn

}dμ(y)

=∞∑

n=1

uρn(x).

Hence, ∫

K

φ(d(y−1 ◦ x)) dμ(y) ≤ 1 for every x ∈ B1.

Let now n be large enough so that

ρn < sup{d(a−1 ◦ b) : a ∈ K, b ∈ G \ B1}. (13.6)

Let us consider a ρn-covering of K , say {Bd(xk, rk)}k , consisting of d-balls all inter-secting K . By (13.6), it is easily seen that xk ∈ B1 for every k ∈ N. This gives

1 ≥∫

K

φ(d(y−1 ◦ xk)) dμ(y) ≥∫

Bd(xk,rk)

[· · ·]

(φ is decreasing) ≥∫

Bd(xk,rk)

φ(rk) dμ(y) = φ(rk) μ(Bd(xk, rk))

≥ n

2(rk)

2−Qμ(Bd(xk, rk)). (13.7)

The last inequality follows from rk < ρn and the very definition of φ. From (13.7)we derive the second inequality in the following chain of inequalities:

μ(K) ≤∑

k

μ(Bd(xk, rk)) ≤ 2

n

∑

k

(rk)Q−2.

This gives (from E ⊇ K and again rk < ρn)

M(ρn)

(Q−2)(E) ≥ M(ρn)

(Q−2)(K) ≥∑

k

(rk)Q−2 ≥ n

2μ(K) → ∞ as n → ∞.

This gives m(Q−2)(E) = ∞, contradicting the assertion of the theorem. ��

13.2 d-Hausdorff Measure and L-capacity

The aim of this section is to prove Theorem 13.2.5 (page 568). In order to do it, weneed some lemmas. The first one is a simple result from the real analysis.


Lemma 13.2.1. Let μ be a Radon measure on RN . Let g : (0, R] → [0,∞) be a

continuous decreasing function. Finally, let f : RN → [0,∞) be a μ-measurable

function. Set

n(t) :=∫

f (y)<t

dμ(y). (13.8)

Suppose n(t) is finite-valued on (0,∞). Then g is integrable in the generalizedRiemann–Stieltjes sense on (0, R] with respect to n(t) if and only if g ◦ f is μ-integrable on the set {y ∈ R

N : 0 < f (y) < R}. Moreover,

∫

0<f (y)<R

g(f (y)) dμ(y) =∫ R

0g(t) dn(t) (13.9)

whenever one of the integrals is finite.

Proof. Since t �→ n(t) is decreasing, it is of bounded variation on every compactsub-interval of (0, R]. Hence g(t) (which is continuous) is dn(t)-integrable on everycompact sub-interval of (0, R].

Suppose first that g is integrable in the generalized Riemann–Stieltjes sense on(0, R] with respect to n(t). If ε > 0, there exists δ = δ(ε) > 0 such that

∣∣∣∣∫ R

0g(t) dn(t) −

∫ R

δ

g(t) dn(t)

∣∣∣∣ <ε

2.

We can assume δ(ε) < ε. Moreover, there exists a partition of [δ, R], say δ = t0 <

t1 < · · · < tp = R, such that

∣∣∣∣∫ R

δ

g(t) dn(t) −p−1∑

k=0

g(tk) (n(tk+1) − n(tk))

∣∣∣∣ <ε

2. (13.10)

We remark that n(tk+1)−n(tk) = ∫tk≤f (y)<tk+1

dμ(y). Moreover, from tk ≤ f (y) <

tk+1 it follows g(tk) ≥ g(f (y)) > g(tk+1), whence

p−1∑

k=0

g(tk)

∫

tk≤f (y)<tk+1

dμ(y) ≥p−1∑

k=0

∫

tk≤f (y)<tk+1

g(f (y)) dμ(y)

=∫

δ(ε)≤f (y)<R

g(f (y)) dμ(y).

As a consequence, it holds

∫ R

0g(t) dn(t) >

∫ R

δ(ε)

g(t) dn(t) − ε

2≥

p−1∑

k=0

g(tk) (n(tk+1) − n(tk)) − ε

≥∫

δ(ε)≤f (y)<R

g(f (y)) dμ(y) − ε.


Making use of (13.10), we have∣∣∣∣∫

δ(ε)≤f (y)<R

g(f (y)) dμ(y) −∫ R

0g(t) dn(t)

∣∣∣∣ < ε. (13.11)

Being g ≥ 0 and δ(ε) < ε, from Beppo Levi’s theorem we have∫

δ(ε)≤f (y)<R

g(f (y)) dμ(y) ↗∫

0<f (y)<R

g(f (y)) dμ(y) if ε → 0+.

Hence (13.11) proves the assertion.Let us suppose that g(f (y)) is μ-integrable on {y ∈ R

N : 0 < f (y) < R}.Let δ ∈ ]0, R[ be fixed and ε > 0. Since g is continuous on [δ, R], there existsσ = σ(δ, ε) such that | g(x′) − g(x′′)| < ε/n(R)

for every x′, x′′ ∈ [δ, R] with |x′ − x′′| < σ . Let us now fix any arbitrary partitionδ = t0 < t1 < · · · < tp = R of [δ, R] such that tk+1 − tk < σ , and for k =0, . . . , p − 1, let us choose τk ∈ [tk, tk+1]. We have

p−1∑

k=0

g(τk)(n(tk+1) − n(tk))

≥ −ε/n(R) ·p−1∑

k=0

(n(tk+1) − n(tk)) +p−1∑

k=0

g(tk) (n(tk+1) − n(tk))

≥ −ε +∫

δ≤f (y)<R

g(f (y)) dμ(y).

Analogously,

p−1∑

k=0

g(τk)(n(tk+1) − n(tk)) ≤ ε +∫

δ≤f (y)<R

g(f (y)) dμ(y).

This proves that

∣∣∣∣∫

δ≤f (y)<R

g(f (y)) dμ(y) −p−1∑

k=0

g(τk) (n(tk+1) − n(tk))

∣∣∣∣ < ε

for every partition {tk}k of [δ, R] with tk+1 − tk < σ and any τk ∈ [tk, tk+1]. In otherwords, g(t) is dn(t)-integrable on [δ, R] and

∫ R

δ

g(t) dn(t) =∫

δ≤f (y)<R

g(f (y)) dμ(y).

Being δ arbitrary and taking into account the hypothesis

limδ→0+

∫

δ≤f (y)<R

g(f (y)) dμ(y) =∫

0<f (y)<R

g(f (y)) dμ(y),

we derive that g is integrable in the generalized sense of Riemann–Stieltjes on (0, R]w.r.t. n(t) and (13.9) holds. ��


If we take f (y) := d(y−1 ◦ z) and g(t) := t2−Q in (13.9), we obtain

∫

0<d(y−1◦z)<1d2−Q(y−1 ◦ z) dμ(y) =

∫ 1

0t2−Q dmz(t), (13.12)

where mz(t) = μ(Bd(z, t)).

Lemma 13.2.2. Let μ be any compactly supported Radon measure on G. Let also φ

be a strictly positive measure function such that

I (φ,Q) :=∫ 1

0

φ(t)

tQ−1dt < ∞. (13.13)

Then there exists a constant C > 0 (precisely, it holds C = (Q − 2) I (φ,Q)) suchthat

(Γ ∗ μ)(z) ≤ μ(G) + C sup0<r<1

μ(Bd(z, r))

φ(r)∀ z ∈ G. (13.14)

Proof. We split the proof in three steps.(I): z ∈ G is such that (Γ ∗ μ)(z) < ∞. This case ensures that μ({z}) = 0. Let

us setmz(r) := μ(Bd(z, r)), r > 0.

First of all, we claim that

limε→0+ ε2−Q mz(ε) = 0. (13.15)

Indeed, from μ({z}) = 0 we have mz(0+) = 0, so that

ε2−Q mz(ε) = ε2−Q (mz(ε) − mz(0+)) =

∫ ε

0ε2−Q dmz(t) ≤

∫ ε

0t2−Q dmz(t),

where the far right-hand side vanishes as ε → 0, for∫ 1

0 t2−Q dmz(t) < ∞. Indeed,making use of (13.12), we have

∫ 1

0t2−Q dmz(t) =

∫

0<d(y−1◦z)<1d2−Q(y−1 ◦ z) dμ(y) ≤ (Γ ∗ μ)(z),

where the far right-hand side is finite, by our assumption. Then

(Γ ∗ μ)(z) =( ∫

{y=0}+

∫

0<d(y−1◦z)<1+

∫

d(y−1◦z)≥1

)d2−Q(y−1 ◦ z) dμ(y)

≤ 0 +∫

0<d(y−1◦z)<1d2−Q(y−1 ◦ z) dμ(y) + μ(G \ Bd(z, 1))

(see (13.12)) =∫ 1

0t2−Q dmz(t) + μ(G \ Bd(z, 1))


(by parts) = limε→0+

(mz(1) − ε2−Q mz(ε) −

∫ 1

ε

(2 − Q)t1−Q mz(t) dt

+ μ(G \ Bd(z, 1))

)

(see (13.15)) = mz(1) + (Q − 2)

∫ 1

0t1−Q mz(t) dt + μ(G \ Bd(z, 1))

= μ(G) + (Q − 2)

∫ 1

0t1−Q μ(Bd(z, t))

φ(t)φ(t) dt

≤ μ(G) + (Q − 2) I (φ,Q) sup0<t<1

μ(Bd(z, t))

φ(t),

where I (φ,Q) is as in (13.13). This proves (13.14).(II): z ∈ G is such that (Γ ∗ μ)(z) = ∞ and μ({z}) = 0. Let R � 1 be such

that supp(μ) ⊂ Bd(0, R). We have (also using μ({z}) = 0)

∞ = (Γ ∗ μ)(z) =∫

0<d(y−1◦z)<R

d2−Q(y−1 ◦ z) dμ(y) =∫ R

0t2−Q dmz(t).

Since μ is a Radon measure, this also ensures that∫ 1

0t2−Q dmz(t) = ∞. (13.16)

(13.14) is proved in case (II), if we show that

sup0<r<1

μ(Bd(z, r))

φ(r)= ∞.

Suppose by contradiction that there exists M > 0 such that

mz(r) = μ(Bd(z, r)) ≤ M φ(r) for all r ∈ ]0, 1[. (13.17)

This gives (see also (13.16) and (13.14))

∞ =∫ 1

0t2−Q dmz(t) = lim

ε→0+

∫ 1

ε

t2−Q dmz(t)

= limε→0+

(mz(1) − ε2−Q mz(ε) + (Q − 2)

∫ 1

ε

t1−Q mz(t) dt

)

(see (13.17)) ≤ mz(1) + (Q − 2)M

∫ 1

0t1−Q φ(t) dt < ∞,

a contradiction.(III): z ∈ G is such that (Γ ∗ μ)(z) = ∞ and μ({z}) > 0. From φ(0+) = 0 we

derive

sup0<r<1

μ(Bd(z, r))

φ(r)≥ μ(Bd(z, 1/n))

φ(1/n)≥ μ({z})

φ(1/n)−→ ∞

as n → ∞, so that both sides of (13.14) equal ∞. ��


We have the following general covering lemma.

Lemma 13.2.3 (Covering). Let D = {Bd(xα, rα)}α∈I be any collection of d-ballswith supα∈I rα < ∞. Let E ⊂ G be a bounded set covered by D. Then there exists adisjoint at most countable sub-collection {Bd(xαk

, rαk)}k∈N of D such that

E ⊆⋃

k∈N

Bd(xαk, 3c2rαk

).

Here, c is the constant appearing in the pseudo-triangle inequality (13.1).

Proof. We ignore the d-balls non-intersecting E, and we still denote by I the result-ing sub-family of indices. Let also set Bα := Bd(xα, rα). Let α1 ∈ I be such thatrα1 ≥ 1

2 supα∈I rα . Let α2 ∈ I (if it exists, otherwise the process ends) be such that

Bα2 ∩ Bα1 = ∅, rα2 ≥ 1

2sup

{rα : Bα ∩ Bα1 = ∅}

.

Proceeding by induction, chosen α1, . . . , αk , we let αk+1 ∈ I (if it exists, otherwisethe process ends) be such that

Bαk+1 ∩ Bαj= ∅ for every j = 1, . . . , k,

(13.18)rαk+1 ≥ 1

2sup{rα : Bα ∩ Bαj

= ∅ for every j = 1, . . . , k}.

Set A := {αk : k ∈ N}. We split the proof in two steps.(I): The selection process (13.18) is finite. Let α′ /∈ A. Since the selection process

is finite, we have Bα′ ∩ Bαj�= ∅ for some j . Consequently, a simple application of

the pseudo-triangle inequality (13.1) proves that Bα′ ⊆ Bd(xαj, 3 c2 rαj

), and theproof is complete.

(II): The selection process (13.18) is infinite. Since E ∩ Bα �= ∅ for every α ∈ I,the Bαk

’s are mutually disjoint and E is bounded, we claim that in this case

rαk−→ 0 as k → ∞. (13.19)

Suppose by contradiction that (13.19) is false. This means that there exist ε0 > 0 anda sequence {kj }j in N such that rαkj

> ε0 for every j ∈ N. Consequently, we have

∞ =∑

j∈N

meas(Bd(xαkj, ε0)) ≤

∑

j∈N

meas(Bαkj) = meas

( ⋃

j∈N

Bαkj

).

This is impossible since⋃

k Bαkis bounded. This last fact is proved as follows. Since

E is bounded, there exists H > 0 such that E ⊂ Bd(0,H); if ξ ∈ Bαk∩ E, we have

d(z) ≤ c(d(z, ξ) + d(ξ)) ≤ c2d(z, xk) + c2d(xk, ξ) + c d(ξ)

≤ 2c2 rαk+ c H ≤ 2c2 sup

αrα + c H < ∞

for every k and every z ∈ Bαk. The claimed (13.19) is proved.


If I = A, the proof is complete. Otherwise, let α′ /∈ A. If rαk+1 ≥ 12 rα′ for every

k ∈ N, then (thanks to (13.19)) the selection process is finite, which is not the case.Hence, there exists k (necessarily k �= 0, for rα1 ≥ 1

2 rα′ ) such that rαk+1 < 12 rα′ . Let

k0 be the least positive integer such that

rαk0+1 <1

2rα′ . (13.20)

Then there exists j ∈ {1, . . . , k0} such that

Bα′ ∩ Bαj�= ∅ (13.21)

(indeed, if it were Bα′ ∩ Bαj= ∅ for every j = 1, . . . , k0, we would have rαk0+1 ≥

12 rα′ by the very selection process (13.18), contradicting (13.20)). Then (13.21),together with rαj

≥ 12 rα′ (recall that k0 is the least index satisfying (13.20)), proves

Bα′ ⊆ Bd(xαj, 3c2 rαj

)

by a simple application of the pseudo-triangle inequality (13.1). This ends theproof. ��Lemma 13.2.4. Let E ⊂ G be a bounded L-polar set. Then, for every measurefunction φ such that ∫ 1

0t1−Q φ(t) dt < ∞, (13.22)

we have mφ(E) = 0.

Proof. From Corollary 11.2.5 (page 492), being E bounded, there exists μ ∈ M0such that Γ ∗ μ ≡ ∞ on E. If φ is as in the assertion of the present lemma, we set

φ1(t) := min{φ(3 c2 t), φ(1/2)

}.

Here c≥1 is the constant appearing in the pseudo-triangle inequality (13.1). It iseasily seen that

∫ 10 t1−Q φ1(t) dt is finite, thanks to (13.22). Fix a > 0 such that

μ(G)

a< φ(1/2). (13.23)

From Lemma 13.2.2 applied to the measure function φ1 we get

(Γ ∗ μ)(x) ≤ μ(G) + C(Q, φ1) sup0<r<1

μ(Bd(x, r))

φ1(r)

for any x ∈ G. If, in particular, we take x ∈ E (so that Γ ∗ μ(x) = ∞), beingμ(G) < ∞, we derive the existence of r(x) ∈]0, 1[ such that

μ(Bd(x, r(x)))

φ1(r(x))> a. (13.24)


By the covering Lemma 13.2.3 applied to the family{Bx := Bd(x, r(x)) : x ∈ E

},

there exists a sequence xk ∈ E such that the Bxk’s are mutually disjoint and

E ⊆⋃

k

Bd(xk, 3c2 r(xk)). (13.25)

This gives (using (13.24) in the first inequality below)

∑

k

φ1(r(xk)) <1

a

∑

k

μ(Bxk) = 1

aμ

( ⋃

k

Bxk

)≤ 1

aμ(G). (13.26)

Taking into account the choice (13.23) of a, from (13.26) it follows that no summandφ1(r(xk)) coincides with φ(1/2). By the very definition of φ1, this gives

φ1(r(xk)) = φ(3 c2 r(xk)) ∀ k ∈ N. (13.27)

This yields

M(3c2)φ (E) ≤

∑

k

φ(3c2 r(xk)) =∑

k

φ1(r(xk)) ≤ μ(G)/a. (13.28)

In the first inequality we used (13.25), in the second we exploited (13.27), in the thirdwe took into account (13.26).

Since we can take any a � 1 satisfying (13.23), we can let a tend to ∞ in (13.28)to derive that

M(3c2)φ (E) = 0.

From the fact that φ is increasing and positive it easily1 follows that M(ρ)φ (E) = 0

for any ρ > 0, whence mφ(E) = 0. The proof is complete. ��As a consequence of Lemma 13.2.4, we have a partial converse of Theo-

rem 13.1.7.

Theorem 13.2.5 (d-Hausdorff measure and L-polarity. II). Let E ⊂ G be abounded L-polar set. Then we have

m(α)(E) = 0 for every α > Q − 2. (13.29)

Hence, the d-Hausdorff dimension of a bounded L-polar set is ≤ Q − 2.

Proof. This follows from Lemma 13.2.4, by taking φ(t) = tα with α > Q − 2 andthe very definition of d-Hausdorff dimension (see Lemma 13.1.3). ��

Collecting together Theorems 13.1.7 and 13.2.5, we have the following result.

1 See Ex. 4 at the end of the chapter.

13.3 New Phenomena Concerning the d-Hausdorff Dimension 569

Remark 13.2.6. Let E be any bounded and L-capacitable subset of G, and letdimH (E) be the d-Hausdorff dimension of E. We have:

(i) If dimH (E) < Q − 2, then E is L-polar,(ii) If dimH (E) > Q − 2, then E is not L-polar,

(iii) If dimH (E) = Q − 2 and m(Q−2)(E) ∈ [0,∞[, then E is L-polar,(iv) If dimH (E) = Q − 2 and m(Q−2)(E) = ∞, we cannot conclude anything about

the L-polarity of E.

13.3 New Phenomena Concerning the d-Hausdorff Dimension:The Case of the Heisenberg Group

The aim of this section is to provide a brief overview (without any proof) of someremarkable recent results on the Hausdorff dimension in the Heisenberg group, byZ.M. Balogh, M. Rickly and F. Serra Cassano [BRSC03].

Let H1 be the Heisenberg–Weyl group on R

3. We denote its points by z =(x, y, t). The relevant dilation is δλ(z) = (λx, λy, λ2t), and the composition lawis given by

z ◦ z′ = (x + x′, y + y′, t + t ′ + 2(x′y − y′x)

).

The homogeneous norm d(z) = ((x2 + y2)2 + t2)1/4 is an L-gauge, where L is thecanonical sub-Laplacian on H

1. We denote by e the usual Euclidean norm on R3, i.e.

e(x, y, t) = (x2 + y2 + t2)1/2.

As it is expected, the Hausdorff measure and the Hausdorff dimension of a subsetof H

1 ≡ R3 w.r.t. d and e may be different. For example, the d-Hausdorff dimen-

sion of H1 is 4 (see Example 13.1.4), whereas its e-Hausdorff dimension is 3. Fur-

thermore, the d-Hausdorff dimension of a regular surface is 3, and the d-Hausdorffdimension of a regular curve can be 1 or 2 (see [Gro96,Str92]).

In the sequel of this section, we use the notation Hαe and H

βd to denote, respec-

tively, m(α) when the relevant gauge is the Euclidean norm e and m(β) when therelevant gauge is the above d .

We fix some notation. We define the following functions:

f, ϕ, g, γ : [0,∞) → [0,∞), where

⎧⎪⎪⎨

⎪⎪⎩

f (α) := min{2α, α + 1},ϕ(β) := max{ 1

2β, β − 1},g(β) := min{β, 1 + β/2},γ (α) := max{α, 2α − 2}.

(Note that f = ϕ−1 and g = γ −1.) The following theorem holds.

Theorem 13.3.1 (Dimension jump theorem, [BRSC03]). For every real numbersα, β > 0, we have

Hf (α)d � Hα

e � Hγ(α)

d ,

Hg(β)e � H

βd � Hϕ(β)

e .


(Here, μ � λ means that the measure μ is absolutely continuous with respect tothe measure λ.) For example, the above theorem (and the very definition of absolutelycontinuous measures) give

Hαe (E) = 0 �⇒ H

f (α)d (E) = 0,

Hαe (E) > 0 �⇒ H

γ(α)

d (E) > 0,

Hβd (E) = 0 �⇒ H

g(β)e (E) = 0,

Hβd (E) > 0 �⇒ Hϕ(β)

e > 0.

Together with Theorem 13.3.1, Balogh, Rickly and Serra Cassano proved the follow-ing theorem.

Theorem 13.3.2 (Sharpness of the dimension jump, [BRSC03]). The followingassertions hold:

(i) Given α ∈ (0, 3], there exists a compact set Eα ⊂ H1 such that

Hαe (Eα) < ∞ and H

f (α)d (Eα) > 0

or, equivalently, given β ∈ (0, 4], there exists a compact set Eβ ⊂ H1 such that

Hϕ(β)e (Eβ) < ∞ and H

βd (Eβ) > 0;

(ii)’ For β ∈ (0, 2) ∪ {4}, there exists a compact set Eβ ⊂ H1 such that

Hβd (Eβ) < ∞ and H

g(β)e (Eβ) > 0

or, equivalently, given α ∈ (0, 2)∪{3}, there exists a compact set Eα ⊂ H1 such

that

Hγ(α)

d (Eα) < ∞ and Hαe (Eα) > 0;

(ii)” For β ∈ [2, 4) and ε ∈ (0, 1), there exists a compact set Eβ,ε ⊂ H1 such that

Hβd (Eβ,ε) = 0 and H

g(β)−εe (Eβ,ε) > 0.

We explicitly describe some of the sets Eα’s in the previous theorem (see[BRSC03] for the details).

(i) Let 0 < s < 1. We consider the Cantor2 set Cs ⊂ [0, 1] of (Euclidean) Hausdorffdimension s.

2 We recall the construction of Cs . Let s ∈ (0, 1) be fixed and set σ := 2−1/s . Let k ∈ N∪{0}and j ∈ {1, . . . , 2k}. We define I s

k,jin the following way:

I s0,1 = [0, 1],

I s1,1 = [0, σ ], I s

1,2 = [1 − σ, 1],

13.3 New Phenomena Concerning the d-Hausdorff Dimension 571

(a) If α ∈ (0, 1], we let Eα = {0} × {0} × Cα (here C1 := [0, 1]). Obviously,Hα

e (Eα) is positive and finite and the same is true of H 2αd (Eα) = H

f (α)d (Eα)

for, roughly speaking, the distance induced by the gauge d on the t-axis coin-cides with the 1

2 -power of the Euclidean distance. Note that the vertical seg-ment E1 = {0}× {0}× [0, 1] has d-Hausdorff dimension 2, but e-Hausdorffdimension 1. Since it also has finite H 2

d -measure, by Theorem 13.1.7 we in-fer that E1 is L-polar, where L is the canonical sub-Laplacian of H

1 (recallthat here Q − 2 = 2).

(b) If α ∈ (1, 2), the set Eα = Cα−1 × {0} × [0, 1] (a Cantor set of verticalsegments) has the requisites in Theorem 13.3.2-(i).

(c) If α = 2, it suffices to take E2 = {x2 +y2 + t2 = 1} (see also [Pan82]). Notethat E2 has positive and finite 2-dimensional Euclidean Hausdorff measureand positive 3-dimensional d-Hausdorff measure.

(d) If α ∈ (2, 3), then the choice

Eα =⋃

r∈Cα−2

∂Bd(0, r)

(a Cantor set of d-spheres) has the requisites in Theorem 13.3.2-(i).(e) If α = 3, take E3 = Bd(0, 1). Note that E3 has positive and finite

3-dimensional Euclidean Hausdorff measure and the same is true of its4-dimensional d-Hausdorff measure.

(ii) If β ∈ (0, 1], the set Eβ = Cβ ×{0}×{0} (here C1 stands for [0, 1]) has positive

and finite Hβd - and H

βe -measures. If β = 4, it suffices to take E4 = Bd(0, 1),

as already discussed. The case β ∈ [2, 4) is more involved, and the reader isreferred directly to [BRSC03].

Bibliographical Notes. In the case of the classical Laplace operator, the relation-ship between the Hausdorff measure and capacity can be found, e.g. in [AG01,Section 5.9] (whose exposition we followed here) and [HK76, Section 5.4]. See[BRSC03] for recent results on the Hausdorff measure in the context of the Heisen-berg group. See also [Gro96,Str92].

I s2,1 = [0, σ 2], I s

2,2 = [σ − σ 2, σ ], I s2,3 = [1 − σ, 1 − σ + σ 2], I s

1,2 = [1 − σ 2, 1],...

if we are given I sk,j

= [ask,j

, bsk,j

] for j = 1, . . . , 2k , then

I sk+1,2j−1 = [as

k,j, as

k,j+ σk+1], I s

k+1,2j= [bs

k,j− σk+1, bs

k,j].

Now, we define Cs as

Cs :=∞⋂

k=0

2k⋃

j=1

Ik,j =:∞⋂

k=0

Csk.

Note that Csk+1 ⊂ Cs

k, so that Cs = limk→∞ Cs

kwith the obvious meaning. Moreover,

Csk

is the union of 2k disjoint closed intervals of length σk . It can be proved that Cs is acompact subset of [0, 1] with no interior points and such that 0 < Hs

e (Cs) < ∞.



Ex. 1) Consider the Heisenberg–Weyl group (H1, ◦) on R3. Let z0 = (x0, y0, t0) ∈

H1 be fixed, with t0 �= 0. Consider the following “segment of δλ-ray”

R := {δλ(z0) : λ ∈ [a, b]},where a < b are fixed positive real numbers. Finally, let ΔH1 be the canonicalsub-Laplacian on H

1. Prove the following facts:• If Γ is the fundamental solution for ΔH1 , the function

u(z) :=∫ b

a

Γ (δλ(z−10 ) ◦ z) dλ

belongs to S(H1) (w.r.t. ΔH1 ) and u(0) < ∞;• u ≡ ∞ on R;• Derive that R is a bounded ΔH1 -polar set. If d is a ΔH1 -gauge, what is

the d-Hausdorff dimension of R? (Use the remarks in Section 13.3.)Hint: u(0) = Γ (z0)

∫ b

aλ−2dλ; u ∈ S(H1) by Theorem 8.2.20, page 410; by

the explicit form of ◦ and Γ , if λ0 ∈ [a, b] is fixed, it holds

u(δλ0(z0)) = [· · ·] =∫ b

a

1

{(λ0 − λ)4 (x20 + y2

0)2 + (λ20 − λ2)2 t2

0 }1/2dλ

≈∫ λ0+ε

λ0−ε

1

|λ − λ0| 2λ0 |t0| dλ = ∞.

Ex. 2) Let Ω ⊂ G be any bounded open set, and let ε > 0 be any positive realnumber. Prove that there exists a countable collection of disjoint d-balls

Bk := Bd

(xk,

rk

4

)

with rk2 < ε for every k and a set N such that

⎧⎨

⎩

Ω = N ∪ ⋃k Bk,

⋃k Bd(xk, c rk) ⊇ Ω.

Ex. 3) Let μ be any compactly supported Radon measure on G. Suppose z ∈ G issuch that (Γ ∗ μ)(z) < ∞ and prove that μ({z}) = 0. (Compare to Exer-cise 22, Chapter 9.)

Ex. 4) Let E ⊂ G be any set and φ be a measure function. Suppose there existsρ0 > 0 such that

M(ρ0)φ (E) = 0.

Prove that M(ρ)φ (E) = 0 for every ρ > 0, whence mφ(E) = 0.


(Hint: Since the map ]0,∞] ρ �→ M(ρ)φ is non-negative and decreasing

(prove this!), we have M(ρ)φ (E) = 0 for every ρ ≥ ρ0. Let now ρ ∈]0, ρ0[ be

fixed. By the very definition of M(ρ0)φ (E) = 0, we have

(�) ∀ ε > 0 ∃ {Bd(xk(ε), rk(ε))}k∈N :⎧⎨

⎩

rk(ε) < ρ0,

E ⊆ ⋃k Bd(xk(ε), rk(ε)),∑

k φ(rk(ε)) < ε.

Since φ is positive on ]0,∞[, in (�) we can take any ε ∈]0, φ(ρ)[.Hence, the third condition in the right-hand side of (�) gives

φ(rk(ε)) ≤∑

k

φ(rk(ε)) < ε < φ(ρ) ∀ k ∈ N.

Since φ is increasing, this gives rk(ε) < ρ, so that M(ρ)φ (E) = 0.)

Ex. 5) Prove the assertions of Example 13.1.4 and Remark 13.1.5. (Hint: The fol-lowing facts may help. For every x ∈ G and r > 0, we have rQ =c HN(Bd(x, r)), where c = HN(Bd(0, 1)). If E ⊆ ⋃

k Bd(xk, rk), then

x ◦ E ⊆⋃

k

Bd(x ◦ xk, rk), δλ(E) ⊆⋃

k

Bd(δλ(xk), λrk),

whence H(ρ)

(α)(x ◦ E) = H

(ρ)

(α)(E) and H

(λρ)

(α)(δλ(E)) = λα H

(ρ)

(α)(E).)

Ex. 6) Prove the assertion in Remark 13.1.2. More precisely, we have the followingresults. For any E ⊆ G, set

diamd(E) := sup{d(x−1 ◦ y) : x, y ∈ E}, diamd(∅) := 0.

If φ is a measure function as in Definition 13.1.1, we let

H(ρ)φ (E) := inf

{∑

k

φ(diamd(Ek)) : E ⊆⋃

k

Ek, diamd(Ek) < ρ ∀ k

},

M(ρ)φ (E) := inf

{∑

k

φ(rk) : E ⊆⋃

k

Bd(xk, rk), rk < ρ ∀ k

}.

• Prove that H(ρ)φ ,M

(ρ)φ , hφ, mφ are outer measures on G, where

hφ(E) := supρ>0

H(ρ)φ (E), mφ(E) := sup

ρ>0M

(ρ)φ (E).

• Let A, B be arbitrary subsets of G. Prove that

distd(A,B) > 0 �⇒{

hφ(A ∪ B) = hφ(A) + hφ(B),

mφ(A ∪ B) = mφ(A) + mφ(B).(13.30)

Here, we have set


distd(A,B) := inf{d(a−1 ◦ b) : a ∈ A, b ∈ B}.Note that distd(A,B) > 0 iff the usual Euclidean distance between A andB is positive.3 Hence, derive from (13.30) and from very general results ofmeasure theory, that every (open and then every) Borel set in G is both hφ

and mφ measurable. Hence, if restricted to their related measurable sets, hφ

and mφ are Borel measures.Now, we take φ(t) = tα with α > 0, and write h(α) and m(α) instead of htα

and mtα .• Prove that

2−αm(α)(E) ≤ h(α)(E) ≤ (2c)α m(α)(E) for every E ⊆ G, (13.31)

where c is the constant in the pseudo-triangle inequality for d . The fol-lowing facts might help: r ≤ diam

(Bd(x, r)

) ≤ 2c r; if e ∈ E, thenE ⊆ Bd(e, diam(E)) ⊂ Bd(e, 2diam(E));

Hεtα (E) ≤ (2c)αH

ε/(2c)tα (E); M2ε

tα (E) ≤ 2αHεtα (E).

• Derive from (13.31) that, for every bounded set E ⊂ G, there exists anumber α(E) ∈ [0,Q] (Q is the homogeneous dimension of G) such that

h(α)(E) = m(α)(E) = ∞ for all α < α(E),

h(α)(E) = m(α)(E) = 0 for all α > α(E).

Thus, we have

α(E) = inf{α > 0 : h(α)(E) = 0} = inf{α > 0 : m(α)(E) = 0}.

3 Indeed, if aj ∈ A, bj ∈ B, then observe that a−1j

◦ bj → 0 in the Euclidean metric iff

d(a−1j

◦ bj ) → 0.

Part III

Further Topics on Carnot Groups

14

Some Remarks on Free Lie Algebras

The aim of this chapter is to investigate a few properties of the free nilpotent Liealgebras. After recalling the definition of the free Lie algebra fm,r with m generatorsand nilpotent of step r , we provide an algorithm (due to M. Hall) to construct abasis for fm,r , jointly with several examples. Furthermore, we describe another usefulalgorithm concerning with free Lie algebras in an analytic context, an algorithm dueto M. Grayson and R. Grossman.

In Section 14.2, we furnish a canonical way to construct free Carnot groups (i.e.Carnot groups whose Lie algebras is free and nilpotent) by means of the Campbell–Hausdorff formula.

As we shall see in Chapter 16, free Carnot groups are the appropriate generaliza-tions of the usual Euclidean group, when we are concerned with the “equivalence”of all sub-Laplacians.

14.1 Free Lie Algebras and Free Lie Groups

We first recall the definition of free Lie algebra with m generators and nilpotent ofstep r (see, e.g. [VSC92, p. 45]; see also [Var84, p. 174]).

Definition 14.1.1 (The free Lie algebra fm,r ). Let m ≥ 2 and r ≥ 1 be fixed in-tegers. We say that fm,r is the free Lie algebra with m generators x1, . . . , xm andnilpotent of step r if:

(i) fm,r is a Lie algebra generated by its elements x1, . . . , xm, i.e.

fm,r = Lie{x1, . . . , xm};(ii) fm,r is nilpotent of step r;

(iii) for every Lie algebra n nilpotent of step r and for every map ϕ from the set{x1, . . . , xm} to n, there exists a (unique) homomorphism of Lie algebras ϕ fromfm,r to n which extends ϕ.

578 14 Some Remarks on Free Lie Algebras

Remark 14.1.2. It is easy to see that, fixed m and r , the Lie algebra fm,r is uniqueup to isomorphism: it is enough to compare two algebras f, f′ satisfying the abovedefinition and recognize that the morphism as in (iii) which takes the generators of f

into the generators of f′ is the inverse of the morphism taking the generators of f′ intothose of f. Moreover, if fm,r is isomorphic to fm′,r ′ , then it necessarily holds m = m′and r = r ′.

We explicitly remark that it is not trivial to prove the existence of such an alge-bra fm,r . We refer the reader to [Var84] for this topic.

Definition 14.1.3 (Free Carnot group). We say that a Carnot group G is a freeCarnot group if its Lie algebra g is isomorphic to fm,r for some m and r .

The following lemma concerning with free Lie algebras fm,r will be useful in thesequel.

Lemma 14.1.4. Let fm,r be the free Lie algebra with m generators x1, . . . , xm, nilpo-tent of step r . Let

ϕ : span{x1, . . . , xm} −→ span{x1, . . . , xm}be a linear bijective map. Then there exists one (and only one) isomorphism of Liealgebras

ϕ : fm,r −→ fm,r

extending ϕ.

Proof. We consider the maps

ϕ, ϕ−1 : span{x1, . . . , xm} −→ span{x1, . . . , xm}.By the very definition of fm,r , there exist two homomorphisms of Lie algebras ϕ,

(ϕ−1) defined on fm,r into itself which extend ϕ and ϕ−1, respectively. We considerthe map

ψ := (ϕ−1) ◦ ϕ.

The proof is complete if we show that ψ is the identity map. To this end, we notethat, for every i = 1, . . . , m, one has

ψ(xi) = (ϕ−1)(ϕ(xi)) = ϕ−1(ϕ(xi)) = xi,

since ϕ(xi) ∈ span{x1, . . . , xm}. By the definition of fm,r , there exists a unique ho-momorphism of the Lie algebra fm,r into itself which extends

ψ |span{x1,...,xm}.

Since both ψ and the identity map satisfy this requirement, the assertion immediatelyfollows. ��

14.1 Free Lie Algebras and Free Lie Groups 579

We next give a model for fm,r . It is known that, setting

H(m, r) := dim(fm,r ),

fm,r may be represented by upper triangular matrices of order H(m, r) (see [Var84]).There exists another remarkable representation of fm,r via vector fields, useful in theapplications and easy to be dealt with, due to M. Grayson and R. Grossman (see[GG90]), which we now briefly recall. This representation makes use of the so-calledHall basis for fm,r .

Definition 14.1.5 (The Hall basis for fm,r ). Let fm,r be the free Lie algebra with m

generators x1, . . . , xm, nilpotent of step r . We define the elements of the Hall basisfor fm,r in the following way:

• x1, . . . , xm are the first m elements of the Hall basis and they are referred to asthe standard monomials of height 1;

• We now let 2 ≤ n ≤ r , and we define the standard monomials of height n:suppose we have already defined the standard monomials of height 1, . . . , n − 1,and suppose we have ordered them in such a way that u precedes v (we writeu < v) if the height of u is strictly lower than that of v; then the Lie bracket[u, v] is a standard monomial of height n if the sum of the heights of u and v

equals n and if, moreover, the following conditions are satisfied:1. u and v are standard monomials such that u > v;2. if u = [x, y] is the form of the standard monomial u, then v ≥ y.

The collection of all the standard monomials of height 1, . . . , r is referred to as theHall basis for fm,r .


Theorem 14.1.6 (Hall, [Hal50]). The Hall basis for fm,r is a basis (in the sense ofvector fields) for fm,r .

Proof. The assertion follows from the arguments in the paper by M. Hall [Hal50,Theorem 3.1], in which a general basis for free Lie rings is constructed exactly as inDefinition 14.1.5 (but with no restriction on r , i.e. r = ∞). Since fm,r is nilpotentof step r , it is enough to consider the elements of the basis constructed in [Hal50]which are brackets of height r at most. ��Example 14.1.7. We next give some examples of Hall bases:

• The Hall basis for f3,4 is given by:

height 1: x1, x2, x3,

height 2: [x2, x1], [x3, x1], [x3, x2],

height 3:

⎧⎨

⎩

[[x2, x1], x1], [[x3, x1], x1],[[x2, x1], x2], [[x3, x1], x2], [[x3, x2], x2],[[x2, x1], x3], [[x3, x1], x3], [[x3, x2], x3],


height 4:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[[[x2, x1]x1]x1], [[[x3, x1]x1]x1],[[[x2, x1]x1]x2], [[[x3, x1]x1]x2],[[[x2, x1]x2]x2], [[[x3, x1]x2]x2], [[[x3, x2]x2]x2],[[[x2, x1]x1]x3], [[[x3, x1]x1]x3],[[[x2, x1]x2]x3], [[[x3, x1]x2]x3], [[[x3, x2]x2]x3],[[[x2, x1]x3]x3], [[[x3, x1]x3]x3], [[[x3, x2]x3]x3],[[x3, x1], [x2, x1]], [[x3, x2], [x2, x1]],[[x3, x2], [x3, x1]].

• The Hall basis for f4,2 is given by:

height 1: x1, x2, x3, x4,

height 2:

⎧⎪⎨

⎪⎩

[x2, x1], [x3, x1], [x4, x1],[x3, x2], [x4, x2],

[x4, x3].• The Hall basis for fm,2 is given by:

height 1: x1, x2, x3, . . . , xm,

height 2:

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

[x2, x1], [x3, x1], · · · [xm, x1],[x3, x2], · · · [xm, x2],

. . ....

[xm, xm−1].Remark 14.1.8. We explicitly remark that

H(m, 2) = dim(fm,2) =m∑

j=1

j = m(m + 1)

2.

In particular, this implies, the following result.

Remark 14.1.9. The Heisenberg group HN is a free Carnot group if and only if

N = 1.

Indeed, since the Lie algebra hN of HN has step 2 and 2N generators, a necessary

condition for HN to be free is that 2N +1 = dimhN = H(2N, 2) = 2N(2N +1)/2,

i.e. that N = 1. On the other hand, it is immediate to recognize that h1 is isomorphicto f2,2. ��

As we shall see more closely in Chapter 17, this shows that, even if the vectorfields related to the Heisenberg group H

N are naturally defined on R2N+1, the free

(lifted) counterpart of such vector fields acts on R2N(2N+1)/2.


We finally turn our attention to the cited model for fm,r by Grayson–Grossman,which we now describe. We order the elements of the Hall basis for fm,r according1

to Definition 14.1.5. We set H := dimfm,r , and we denote

E1, . . . , EH

the elements of the Hall basis with such a fixed ordering. We fix an element Ei .According to the construction of the Hall basis in Definition 14.1.5, it certainly hasthe following form (if i > m)

Ei = [Ej1 , Ek1], with j1 > k1.

Taking Ek1 fixed (whatever its height is) and arguing in the same way for Ej1 , onehas

Ei = [[Ej2, Ek2 ]; Ek1], with j2 > k2.

After finitely many steps, we have obtained the expression

Ei = [Ej1, Ek1]= [[Ej2, Ek2]; Ek1 ]= [[[Ej3, Ek3 ], Ek2 ]; Ek1]= [[[[Ej4 , Ek4], Ek3 ], Ek2 ]; Ek1]= · · ·= [[· · · [[Ejn, Ekn], Ekn−1 ], . . . , Ek2 ]; Ek1],

with1 ≤ kn < jn ≤ m and kl+1 ≤ kl for 1 ≤ l ≤ n − 1.

This maximal expression for Ei involves n commutations (or, equivalently, there aren indices of the type k’s), and then we set, by definition,

d(i) := n

with the conventiond(1) = d(2) = · · · = d(m) := 0.

1 A somewhat “canonical” way to order the elements of the Hall basis is the following one:

1. x1, x2, . . . , xm is a fixed ordering of the standard monomials of height 1.2. If 2 ≤ n ≤ r , we now show how to canonically order the standard monomials of height n.

Suppose the standard monomials of heights 1, . . . , n − 1 have already been ordered. Letu and v be two standard monomials of height n in the form

u = [x1, y1], v = [x2, y2].We then follow the rules:• if y1 �= y2, then u precedes v if and only if y1 precedes y2;• if y1 = y2, then u precedes v if and only if x1 precedes x2.


This process associates in a natural way a multi-index I (i) = (a1, a2, . . . , aH ) to Ei ,where

as := cardinality of the set {t | kt = s}.In other words, for every s = 1, . . . , H , as is the number of times the element Es

appears withinEkn, Ekn−1 , . . . , Ek2; Ek1 .

By definition, we set

I (1) = I (2) = · · · = I (m) := (0, . . . , 0).

Moreover, we say that Ei is a direct descendant of all the elements of the type Ejl

Ej1, Ej2, . . . , Ejn−1 , Ejn,

and we writejl ≺ i.

We remark that ≺ is a partial ordering, and that if a ≺ b, then every entry of I (a)

is ≤ of the corresponding entry of I (b). For every pair a, b satisfying a ≺ b, wedefine the monomial

a ≺ b ⇒ Pa,b := (−1)d(b)−d(a)

(I (b) − I (a))! xI (b)−I (a).

For example, if

Ei = [[[[[[[E2, E1], E1], E1], E2], E4], E4]; E7] and

Ek = [[[E2, E1], E1]; E1],then d(i) = 7, d(k) = 3, d(2) = 0 and

I (i) = (3, 1, 0, 2, 0, 0, 1, 0, . . . , 0),

I (k) = (3, 0, 0, 0, 0, 0, 0, 0, . . . , 0),

whereas

P2,i = −x31x2x

24x7

3! 2! , Pk,i = x2x24x7

2! .

The following remarkable result holds (see [GG90, Theorem 2.1]).

Theorem 14.1.10 (Grayson–Grossman, [GG90]). Let r ≥ 1, m ≥ 2, and let H bethe dimension of fm,r . Then the vector fields

E1 := ∂

∂x1,

E2 := ∂

∂x2+

∑

j�2

P2,j

∂

∂xj

,

...

Em := ∂

∂xm

+∑

j�m

Pm,j

∂

∂xj

(14.1)

with polynomial coefficients on RH have the following properties:


(1) E1, . . . , Em are homogeneous of degree one with respect to the dilation canoni-cally induced on R

H by the stratification

RH ≡ fm,r = V1 ⊕ · · · ⊕ Vr,

where Vi is the vector space spanned by the standard monomials of the Hallbasis with height i;

(2) The elements of the Hall basis generated by the vector fields E1, . . . , Em satisfy

Ei(0) = ∂

∂xi

, i = 1, . . . , H ;

(3) The Lie algebra of vector fields on RH generated by E1, . . . , Em is isomorphic

to fm,r .

Remark 14.1.11. We explicitly remark that the vector fields in (14.1) satisfy condi-tions (H0)–(H1)–(H2) in Section 4.2, page 191. Then they naturally define, as de-scribed in Section 4.2 (see, in particular, Theorem 4.2.10), a Carnot group G whichis free (of step r and with m generators), according to Definition 14.1.3: indeed,by (3) in the above Theorem 14.1.10, we immediately infer that the Lie algebra of G

is isomorphic to fm,r . Moreover, by (2) in the above Theorem 14.1.10, the canonicalsub-Laplacian ΔG is simply given by ΔG = ∑m

j=1 E2j . ��

Example 14.1.12. To end this section, we give an example of a model for f4,2. LetE1, E2, E3, E4 be the generators of f4,2. We set

E5 := [E2, E1], E6 := [E3, E1], E7 := [E4, E1],E8 := [E3, E2], E9 := [E4, E2], E10 := [E4, E3].

With the above notation, one has

d(1) = d(2) = d(3) = d(4) = 0,

d(5) = d(6) = d(7) = d(8) = d(9) = d(10) = 1,

I (1) = I (2) = I (3) = I (4) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0),

I (5) = I (6) = I (7) = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0),

I (8) = I (9) = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0),

I (10) = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0),

2 ≺ 5, 3 ≺ 6, 4 ≺ 7, 3 ≺ 8, 4 ≺ 9, 4 ≺ 10,

P2,5 = P3,6 = P4,7 = −x1,

P3,8 = P4,9 = −x2,

P4,10 = −x3.

Then, according to Theorem 14.1.10, a model for f4,2 is given by the Lie algebra ofvector fields generated by the following vector fields on R

10:


E1 := ∂

∂x1,

E2 := ∂

∂x2− x1

∂

∂x5,

E3 := ∂

∂x3− x1

∂

∂x6− x2

∂

∂x8,

E4 := ∂

∂x4− x1

∂

∂x7− x2

∂

∂x9− x3

∂

∂x10

with the commutator identities

E5 = [E2, E1] = ∂

∂x5,

E6 = [E3, E1] = ∂

∂x6,

E7 = [E4, E1] = ∂

∂x7,

E8 = [E3, E2] = ∂

∂x8,

E9 = [E4, E2] = ∂

∂x9,

E10 = [E4, E3] = ∂

∂x10.

As observed in Remark 14.1.11, the following second order differential operator onR

10

E21 + E2

2 + E23 + E2

4

= (∂x1)2 + (∂x2 − x1 ∂x5)

2

+ (∂x3 − x1 ∂x6 − x2 ∂x8)2 + (∂x4 − x1 ∂x7 − x2 ∂x9 − x3 ∂x10)

2

is then the canonical sub-Laplacian related to a suitable free homogeneous Carnotgroup on R

10, with 4 generators and nilpotent of step 2. Finally, we explicitly remarkthat the above is not the only possible model for f4,2 (see, e.g. Example 14.2.5 inSection 14.2).

14.2 A Canonical Way to Construct Free Carnot Groups

14.2.1 The Campbell–Hausdorff Composition �First of all, we recall some references on the so-called Campbell–Hausdorff for-mula. Roughly speaking, if X and Y are two non-commuting indeterminates, the(Baker–)Campbell–(Dynkin–)Hausdorff formula states that the formal expression

14.2 A Canonical Way to Construct Free Carnot Groups 585

“ log(exp(X) exp(Y )) ”

can be expressed in terms of an infinite sum of iterated commutators of X and Y .This statement can be made precise in many contexts such as for formal power series,for matrix algebras, for general normed Banach algebras, for finite-dimensional Liegroups, for solutions of differential equations, etc. (See Bibliographical Notes at theend of the chapter.)

Since we are mainly interested in the setting of Lie groups and Lie algebras, webriefly recall how the Campbell–Hausdorff formula naturally arises in this context(see also Section 2.2, page 121, where we discussed this topic; for the reader’s con-venience, the main results we need here are reproduced). Let (n, [·,·]) be an abstractnilpotent Lie algebra. For X, Y ∈ n, we set2

X � Y :=∑

n≥1

(−1)n+1

n

∑

pi+qi≥11≤i≤n

(ad X)p1(ad Y)q1 · · · (ad X)pn(ad Y)qn−1Y

(∑n

j=1(pj + qj )) p1! q1! · · ·pn! qn!

= X + Y + 1

2[X, Y ]

+ 1

12[X, [X, Y ]] − 1

12[Y, [X, Y ]]

− 1

48[Y, [X, [X, Y ]]] − 1

48[X, [Y, [X, Y ]]]

+ {brackets of height ≥ 5}. (14.2)

Since n is nilpotent, (14.2) is a finite sum and � determines a binary operation onn, which is defined in a universal way by a sum of Lie monomials with rationalcoefficients.

Now, let (F, ∗) be a Lie group with the Lie algebra f. Suppose the exponentialmap Exp : f → F has an inverse function Log globally defined. Then the operation

f × f � (X, Y ) �→ Log(Exp (X) ∗ Exp (Y )

) ∈ f

is well posed on f. With this last composition law, f is obviously a Lie group isomor-phic to (F, ∗) via Exp . The following result states that this last composition law is(under suitable hypotheses on F) precisely the universal one defined in (14.2), i.e.the following Campbell–Hausdorff formula holds:

Log(Exp (X) ∗ Exp (Y )

) = X � Y for all X, Y ∈ f. (14.3)

2 Here, we use the notation (ad A)B = [A, B]. Moreover, if qn = 0, the term in the sum(14.2) is, by convention,

(ad X)p1(ad Y)q1 · · · (ad X)pn−1(ad Y)qn−1(ad X)pn−1X.

Clearly, if qn > 1, or qn = 0 and pn > 1, the term is zero.


Theorem 14.2.1 (Corwin and Greenleaf, [CG90], Theorem 1.2.1). Let (F, ∗) bea connected and simply connected Lie group. Suppose that the Lie algebra f of F

is nilpotent. Let � be the operation defined in (14.2). Then � defines a Lie groupstructure on f and Exp : (f,�) → (F, ∗) is a Lie group isomorphism. In particular,if Log is the inverse function of Exp , then Log : (F, ∗) → (f,�) is a Lie groupisomorphism and (14.3) holds.

We then recall the third fundamental theorem of Lie, which is another deep resultin the theory of Lie groups.

Theorem 14.2.2 (Varadarajan, [Var84], Theorem 3.15.1). Let f be a finite-dimens-ional Lie algebra. Then there exists a connected and simply connected Lie groupwhose Lie algebra is isomorphic to f.

From the above two theorems we obtain the following result (for its proof, seeCorollary 2.2.15, page 130).

Theorem 14.2.3. If f is a finite-dimensional nilpotent Lie algebra, then the operation� introduced in (14.2) defines a Lie group structure on f. We call � the Campbell–Hausdorff operation on f.

14.2.2 A Canonical Way to Construct Free Carnot Groups

We now apply the results of the previous section in order to show a standard way toconstruct Carnot groups on R

N . This construction makes also use of fm,r , the freenilpotent Lie algebra of step r with m generators, which we treated in Section 14.1.

Let r ≥ 1 and m ≥ 2 be given, and set H := dim(fm,r ). Let E1, . . . , Em be asystem of generators for fm,r . Suppose that {Ei}i≤H is an enumeration of the Hallbasis for fm,r which preserves the natural ordering ≤ given in Definition 14.1.5. Wedefine a family of dilations {Dλ}λ>0 on fm,r by

Dλ

(H∑

i=1

xiEi

)=

H∑

i=1

λαi xiEi,

where αi is the height of Ei . Finally, we identify fm,r with RH in the natural way by

introducing the linear isomorphism

π : RH → fm,r such that x �→

H∑

i=1

xi Ei .

It is now not difficult to prove the following result (we explicitly observe that, byTheorem 14.2.3, (fm,r ,�) is a Lie group).

Theorem 14.2.4 (Free homogeneous Carnot groups). With the above notation, forevery x, y ∈ R

H and λ > 0, we set

x ◦ y := π−1(π(x) � π(y)), δλ(x) := π−1(Dλ(π(x))

).

Then G := (RH , ◦, δλ) is a free homogeneous Carnot group.

14.2 A Canonical Way to Construct Free Carnot Groups 587

Example 14.2.5. We give a simple descriptive example of this construction for a freehomogeneous Carnot group G = (R10, ◦, δλ) of step 2 and with 4 generators. Sup-pose f4,2 is generated by E1, . . . , E4. The Hall basis E1, . . . , E10 for f4,2 is givenby

E1, E2, E3, E4; [E2, E1], [E3, E1], [E4, E1], [E3, E2], [E4, E2], [E4, E3].The group of dilations on G is given by

δλ(x) = (λx1, . . . , λx4, λ2x5, . . . , λ

2x10). (14.4)

The Campbell–Hausdorff formula on a Lie algebra nilpotent of step 2 is given by

X � Y = X + Y + 1

2[X, Y ].

The group law x ◦ y is simply obtained by explicitly writing( ∑

i≤10

xiEi

)�

( ∑

i≤10

yiEi

)

with respect to the Hall basis, making use of bilinearity, skew-symmetry and theJacobi identity

∑

i≤10

xiEi �∑

i≤10

yiEi

=∑

i≤10

xiEi +∑

i≤10

yiEi + 1

2

∑

i,j≤10

xiyj [Ei,Ej ] = · · ·

=∑

i≤4

(xi + yi)Ei +(

x5 + y5 + 1

2(x2y1 − x1y2)

)E5 + · · ·

+(

x10 + y10 + 1

2(x4y3 − x3y4)

)E10.

This yields, for every x, y ∈ G,

x ◦ y =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x1 + y1x2 + y2x3 + y3x4 + y4

x5 + y5 + 12 (x2 y1 − x1 y2)

x6 + y6 + 12 (x3 y1 − x1 y3)

x7 + y7 + 12 (x4 y1 − x1 y4)

x8 + y8 + 12 (x3 y2 − x2 y3)

x9 + y9 + 12 (x4 y2 − x2 y4)

x10 + y10 + 12 (x4 y3 − x3 y4)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.


With the above composition law ◦ and the dilation δλ defined in (14.4), G =(R10, ◦, δλ) is a free homogeneous Carnot group of step 2 and with 4 generators.

Now, by making use of (1.34) (page 19), it is very simple to find the first 4elements Z1, . . . , Z4 of the Jacobian basis of the algebra of G: given ϕ ∈ C∞(R10),we have

(Z1ϕ)(x) = (∂/∂y1)|y=0ϕ(x ◦ y)

= ∂x1ϕ(x) + 1

2x2 ∂x5ϕ(x) + 1

2x3 ∂x6ϕ(x) + 1

2x4 ∂x7ϕ(x),

(Z2ϕ)(x) = (∂/∂y2)|y=0ϕ(x ◦ y)

= ∂x2ϕ(x) − 1

2x1 ∂x5ϕ(x) + 1

2x3 ∂x8ϕ(x) + 1

2x4 ∂x9ϕ(x),

(Z3ϕ)(x) = (∂/∂y3)|y=0ϕ(x ◦ y)

= ∂x3ϕ(x) − 1

2x1 ∂x6ϕ(x) − 1

2x2 ∂x8ϕ(x) + 1

2x4 ∂x10ϕ(x),

(Z4ϕ)(x) = (∂/∂y4)|y=0ϕ(x ◦ y)

= ∂x4ϕ(x) − 1

2x1 ∂x7ϕ(x) − 1

2x2 ∂x9ϕ(x) − 1

2x3 ∂x10ϕ(x).

As a consequence, the following second order differential operator on R10

Z21 + Z2

2 + Z23 + Z2

4

=(

∂x1 + 1

2x2 ∂x5 + 1

2x3 ∂x6 + 1

2x4 ∂x7

)2

+(∂x2 − 1

2x1 ∂x5 + 1

2x3 ∂x8 + 1

2x4 ∂x9

)2

+(

∂x3 − 1

2x1 ∂x6 − 1

2x2 ∂x8 + 1

2x4 ∂x10

)2

+(

∂x4 − 1

2x1 ∂x7 − 1

2x2 ∂x9 − 1

2x3 ∂x10

)2

is the canonical sub-Laplacian related to a free homogeneous Carnot group on R10,

with 4 generators and nilpotent of step 2 (for another example, see Example 14.1.12).

Bibliographical Notes. The motivation for this brief investigation on free Lie alge-bras is primarily due to the Rothschild and Stein lifting theorem, according to whichany sum of squares of Hörmander vector fields can be “approximated as close as wewant” (in a suitable sense; see [RS76]) by a sub-Laplacian on a (larger) manifold,which is a stratified group whose Lie algebra is free.



Ex. 1) Prove that fm,r is isomorphic to fm′,r ′ if and only if m = m′ and r = r ′.Ex. 2) Write down the Hall bases for f2,3, f2,4 and f4,3.Ex. 3) This exercise concerns with the canonical way of constructing Carnot groups

via the Campbell–Hausdorff formula described in Section 14.2. Suppose itis given an algebra generated by the two elements X1, X2 with the fol-lowing commutator relations (indeed, it should be noted that these relationsare consistent with the properties of the bracket operation: bilinearity, skew-symmetry and the Jacobi identity; see Definition 2.1.39): X1, X2, [X1, X2]and [X1, [X1, X2]] are linearly independent, whereas there exists α ∈ R suchthat

[X2, [X1, X2]] = α [X1, [X1, X2]]and all commutators of height ≥ 4 vanish.Consider the Campbell–Hausdorff operation for an algebra nilpotent of step 3

X � Y = X + Y + 1

2[X, Y ] + 1

12[X, [X, Y ]] − 1

12[Y, [X, Y ]].

Write down explicitly the Campbell–Hausdorff operation(x1 X1 + x2 X2 + x3 [X1, X2] + x4 [X1, [X1, X2]]

)

� (y1 X1 + y2 X2 + y3 [X1, X2] + y4 [X1, [X1, X2]]

)

and write the result with respect to the basis X1, X2, [X1, X2] and[X1, [X1, X2]]. Denote the 4-tuple of the coordinates thus obtained as(x1, x2, x3, x4) ◦ (y1, y2, y3, y4). Then, arguing as in Theorem 14.2.4, ◦ de-fines on R

4 a homogeneous Carnot group. Verify that ◦ is given by

⎛

⎜⎜⎜⎜⎜⎝

x1 + y1x2 + y2

x3 + y3 + 12 (x1 y2 − x2 y1)

x4 + y4 + 12 (x1 y3 − x3 y1) + α

2 (x2 y3 − x3 y2)

+ 112 (x1 − y1) (x1 y2 − x2 y1) + α

12 (x2 − y2) (x1 y2 − x2 y1)

⎞

⎟⎟⎟⎟⎟⎠.

Ex. 4) We follow the ideas in Ex. 1 to re-derive the Heisenberg–Weyl group on R3:

the Campbell–Hausdorff operation for a Lie algebra h of step 2 has the form

X � Y = X + Y + 1

2[X, Y ].

We consider an algebra of step two with the basis Z1, Z2 and Z3 := [Z1, Z2].Then we have

(ξ1 Z1 + ξ2 Z2 + ξ3 Z3

) � (η1 Z1 + η2 Z2 + η3 Z3

)

= (ξ1 Z1 + ξ2 Z2 + ξ3 Z3) + (

η1 Z1 + η2 Z2 + η3 Z3)


+ 1

2

[ξ1 Z1 + ξ2 Z2 + ξ3 Z3, η1 Z1 + η2 Z2 + η3 Z3

]

([Z1, Z2] = Z3, [Z2, Z1] = −Z3 and commutators of height > 2 vanish)

= (ξ1 + η1)Z1 + (ξ2 + η2)Z2 +(

ξ3 + η3 + 1

2ξ1 η2 − 1

2η1 ξ2

)Z3.

By the natural identification

h � ξ1 Z1 + ξ2 Z2 + ξ3 Z3 ←→ (ξ1, ξ2, ξ3) ∈ R3,

we have re-obtained the Heisenberg–Weyl group H1 on R

3 whose composi-tion law is

ξ ◦ η =(

ξ1 + η1, ξ2 + η2, ξ3 + η3 + 1

2ξ1 η2 − 1

2η1 ξ2

).

Carry out a similar exercise with an algebra of step two whose basis is

X1, . . . , XN ; Y1, . . . , YN ; [X1, Y1]with the following commutator identities:a) [Xj , Yj ] = [X1, Y1] for all j = 1, . . . , N ,b) [Xi,Xj ] = 0 = [Yi, Yj ] for all i, j = 1, . . . , N ,c) [Xi, Yj ] = 0 for all i, j = 1, . . . , N such that i �= j .

Recognize the Lie group in (4.16), page 201.Ex. 5) Give a detailed proof of Theorem 14.2.4.Ex. 6) By means of Example 14.1.7 and the ideas in Section 14.2, find a Carnot

group whose Lie algebra is isomorphic to f4,2. Finally, consider the generalcase of fm,2.

Ex. 7) a) Use Grayson–Grossman’s Theorem 14.1.10 to find vector fields generat-ing f5,2.

b) Show that they satisfy conditions (H0)–(H1)–(H2) in Section 4.2 (page 191)and construct the relevant homogeneous Carnot group as described inthat section.

c) Construct a Carnot group whose Lie algebra is isomorphic to f5,2 follow-ing the ideas in Section 14.2. Compare to point (b).

Ex. 8) Let m, r ∈ N (m ≥ 2) and consider fm,r . Set H(m, r) = dim(fm,r). Finally,let f be any finite-dimensional (real) Lie algebra and set H := dim(f). Whatof the following conditions are sufficient in order to have fm,r and f isomor-phic as Lie algebras?a) H(m, r) = H . (Hint: Consider f2,2 and f3,1 or, equivalently, the Lie

algebra of the Heisenberg–Weyl group H1 on R

3 and the Lie algebra ofthe Euclidean group (R3,+).)

b) H(m, r) = H and fm,r and f are nilpotent of step r . (Hint: Con-sider f3,2 and the Lie algebra f, nilpotent of step 2, whose basis is{X, Y,A,B,C, [X, Y ]} such that A,B,C commute with each other andwith X and Y ; roughly speaking, f is the Lie algebra of the sum of H

1

and (R3,+) (see Section 4.1.5, page 190).)


c) fm,r and f are both nilpotent of step r and Lie-generated by m of theirelements. (Hint: Consider f3,2 and the Lie algebra f, nilpotent of step 2,whose basis is {X, Y,A, [X, Y ]} such that A commutes with X and Y ;roughly speaking, f is the Lie algebra of the sum of H

1 and (R1,+).)d) H(m, r) = H and fm,r and f are Lie-generated by m of their elements.

(Hint: Consider f3,2 and the Lie algebra f, nilpotent of step 3, whose basisis

{X1, X2, A,X3 := [X1, X2],X4 := [X1, [X1, X2]], X5 := [X1, [X1, [X1, X2]]]

},

such that A commutes with Xi (i = 1, . . . , 5), [X1, X5] = 0, [Xi,Xj ] =0 for every 1 < i < j ≤ 5. Roughly speaking, f is obtained from thefiliform Lie algebra h in Example 4.3.5 (page 208) when i = 5, by addingan extra generator A commuting with every other generator.)

e) H(m, r) = H and fm,r and f are nilpotent of step r and both are Lie-generated by m of their elements.

Hint: Condition (e) is actually sufficient. Indeed, suppose

f = Lie{F1, . . . , Fm}(here F1, . . . , Fm ∈ f) is nilpotent of step r . Let also x1, . . . , xm denote thegenerators for fm,r , as in Definition 14.1.1.According to this very definition (replacing n with f, what is possible since f

is nilpotent of step r), if we consider the map ϕ sending xi in Fi (for everyi = 1, . . . , m), there exists a homomorphism of Lie algebras ϕ from fm,r to f

which extends ϕ.We are done if we show that ϕ is a vector space isomorphism. Since fm,r andf have the same dimension, it suffices to prove that ϕ is onto. This follows bythe argument below

f = Lie{F1, . . . , Fm} = Lie{ϕ(x1), . . . , ϕ(xm)}= Lie{ϕ(x1), . . . , ϕ(xm)} = ϕ(Lie{x1, . . . , xm}) = ϕ(fm,r).

Provide the proofs of the above equalities.

15

More on the Campbell–Hausdorff Formula

The main aim of this chapter is to sketch a proof of Lemma 4.2.4 in Section 4.2(page 194). This lemma played a crucial rôle in the construction of Carnot groups.We could refer to this lemma as a sort of Campbell–Hausdorff formula for stratifiedvector fields.

Our proof here is articulated in the following way. In Section 15.1, we show in de-tails how to derive Lemma 4.2.4 from a general version of the Campbell–Hausdorffformula for formal power series. Then, in Sections 15.2 and 15.3, we provide twoproofs of this formula for formal power series: the former is due to D. Djokovic[Djo75], the latter is due to M. Eichler [Eic68].

The first proof makes use of results from the theory of formal power series; thesecond one rests on some properties of Lie polynomials. For the sake of brevity,we shall leave to the reader the details of the proofs of these properties, which areactually very intuitive. As a result, the proofs of the Campbell–Hausdorff formulaare surprisingly simple and short, as in the spirit of the cited authors.

Finally, Section 15.4 provides the version of the Campbell–Hausdorff formulamostly cited in analytic contexts. Once again, it is unavoidable to make use of theformula for formal power series. Furthermore, the main task is to show how to controland give precise estimates of the remainder terms in the relevant series expansions.

15.1 A Proof of the Campbell–Hausdorff Formula for StratifiedVector Fields

We recall the notation of Section 4.2: X1, . . . , Xm is a given set of smooth vectorfields on R

N satisfying hypotheses (H0)–(H1)–(H2) of Section 4.2, i.e. the followingconditions are fulfilled:

(H0) X1, . . . , Xm are linearly independent and δλ-homogeneous of degree one withrespect to a suitable family of dilations {δλ}λ>0 of the following type

δλ : RN → R

N, δλ(x) = δλ(x(1), . . . , x(r)) := (λx(1), . . . , λrx(r)),

594 15 More on the Campbell–Hausdorff Formula

where r ≥ 1 is an integer, x(i) ∈ RNi for i = 1, . . . , r , N1 = m and N1 +

· · · + Nr = N .(H1) We set W(k) = span{XJ | J ∈ {1, . . . , m}k}, where

X(j1,...,jk) = [Xj1, . . . [Xjk−1, Xjk] . . .]

if J = (j1, . . . , jk). Then, dim(W(k)) = dim{XI (0) : X ∈ W(k)} for everyk = 1, . . . , r .

(H2) dim(Lie{X1, . . . , Xm}I (0)) = N .

Leta = Lie{X1, . . . , Xm}.

For every k = 1, . . . , r , Z(k)1 , . . . , Z

(k)Nk

will be a fixed basis for W(k). We know that

{Z1, . . . , ZN } := {Z(1)1 , . . . , Z

(1)N1

, . . . , Z(r)1 , . . . Z

(r)Nr

}is a basis for a. For every ξ = (ξ1, . . . , ξN) = (ξ (1), . . . , ξ (r)) we set

ξ · Z =N∑

j=1

ξj Zj =r∑

k=1

Nk∑

j=1

ξ(k)j Z

(k)j .

We know that1 the map (x, t) �→ exp(t ξ · Z)(x) is well defined for every x ∈ RN

and t ∈ R. Moreover,

Exp : RN −→ R

N, Exp (ξ) := exp(ξ · Z)(0)

is a global diffeomorphism.2 Its inverse function is denoted by Log . In Section 4.2,we defined a composition law on R

N by means of

x, y ∈ RN, x ◦ y := exp(Log (y) · Z)(x).

Our aim there was to show that G := (RN, ◦, δλ) is a Carnot group whose Liealgebra g coincides with a. The main task of the proof definitely was to show that ◦is associative. To this end, we needed the cited Lemma 4.2.4, which stated that

For every X, Y ∈ a, there exists a unique V ∈ a such that

exp(Y )(

exp(X)(x)) = exp(V )(x) (15.1)

for every x ∈ RN . In this section, we at last prove (15.1) as a consequence of the

following theorem.

1 We recall that, given a smooth vector field X on RN , exp(tX)(x) := γ (t, x) denotes the

solution γ (·) = γ (·, x) to the ordinary differential system{

γ = XI (γ ),

γ (0) = x.

2 We explicitly remark that we are using the notation Exp to denote a map on RN instead of

a map on an algebra of vector fields.

15.1 The Campbell–Hausdorff Formula for Stratified Fields 595

Theorem 15.1.1 (The Campbell–Hausdorff formula for stratified vector fields).Let X, Y ∈ a be fixed. Let V be the differential operator defined by the formalexpansion

r∑

j=1

(−1)j+1

j

(r∑

k1+k2=1

Xk2 Y k1

k2! k1!

)j

= V +{

summands of the type Yyn Xxn · · · Yy1 Xx1 withn∑

i=1

(yi + xi) > r

}.

(15.2)

Then V turns out to be a vector field belonging to Lie{X, Y } (hence belonging to a)such that

exp(Y )(

exp(X)(x)) = exp(V )(x) ∀ x ∈ R

N. (15.3)

We explicitly remark that (15.3) proves (15.1), i.e. it proves Lemma 4.2.4. For exam-ple, a direct computation shows that

V = X � Y := X + Y + 1

2[X, Y ] + 1

12[X, [X, Y ]] − 1

12[Y, [X, Y ]]

− 1

48[Y, [X, [X, Y ]]] − 1

48[X, [Y, [X, Y ]]] + · · · .

We recognize here the Campbell–Hausdorff composition law X�Y defined in (14.2),page 585.

Before proceeding with the proof of Theorem 15.1.1, we give an example of howthe formal expansion in (15.2) behaves. For instance, taking r = 3, we have

3∑

j=1

(−1)j+1

j

(3∑

k1+k2=1

Xk1 Y k2

k1! k2!

)j

=3∑

j=1

(−1)j+1

j

(X0Y 1

0! 1! + X1Y 0

1! 0! + X0Y 2

0! 2! + X1Y 1

1! 1! + X2Y 0

2! 0!

+ X0Y 3

0! 3! + X1Y 2

1! 2! + X2Y 1

2! 1! + X3Y 0

3! 0!)j

=3∑

j=1

(−1)j+1

j

(Y + X + Y 2

2+ XY + X2

2+ Y 3

6+ XY 2

2+ X2Y

2+ X3

6

)j

=(

Y + X + Y 2

2+ XY + X2

2+ Y 3

6+ XY 2

2+ X2Y

2+ X3

6

)1

− 1

2

(Y + X + Y 2

2+ XY + X2

2+ Y 3

6+ XY 2

2+ X2Y

2+ X3

6

)2

+ 1

3

(Y + X + Y 2

2+ XY + X2

2+ Y 3

6+ XY 2

2+ X2Y

2+ X3

6

)3


= Y + X + Y 2

2+ XY + X2

2+ Y 3

6+ XY 2

2+ X2Y

2+ X3

6

− 1

2

(Y 2 + YX + Y 3

2+ YXY + YX2

2+ XY + X2 + XY 2

2+ X2Y

+ X3

2+ Y 3

2+ Y 2X

2+ XY 2 + XYX + X2Y

2+ X3

2

)

+ 1

3(Y 3 + Y 2X + YXY + YX2 + XY 2 + XYX + X2Y + X3)

+ {summands of height ≥ 4}= X + Y + X2

(1

2− 1

2

)+ Y 2

(1

2− 1

2

)+ XY

(1 − 1

2

)+ YX

(−1

2

)

+ X3(

1

6− 1

4− 1

4+ 1

3

)+ Y 3

(1

6− 1

4− 1

4+ 1

3

)

+ X2Y

(1

2− 1

2− 1

4+ 1

3

)

+ Y 2X

(−1

4+ 1

3

)+ XY 2

(1

2− 1

4− 1

2+ 1

3

)+ YX2

(−1

4+ 1

3

)

+ YXY

(−1

2+ 1

3

)+ XYX

(−1

2+ 1

3

)

+ {summands of height ≥ 4}= X + Y + 1

2XY − 1

2YX + 1

12X2Y + 1

12Y 2X + 1

12XY 2

+ 1

12YX2 − 1

6YXY − 1

6XYX

+ {summands of height ≥ 4}= X + Y + 1

2[X, Y ] + 1

12[X, [X, Y ]] − 1

12[Y, [X, Y ]]

+ {summands of height ≥ 4}.Proof (of Theorem 15.1.1). Throughout the proof, X, Y ∈ a are fixed. We beginby noticing that (see the arguments in Remark 1.1.3, page 9) the map R × R

N (t, x) �→ exp(tX)(x) has polynomial component functions. Moreover, we have

n∑

i=1

(yi + xi) > r ⇒ Yyn Xxn · · · Yy1 Xx1I ≡ 0. (15.4)

This follows by recalling that any field in a is a sum of vector fields δλ-homogeneousof degree at least 1 and by observing that the component functions of the identitymap I are δλ-homogeneous monomials of degree at most r . Since exp(tX)(x) is ananalytic function of t , (1.7) (page 7) gives

exp(X) ≡r∑

k=0

1

k!XkI.

15.1 The Campbell–Hausdorff Formula for Stratified Fields 597

We henceforth fix x ∈ RN and set

Φ(t1, t2) := exp(t1Y)(

exp(t2X)(x)), t1, t2 ∈ R.

Clearly, any component function of Φ is a polynomial in t1, t2. From (1.15) (page 10),for every smooth vector field Z on R

N , f ∈ C∞(RN, RN), k ∈ N and every z ∈ R

N ,we have (

dk

dt k

)

t=0

(f (exp(tZ)(z))

) = (Zkf )(z). (15.5)

This gives, for every k1, k2 ∈ N,

(∂k1+k2

∂tk11 ∂t

k22

)∣∣∣∣(t1,t2)=(0,0)

Φ(t1, t2)

=(

∂k2

∂tk22

)

t2=0

(∂k1

∂tk11

)

t1=0I(

exp(t1Y)(

exp(t2X)(x)))

=(

∂k2

∂tk22

)

t2=0(Y k1I )

(exp(t2X)(x)

)

= (Xk2Y k1I )(x). (15.6)

Indeed, in the second equality we used (15.5) with f = I , k = k1, Z = Y , z =exp(t2X)(x), whereas in the third equality we used (15.5) with f = Y k1I , k = k2,Z = X, z = x.

We now use Taylor’s formula for Φ. Since Φ is a polynomial, exploiting (15.4),we derive

exp(Y )(

exp(X)(x)) = Φ(1, 1) =

r∑

k1+k2=0

1

k1! k2! (Xk2Y k1I )(x). (15.7)

We now introduce the following higher order differential operator

W(t,X, Y ) :=r∑

j=1

(−1)j+1

j

(r∑

k1+k2=1

tk1+k2

k1! k2!Xk2 Y k1

)j

.

We formally expand W(t,X, Y ), and we order it as a polynomial in t setting

W(t,X, Y ) =r∑

k=1

tkZk(X, Y ) +r2∑

k=r+1

tkZk(X, Y )

=: V (t,X, Y ) + R(t,X, Y ).

We explicitly remark that the differential operator V appearing in (15.2) in the state-ment of the theorem is simply given by V (t,X, Y ) with t = 1. It is easy to recognizethat, by (15.4), any power of R(t,X, Y ) annihilates the identity map, i.e.


(R(t,X, Y ))kI ≡ 0 for every k ≥ 0. (15.8)

We claim thatZk(X, Y ) ∈ Lie{X, Y } for every k ≥ r. (15.9)

Before proving the claimed (15.9), we first show that (15.9) implies the assertion ofthe theorem. Indeed, if we set

V := V (1, X, Y ),

then from the definition of V (t,X, Y ) and (15.9) we derive that

V ∈ Lie{X, Y }.Finally, we have to prove that (15.3) is satisfied with the above choice of V . Indeed,we have

exp(V )(x) =r∑

k=0

1

k!VkI (x)

=r∑

k=0

1

k!(V (1, X, Y ) + R(1, X, Y )

)kI (x)

=r∑

k=0

1

k!(W(1, X, Y )

)kI (x)

=r∑

k=0

1

k!

{r∑

j=1

(−1)j+1

j

(r∑

k1+k2=1

1

k1! k2!Xk2 Y k1

)j}k

I (x)

= I (x) +r∑

k1+k2=1

1

k1! k2!Xk2 Y k1I (x) = exp(Y )

(exp(X)(x)

).

We now motivate every equality appearing above:

1) the first equality follows from the fact that V ∈ a and from (1.7);2) the second one follows from (15.8) and simple homogeneity arguments;3) the third one is the very definition of W(1, X, Y );4) the fourth equality is a consequence of the formal power series expansion of the

identity 1 + x = exp(log(1 + x)), jointly with (15.4): more precisely, the fourthequality is a consequence of the identity

r∑

k=0

1

k!

{r∑

j=1

(−1)j+1

jHj

}k

= I + H +r2∑

j=r+1

cj Hj ∀ H ∈ a

and of (15.4);5) the last equality follows from (15.7).

15.2 The Campbell–Hausdorff Formula for Formal Power Series–1 599

Finally, we are left with the proof of the claimed (15.9). A proof of it can bederived from a general result by Djokovic [Djo75], which we present in Section 15.2.Indeed, in [Djo75] an assertion analogous to (15.9) is derived in the more generalcontext3 of the formal power series in two non-commuting indeterminates X and Y .This completes the proof of the theorem. �Remark 15.1.2. An odd fact, arising from the proof of Theorem 15.1.1, has to be re-marked: from (15.7) and the fact that exp(X) = ∑r

k=01k!X

kI (see (1.7)) we derived

r∑

k1=0

Y k1

k1! I

(r∑

k2=0

Xk2

k2! I (x)

)=

r∑

k1+k2=0

1

k2! k1! Xk2Y k1I (x).

The order of composition of X and Y seems reversed, on the face of it. Thus, formalcomputations such as

“∑

k1

Y k1

k1!∑

k2

Xk2

k2! =∑

k1+k2

1

k1! k2! Y k1Xk2 ”

are erroneous in our context of composition of exponential maps. This is anotherreason why we proved in detail our Campbell–Hausdorff-type Theorem 15.1.1 ratherthan reducing it, with no further justification, to formal power series arguments.

15.2 A Proof of the Campbell–Hausdorff Formula for FormalPower Series

The aim of this section is to present the arguments (with few more details) in [Djo75]for a proof of the Campbell–Hausdorff formula for formal power series. Here X

and Y will denote general non-commuting indeterminates in a multiplicative algebraover the field Q of the rational numbers (for instance, X and Y may be vector fieldson R

N , as in the previous section). We shall perform computations in the algebra offormal power series in X and Y .

We first fix a very natural4 notation:

eX :=∑

k≥0

Xk

k! (15.10)

3 At this point, a simple remark has to be given, in order to justify the fact that we canreplace general calculus between two non-commuting indeterminates X and Y with ourvector fields X,Y ∈ a. To this end, we explicitly remark that the identities between formalpower series used in [Djo75] in order to prove (15.9) readily reduce, in our context, toidentities between finite sums of vector fields: it will be enough to make use of argumentssuch as (15.4).

4 As Remark 15.1.2 has shown, though this is a very natural definition, when X is a vectorfield on R

N , eX should not be confused with the exponential exp(X) of a vector field asthe integral curve to the system γ = X(γ ).


denotes the formal expression of the exponential of X. The Campbell–Hausdorffformula simply states that there exists Z which is a formal infinite sum of summandsin Lie{X, Y } such that eY · eX = eZ . More precisely, see Theorem 15.2.2 below.

In the next result and for the future references, we need the definition of Lie-polynomial.

Definition 15.2.1 (Lie polynomial). If K is any field, we denote by K[X, Y ] the vec-tor space (over K) of the polynomials in two non-commuting indeterminates X, Y ,i.e. K[X, Y ] is K-spanned by the formal expressions

Xα1Yβ1 · · ·Xαi Y βi

(called, a monomial of degree α1 + β1 + · · · + αi + βi), where i ∈ N andα1, β1, · · · , αi, βi ∈ N ∪ {0}. The degree of a polynomial is the maximum degreeof its monomials.

Let X and Y be two non-commuting indeterminates in an associative algebra.Set [X, Y ] = XY − YX (which we call a Lie-bracket). Given k ∈ N, k ≥ 2, andZ1, . . . , Zk ∈ {X, Y }, we say that the Lie-bracket

[Z1[Z2[Z3[· · · [Zk−1, Zk] · · ·]]]],is a Lie(-bracket) monomial of length k in X and Y . We also say that X and Y areLie-monomials of length 1.

A linear combination of Lie monomials of length k in X and Y is called a ho-mogeneous Lie-polynomial of length k. Any finite sum of Lie-bracket monomials iscalled a Lie-polynomial. Analogous definitions are given in the setting of Lie alge-bras.

Obviously, K[X, Y ] can be equipped with an associative algebra structure in anobvious way.5 We are now ready to state and prove the following theorem.

Theorem 15.2.2 (The Campbell–Hausdorff formula for formal power series).For every positive integer k, there exists a homogeneous Lie-bracket-polynomial Zk

of length k (with rational coefficients) in two non-commuting indeterminates suchthat, using the notation Zk(·, ·) to express the dependence of Zk on the two indeter-minates, we have:

if Z(X, Y ) =∞∑

k=1

Zk(X, Y ) then eY · eX = eZ (15.11)

for every couple of non-commuting indeterminates X, Y . For example,

Z1(X, Y ) = X + Y,

Z2(X, Y ) = 1

2[Y,X],

5 For example, XY 2 · Y 3X4 = XY 5X4 and XY 2 · X4 = XY 2X4, XY 2X5 · XY 3X4Y =XY 2X6Y 3X4Y , etc.


Z3(X, Y ) = 1

12[Y, [Y,X]] − 1

12[X, [Y,X]],

Z4(X, Y ) = − 1

48[X, [Y, [Y,X]]] − 1

48[Y, [X, [Y,X]]]. (15.12)

Proof. For every t ∈ R, we have

et Y · et X =∑

k2≥0

(t Y )k2

k2! ·∑

k1≥0

(t X)k1

k1! =∑

k1,k2≥0

tk1+k2

k1!k2! Y k2 Xk1

= I +∑

k1+k2≥1

tk1+k2

k1!k2! Y k2Xk1 =: I + HX,Y (t).

If HX,Y (t) is as above, we set

ZX,Y (t) := ln(I +HX,Y (t)) =∑

j≥1

(−1)j+1

j

( ∑

k1+k2≥1

tk1+k2

k1!k2! Y k2Xk1

)j

. (15.13a)

From the formal power series expansion of the identity eln(1+x) = 1 + x, it thenfollows that

eZX,Y (t) = I + HX,Y (t) = et Y · et X. (15.13b)

Now, (15.11) is formally fulfilled with the natural choice

Z(X, Y ) := ZX,Y (1). (15.13c)

The theorem is then proved if we show that ZX,Y (1) can be put in the formZ(X, Y ) = ∑∞

k=1 Zk(X, Y ) with the Zk’s as in the assertion of the theorem. Tothis end, it suffices to prove the following power series expansion w.r.t. t :

ZX,Y (t) =∞∑

k=1

tk · Zk(X, Y ), (15.13d)

whereZk(X, Y ) ∈ Lie{X, Y } for every k ∈ N (15.13e)

and Zk(X, Y ) is homogeneous of length k.

To begin with, we observe that, differentiating the far right-hand side of (15.13b)w.r.t. t (and writing henceforth Z(t) instead of ZX,Y (t)), we have

∂t

(eZ(t)

) = ∂t

(et Y · et X

) = (∂tet Y

) · et X + et Y · (∂tet X

)

= Y · et Y · et X + et Y · et X · X = Y · eZ(t) + eZ(t) · X.

Hence, we obtain

∂t

(eZ(t)

) · e−Z(t) = eZ(t) · X · e−Z(t) + Y. (15.13f)


As a general fact, it holds

eZ · X · e−Z =∑

m≥0

Zm

m! · X ·∑

k≥0

(−1)kZk

k! =∑

m,k≥0

(−1)kZm · X · Zk

m! k! .

Then, setting m + k =: n, the expression for eZ · X · e−Z becomes6

eZ · X · e−Z =∑

n≥0

1

n!n∑

k=0

(−1)k(

n

k

)Zn−k · X · Zk. (15.13g)

Throughout the sequel, we use the following classical notation for the left and rightmultiplication:

LZ(X) := Z · X, RZ(X) := X · Z,

(Ad Z)(X) := (LZ − RZ

)(X) = Z · X − X · Z.

Obviously, LZ and RZ commute.7 As a consequence, we can apply Newton’s bino-mial expansion to get

(Ad Z

)n(X) = (

LZ − RZ

)n(X) =

n∑

k=0

(−1)k(

n

k

)Ln−k

Z · RkZ (X)

=n∑

k=0

(−1)k(

n

k

)Zn−k · X · Zk. (15.13h)

Taking into account (15.13h), the expression (15.13g) turns out to be

eZ · X · e−Z =∑

n≥0

1

n!(Ad Z

)n(X) =: eAd Z(X).

This gives (see (15.13f))

∂t

(eZ(t)

) · e−Z(t) = Y + eAd Z(t)(X). (15.13i)

On the other hand, a direct computation yields

∂t

(eZ(t)

) · e−Z(t) =(

∑

m≥1

1

m! ∂t (Z(t)m)

)·(

∑

k≥0

(−1)k

k! Z(t)k

)

(k =: n − m) =∑

n≥1

1

n!n∑

m=1

(−1)n−m

(n

m

)∂t (Z(t)m)Z(t)n−m =: ().

6 Here we wrote1

m! k! =(

n

k

)1

n! .

7 We obviously have LZ(RZ(X)) = Z · X · Z = RZ(LZ(X)).


Thus, since one has

∂t (Z(t)m) =m−1∑

k=0

Z(t)k · (∂t Z(t)) · Z(t)m−k−1,

we derive that

() =∑

n≥1

1

n!n∑

m=1

m−1∑

k=0

(−1)n−m

(n

m

)Z(t)k · (∂t Z(t)) · Z(t)n−k−1

=∑

n≥1

1

n!n−1∑

k=0

(n∑

m=k+1

(−1)n−m

(n

m

))Z(t)k · (∂t Z(t)) · Z(t)n−k−1

(we set r := n − m and use properties of the binomial coefficients)

=∑

n≥1

1

n!n−1∑

k=0

(n−k−1∑

r=0

(−1)r(

n

r

))Z(t)k · (∂t Z(t)) · Z(t)n−k−1 =: ().

An inductive argument shows that

m∑

r=0

(−1)r(

n

r

)= (−1)m

(n − 1

m

), 0 ≤ m ≤ n − 1.

Consequently, we infer

() =∑

n≥1

1

n!n−1∑

k=0

((−1)n−k−1

(n − 1

k

))Z(t)k · (∂t Z(t)) · Z(t)n−k−1

=∑

n≥1

1

n!n−1∑

j=0

((−1)j

(n − 1

j

))Z(t)n−1−j · (∂t Z(t)) · Z(t)j

(see (15.13h)) =∑

n≥1

1

n! (Ad Z(t))n−1(∂t Z(t)).

Finally, comparing to (15.13i), we have

Y + eAd Z(t)(X) =∑

n≥1

1

n! (Ad Z(t))n−1(∂t Z(t)),

i.e. by the definition of the exponential of Ad Z(t),

Y +∑

n≥0

1

n! (Ad Z(t))n (X) =∑

n≥1

1

n! (Ad Z(t))n−1(∂t Z(t)). (15.13j)

Now, the power series∑

n≥1 xn−1/n! furnishes the formal expansion of ϕ(x) :=(ex − 1)/x. Then, since ϕ(0) = 1 �= 0, there exists a reciprocal series expansion


of∑

n≥1 xn−1/n!, say g(x) := ∑n≥0 an xn. More precisely, the following formal

identity holds (∑

n≥0

an xn

)·( ∑

n≥1

xn−1/n!)

= 1.

As a consequence, multiplying both sides of (15.13j) times g(Ad Z(t)), we obtain

g(Ad Z(t)

)(

Y +∑

n≥0

1

n! (Ad Z(t))n (X)

)

=∑

n≥0

an(Ad Z(t))n ·( ∑

n≥1

1

n! (Ad Z(t))n−1(∂t Z(t)))

= ∂t Z(t).

This proves that Z(t) satisfies the differential equation

∂t Z(t) = g(Ad Z(t))

(Y +

∑

n≥0

1

n! (Ad Z(t))n (X)

). (15.13k)

By reordering the sum in (15.13a) which defines Z(t), we obtain

Z(t) =∑

j≥1

(−1)j+1

j

( ∞∑

k=1

( ∑

k1+k2=k

1

k1! k2! Y k2 Xk1

)tk

)j

.

If we set

bk :=∑

k1+k2=k

1

k1! k2! Y k2 Xk1, k ≥ 1,

by means of the formula for the j -th power of a power series, we derive

Z(t) =∑

j≥1

(−1)j+1

j

∞∑

k=1

tk( ∑

n1+···+nj =k

n1,...,nj ≥1

bn1 · · · bnj

)

=∞∑

k=1

tk

( ∞∑

j=1

(−1)j+1

j

∑

n1+···+nj =k

n1,...,nj ≥1

bn1 · · · bnj

)

=∞∑

k=1

tk

( ∞∑

j=1

(−1)j+1

j

∑

(r1+s1)+···+(rj +sj )=k

r1+s1≥1,...,rj +sj ≥1

Y r1Xs1 · · · Y rj Xsj

r1!s1! · · · rj !sj !

)

=:∞∑

k=1

tk · Zk(X, Y ).


Following this notation, the left-hand side in (15.13k) (i.e. the derivative of Z(t))equals

∞∑

k=0

tk · (k + 1) Zk+1(X, Y ).

Following again the notation Z(t) = ∑k≥1 tk Zk , recalling that g(x) = ∑

n≥0 an xn,and by using the formula for the power of a series in t , the right-hand side of (15.13k)equals

X + Y +∞∑

k=1

tk∞∑

n=1

1

n!∑

m1+···+mn=k

Ad Zm1 · · · Ad Zmn(X)

+∞∑

k=1

tk∞∑

n=1

an

∑

m1+···+mn=k

Ad Zm1 · · · Ad Zmn

(X + Y

+∞∑

l=1

t l∞∑

j=1

1

j !∑

h1+···+hj =l

Ad Zh1 · · · Ad Zhj(X)

).

It has to be noticed that the coefficient of tk in this last expression is a linear combi-nation of terms of the type

Ad Zm1 · · · Ad Zmn(X) and Ad Zm1 · · · Ad Zmn(Y )

with m1 + · · · + mn = k. In particular, by equating the coefficients of tk from bothsides of (15.13k), one has

Z1(X, Y ) = X + Y,

whereas Z2(X, Y ) is a linear combination of commutators of height 2 of X +Y withX (and Y ); moreover Z3(X, Y ) is a linear combination of commutators of height3 of Z2(X, Y ) (or X + Y ) and X (or Y ) and so on. An inductive argument nowproves that Zk(X, Y ) is a linear combination of commutators of height k of X andY . In particular, Zk(X, Y ) ∈ Lie{X, Y } for every k ∈ N. This actually demonstrates(15.13e), and the proof is complete. �

15.3 Another Proof of the Campbell–Hausdorff Formula forFormal Power Series

In this section, we provide another proof of the Campbell–Hausdorff formula forformal power series, as given by M. Eichler in [Eic68]. This proof essentially restson some properties of Lie polynomials. (In due course of the proof, we shall leave tothe reader the details for these properties, which are indeed very intuitive.)

In the rest of the section, we consider the associative algebra of formal powerseries in two non-commuting indeterminates A,B over the field of the rational num-bers Q. In particular, we shall use the definition of the exponential as in (15.10)and the notion of Lie-polynomial (see Definition 15.2.1). We are now ready to giveanother proof of Theorem 15.2.2.


Proof (of Theorem 15.2.2). We aim to prove that

eA · eB = e∑∞

n=1 Fn(A,B), (15.14)

where, for every n ∈ N, Fn(A,B) is a Lie-polynomial in A,B which is homogeneousof length n.

First of all, we remark that it certainly exist unique homogeneous polynomialsFn’s in A,B satisfying (15.14) (note that we are not yet asking for Lie-polynomials).Indeed, there exists a unique formal power series Z in A,B such that eA · eB = eZ .It suffices to take

Z := log(eA · eB) = log(

1 +∑

n+m≥1

An/n!Bm/m!)

or, more precisely,

Z =∑

j≥1

(−1)j+1

j

( ∑

n+m≥1

An/n!Bm/m!)j

.

Then we reorder the summands defining Z in such a (unique) way that they arehomogeneous polynomials in A,B. Our main task is to prove that the resulting Fn’sare Lie-polynomials in A,B.

We find F1 and F2. F1 results from the choice (j = 1, n = 1,m = 0) or from(j = 1, n = 0,m = 1) thus giving

F1(A,B) = A + B.

F2 results from the choice

(j, n,m) ∈ {(1, 1, 1), (1, 2, 0), (1, 0, 2), (2, 1, 0), (2, 0, 1)},so that

F2(A,B) = AB + A2

2+ B2

2− 1

2(A + B)2

= AB + A2

2+ B2

2− 1

2(A2 + AB + BA + B2)

= 1

2AB − 1

2BA = 1

2[A,B].

Hence, F1 and F2 are homogeneous Lie-polynomials of the desired degree. We nowargue by induction. Let n ≥ 3 and suppose that F1, . . . , Fn−1 are Lie-polynomials ofdegree 1, . . . , n−1, respectively. We are done if we show that Fn is a Lie-polynomial(its degree is obviously n for it is the sum of polynomials of degree n).

The crucial device is to take three non-commuting indeterminates A,B,C andto use associativity to equal the expansions of


(eA · eB) · eC = eA · (eB · eC). (15.15)

By (using twice) (15.14), the left-hand side of (15.15) is

e∑∞

j=1 Fj (A,B) · eC = e∑∞

i=1 Fi

(∑∞j=1 Fj (A,B),C

).

The reader should appreciate that we are here using the universality of the Fj ’s asfunctions of two “arbitrary” non-commuting indeterminates.8 Analogously, the right-hand side of (15.15) is

eA · e∑∞

j=1 Fj (B,C) = e∑∞

i=1 Fi

(A,

∑∞j=1 Fj (B,C)

).

We thus have

W :=∞∑

i=1

Fi

( ∞∑

j=1

Fj (A,B), C

)=

∞∑

i=1

Fi

(A,

∞∑

j=1

Fj (B,C)

). (15.16)

It is easy to verify that whenever Fi and Fj are Lie-polynomials in two indetermi-nates, then Fi(Fj (A,B), C) and Fi(A, Fj (B,C)) are Lie-polynomials in A,B,C.Moreover, it is also not difficult to verify that the homogeneous polynomials whichare the summands into which a Lie-polynomial splits up are homogeneous Lie-polynomials.

We now compare homogeneous terms of degree n (in A,B,C) on both sides of(15.16). Such terms appear in summands of the type

Fi(Fj (A,B), C), Fi(A, Fj (B,C))

if and only if i = 1, j = n; i = n, j = 1; or i + j = n (in the case i > 1,j > 1). Thus, with the remarks of the preceding paragraph at hand, by our inductiveassumption, we see that all homogeneous terms of degree n in both expressions forW are Lie-polynomials, except at most for

Fn(A,B) + Fn(A + B,C)

in the left-hand side of (15.16) and except for

Fn(A,B + C) + Fn(B,C)

in its right-hand side.For the sake of brevity, we introduce the equivalence relation ∼ between poly-

nomials p1, p2 in A,B,C by saying that p1 ∼ p2 whenever p1 − p2 is a Lie-polynomial. The assertion made at the end of the preceding paragraph thus rewritesas

Fn(A,B) + Fn(A + B,C) ∼ Fn(A,B + C) + Fn(B,C). (15.17)

8 After all, the arbitrary nature of the non-commuting indeterminates lies in the very conceptof “indeterminates”.


We now make some trivial remarks on the ∼ relation. If α, β ∈ Q, then (since α A

and β A commute) we have

eα A · eβ A = eα A+β A = eF1(α A,β A).

Consequently, for every n ≥ 2, Fn(α A, β A) = 0, i.e.

Fn(α A, β A) ∼ 0. (15.18)

In particular, this holds for α = 1 and β = 0.Now, we show that (15.17) and (15.18) imply

Fn(A,B) ∼ 0,

which is our goal. First, the choice C := −B in (15.17) yields

Fn(A,B) ∼ −Fn(A + B,−B) + Fn(A, 0) + Fn(B,−B) ∼ −Fn(A + B,−B),

i.e.Fn(A,B) ∼ −Fn(A + B,−B). (15.19)

Analogously, the choice A := −B in (15.17) yields

Fn(B,C) ∼ Fn(−B,B) + Fn(0, C) − Fn(−B,B + C) ∼ −Fn(−B,B + C),

i.e. by renaming B �→ A, C �→ B,

Fn(A,B) ∼ −Fn(−A,A + B). (15.20)

Applying (15.20), (15.19) and again (15.20), we get

Fn(A,B) ∼ −Fn(−A,A + B) ∼ −( − Fn(−A + A + B,−A − B))

= Fn(B,−A − B) ∼ −Fn(−B,B − A − B) = −Fn(−B,−A)

= −(−1)n Fn(B,A).

The last equality is a consequence of the length-homogeneity n of Fn as a polyno-mial. This gives

Fn(A,B) ∼ (−1)n+1 Fn(B,A). (15.21)

Now, we insert C = − 12 B in (15.17). This gives

Fn(A,B) ∼ −Fn

(A + B,−1

2B

)+ Fn

(A,B − 1

2B

)+ Fn

(B,−1

2B

)

∼ −Fn

(A + B,−1

2B

)+ Fn

(A,

1

2B

),

i.e.

Fn(A,B) ∼ Fn

(A,

1

2B

)− Fn

(A + B,−1

2B

). (15.22)


Analogously, taking A := − 12 B in (15.17) and renaming B �→ A, C �→ B, we get

Fn(A,B) ∼ Fn

(1

2A,B

)− Fn

(−1

2A,A + B

). (15.23)

An application of (15.22) to both summands in the right-hand side of (15.23) yields

Fn(A,B) ∼ [· · ·] = Fn

(1

2A,

1

2B

)− Fn

(1

2A + B,−1

2B

)

− Fn

(−1

2A,

1

2A + 1

2B

)+ Fn

(1

2A + B,−1

2A − 1

2B

).

Now, we apply (15.20) to the third summand in the far right-hand of the above ex-pression and (15.19) to the second and fourth summands. We thus get

Fn(A,B) ∼ Fn

(1

2A,

1

2B

)+ Fn

(1

2A + 1

2B,

1

2B

)

+ Fn

(1

2A,

1

2B

)− Fn

(1

2B,

1

2A + 1

2B

)

= 21−n Fn(A,B) + 2−n Fn(A + B,B) − 2−n Fn(B,A + B)

∼ 21−n Fn(A,B) + 2−n(1 + (−1)n) Fn(A + B,B).

In the equality we used the degree-homogeneity n of Fn, and we used (15.21) for thelast ∼ sign. This gives

(1 − 21−n) Fn(A,B) ∼ 2−n(1 + (−1)n) Fn(A + B,B), (15.24)

which proves that Fn(A,B) ∼ 0 whenever n �= 1 in odd. As for the case when n iseven, we argue as follows. We write A − B instead of A in (15.24)

(1 − 21−n) Fn(A − B,B) ∼ 2−n(1 + (−1)n) Fn(A,B),

and we use (15.19) in the above left-hand side getting (after multiplication times(1 − 21−n)−1)

Fn(A,−B) ∼ −(1 − 21−n)−1 2−n(1 + (−1)n) Fn(A,B). (15.25)

We write −B instead of B in (15.25), and we finally apply (15.25) once again ob-taining

Fn(A,B) ∼ (1 − 21−n)−2 2−2n(1 + (−1)n)2 Fn(A,B).

When n is even and n �= 2, this is possible only if Fn(A,B) ∼ 0. This ends theproof. �


15.4 The Campbell–Hausdorff Formula for General SmoothVector Fields

At this point of the exposition, the Campbell–Hausdorff formula has been exhaus-tively proved in the abstract setting of formal power series and in the more concretesetting of the vector fields in the algebra of a Carnot group. As a matter of fact, it iseasy to recognize that our proof of Theorem 15.1.1 can be applied for any couple ofvector fields X, Y which are δλ-homogeneous of positive degree with respect to anydilation δλ : R

N → RN of the form

δλ(x1, . . . , xN) = (λα1x1, . . . , λαN xN)

with positive αi’s. The reason for this fact is that the cited δλ-homogeneity en-sures that exp(X), exp(Y ) are indeed given by finite sums and there exists a largem ∈ N such that XmI, YmI ≡ 0. As a consequence, all the computations with for-mal power series in two indeterminates make precise sense, and the given proof ofthe Campbell–Hausdorff formula for formal power series can be straightforwardlyadapted.

We have so far provided the proof of the Campbell–Hausdorff formula for manyinteresting cases in literature, but we are now concerned in giving the most usefulversion of such an important formula. Theorem 15.4.1 below is the version whichanalysts mostly refer to. For example, it is, almost verbatim, the version needed inthe very recent paper [Mor00, Proposition 2.3] or a restatement of Proposition 4.3in the pioneering paper [NSW85] by Nagel, Stein and Wainger or, furthermore, ananalogue of the statement needed for the celebrated lifting by Rothschild and Stein(see [RS76, Section 10]). The aim of this section is to provide its proof.

As usual, given a smooth vector field X on an open set Ω ⊆ RN and fixed x ∈ Ω ,

we denote byt �→ γ (t) = exp(tX)(x)

the solution to the first order differential system in RN

γ = XI (γ ), γ (0) = x. (∗)

Equivalently (see for example, Ex. 7 at the end of Chapter 1), if D(X, x) is themaximal domain of existence of the solution of (∗), then the solution of the system

ν(s) = (tX)I(ν(s)

), ν(0) = x

exists at s = 1 for every t ∈ D(X, x), and it holds

ν(1) = γ (t).

Theorem 15.4.1 (The Campbell–Hausdorff formula for smooth vector fields).Let X and Y be smooth vector fields on the open set Ω ⊆ R

N . Then the followingformal equality holds:

15.4 The Campbell–Hausdorff Formula for Smooth Vector Fields 611

exp(sY )◦exp(tX) = exp

(sY +tX+ st

2[X, Y ]+

∑

k+j>2

sktj Ck,j (X, Y )

). (15.26)

Here Ck,j (X, Y ) denotes a finite linear combination of commutators of X and Y .Any summand in Ck,j (X, Y ) contains k times the field X and j times the field Y . Theprecise meaning of (15.26) is the following:

For any fixed compact set K ⊂ Ω and any given couple of integers k0, j0, thereexists a real number r0 > 0 (depending on K, k0, j0, X, Y ) such that if |s|, |t | < r0,then we have (see also Figure 15.1)

exp(sY )(

exp(tX)(x)) = exp

(sY + tX + st

2[X, Y ]

+∑

1 ≤ k ≤ k0, 1 ≤j≤j0k+j>2

sktj Ck,j (X, Y )

)(x)

+ O(sk0+1) + O(tj0+1) (15.27)

for every x ∈ K , where the above O’s mean |O(rm)| ≤ c rm for every |r| ≤ r0 anduniformly in x ∈ K (the constant c depends on r0,K, k0, j0, X, Y ).

Finally, the Ck,j ’s are “universal” Lie-polynomials in two non-commuting inde-terminates, as given in Theorem 15.2.2. For example,

C1,1(X, Y ) = 1

2[X, Y ],

C1,2(X, Y ) = 1

12[X, [X, Y ]], C2,1(X, Y ) = − 1

12[Y, [X, Y ]],

C2,2(X, Y ) = − 1

48[Y, [X, [X, Y ]]] − 1

48[X, [Y, [X, Y ]]],

C1,3(X, Y ) = C3,1(X, Y ) = 0.

Proof. Consider the notation in the statement of the Campbell–Hausdorff formulafor formal power series (Theorem 15.2.2). Then replace X and Y in that statementwith, respectively, sY and tX, where X, Y are still two non-commuting indetermi-nates and s, t are real numbers. Within the context of formal power series, formulasfrom (15.11) to (15.13d) then yield

∑

j≥0

(tX)j

j ! ·∑

k≥0

(sY)k

k! =∑

n≥0

(ZsY,tX

)n

n! , (15.28)

where ZsY,tX = ∑∞n=1 Zn(sY, tX), and Zn(·, ·) is a “universal” homogeneous Lie-

bracket-polynomial of length n in two indeterminates. Now, for every fixed k, j ∈N∪{0}, we group together the terms containing sktj in the Zn(sY, tX)’s. Denote theresulting group by Ck,j (X, Y). Thus, ZsY,tX rewrites as


Fig. 15.1. The Campbell–Hausdorff flow

ZsY,tX = sY + tX + st

2[X, Y] +

∑

k+j>2

sktj Ck,j (X, Y). (15.29)

For the future references, we see that (15.28) and (15.29) group together to give

∑

j≥0

(tX)j

j ! ·∑

k≥0

(sY)k

k!

=∑

n≥0

1

n!(

sY + tX +∑

k+j≥2

sktj Ck,j (X, Y)

)n

. (15.30)

The Ck,j (X, Y )’s in the assertion of the present theorem are defined by making useof the above “universal” Ck,j ’s. Note that, obviously, any summand in Ck,j (X, Y )

contains k times the field X and j times the field Y .Throughout the proof, we fix a compact set K ⊂ Ω and a couple of integers

k0, j0. By general results on differential systems of ODE’s (see, e.g. [Har82]), thereexists a positive r0 = r0(X, Y,K) such that the set

{exp(sY )

(exp(tX)(x)

) ∣∣ x ∈ K, |s|, |t | ≤ r0}

makes sense and is a compact subset of Ω . Analogously, using general results onthe dependence on the initial data for ODE’s, it is possible to prove the existence ofr0 > 0 (we may assume it is the same as above) such that the integral curve of thevector field


V (s, t) := sY + tX + st

2[X, Y ] +

∑

1≤k≤k0, 1≤j≤j0k+j>2

sktj Ck,j (X, Y ) (15.31)

exists at unit time, whenever x ∈ K and |s|, |t | ≤ r0. (The Ck,j ’s have been definedin the first part of the proof.)

Thus, the functions

u(s, t) := exp(sY )(

exp(tX)(x)), v(s, t) := exp(V (s, t))(x),

are well-posed on Q0 := (−r0, r0) × (−r0, r0) for every fixed x ∈ K . Our aim isto prove that the maps (s, t) �→ u(s, t), v(s, t) have the same Taylor expansion at(s, t) = (0, 0), up to order k0 w.r.t. s and up to order j0 w.r.t. t .

To begin with, we exhibit the expansion of u. Recalling that, for every functionf ∈ C∞(Ω, R

N), every smooth vector field Z on Ω , every z ∈ Ω and every n ∈N ∪ {0}, it holds

(d

d r

)n

(f (exp(rZ)(z))) = (Znf )(exp(rZ)(z)), (15.32)

we infer(

∂k+j

∂ sk ∂ tj

)u(s, t) =

(∂j

∂ tj

)(∂k

∂ sk

){exp(sY )

(exp(tX)(x)

)}

(use (15.32) with r = s, n = k, f = I , Z = Y , z = exp(tX)(x))

=(

∂j

∂ tj

)(Y kI

)(exp(sY )

(exp(tX)(x)

))

(use (15.32) with r = t , n = j , f = Y kI (exp(sY )(·)), Z = X, z = x)

= Xj(Y kI

(exp(sY )(·)))(exp(tX)(x)).

Thus, we have proved that, for every j, k ∈ N ∪ {0},(

∂k+j

∂ sk ∂ tj

)u(s, t) = Xj

(Y kI

(exp(sY )(·)))(exp(tX)(x)). (15.33)

Now, when s = 0 = t , the map exp(sY )(·) is the identity map, so that the abovecomputation reduces to (15.6), i.e.

(∂k+j

∂ sk ∂ tj

)u(0, 0) = (XjY kI )(x).

As a consequence, the Taylor polynomial of u(s, t) at (0, 0) up to order k0 w.r.t. sand up to order j0 w.r.t. t is given by

I (x) + sY I (x) + tXI (x) +∑

1≤k≤k01≤j≤j0

sktj

k!j ! (XjY kI )(x)

=(

j0∑

j=0

tj

j ! Xj

)(k0∑

k=0

sk

k! Y k

)I (x).


We now estimate the remainder. From the Taylor formula with the Lagrange-remainder up to height j0 + k0 applied to the i-th component of u, say ui , for every(s, t) ∈ Q0 we infer the existence of (σi, τi) ∈ Q0 such that

ui(s, t) =∑

0≤k≤k00≤j≤j0

sktj

k!j ! (XjY kIi)(x) +∑

k>k0 vel j>j0k+j≤k0+j0

sktj

k!j ! (XjY kIi)(x)

+∑

k+j=k0+j0+1

sktj

k!j ! Xj(Y kIi

(exp(σiY )(·)))(exp(τiX)(x)).

Clearly, the second sum in the above right-hand side is given by

O(sk0+1) + O(tj0+1),

uniformly in x ∈ K (due to the smoothness of X, Y and the compactness of K). Thesame estimate is true of the third sum, for (see (15.33))

|Xj(Y kIi(exp(σiY )(·)))(exp(τiX)(x))| ≤ M(Q0,K,X, Y )

uniformly in x ∈ K . All these facts prove that

u(s, t) =(

j0∑

j=0

tj

j ! Xj

)(k0∑

k=0

sk

k! Y k

)I (x) + O(sk0+1) + O(tj0+1). (15.34)

We now turn to the expansion of v(s, t). By the very definition v(s, t) =exp(V (s, t))(x), we have v(s, t) = γs,t (1), where r �→ γs,t (r) is the solution to

γs,t (r) = (V (s, t)I

)(γs,t (r)), γs,t (0) = x.

A simple computation (see the note to Ex. 5) shows that(

d

d r

)n

γs,t (r) = (V (s, t)nI

)(γs,t (r)),

so that, by Taylor’s expansion with an integral remainder, we have

v(s, t) = γs,t (1) =N∑

n=0

1

n!(

d

d r

)n

γs,t (0) + 1

N !∫ 1

0(1 − r)N

(d

d r

)N+1

γs,t (r) dr

=N∑

n=0

1

n!(V (s, t)nI

)(x) + 1

N !∫ 1

0(1 − r)N

(V (s, t)N+1I

)(γs,t (r)) dr.

Let us remark that, for r ∈ [0, 1], (s, t) ∈ Q0 and x ∈ K , all the points γs,t (r)

belong to a fixed compact set, say K0. Hence, if we choose N = k0 + j0, we have(see also (15.31))


∣∣(V (s, t)k0+j0+1I)(γs,t (r))

∣∣

=∣∣∣∣

(sY + tX +

∑

1≤k≤k0, 1≤j≤j0k+j≥2

sktj Ck,j (X, Y )

)k0+j0+1

I (γs,t (r))

∣∣∣∣

≤ c0 (sk0+1 + tj0+1)

× max{

supξ∈KO

|HI (ξ)| : H ∈ R[X, Y ] is a monomial with degree

≤ (k0 + j0)(k0 + j0 + 1)}

= O(sk0+1) + O(tj0+1)

uniformly in x ∈ K . Here we have denoted by R[X, Y ] the set of the polynomials inthe (non-commuting) indeterminates X, Y (see Definition 15.2.1). This shows that

v(s, t) =k0+j0∑

n=0

1

n!V (s, t)nI (x) + O(sk0+1) + O(tj0+1), (15.35)

uniformly in x ∈ K . Collecting together (15.34), (15.35) and the definition (15.31)of V (s, t), we see that the asserted (15.27) will follow if we prove that

(j0∑

j=0

tj

j ! Xj

)(k0∑

k=0

sk

k! Y k

)I (x)

=k0+j0∑

n=0

1

n!(

sY + tX +∑

1≤k≤k0, 1≤j≤j0k + j ≥ 2

sktj Ck,j (X, Y )

)n

I (x)

+ O(sk0+1) + O(tj0+1), (15.36)

again uniformly in x ∈ K .Now, let us return to identity (15.30), an identity in the context of formal power

series in two non-commuting indeterminates X, Y. By the very definition of formalpower series, we derive from (15.30) an identity between terms involving X up to j0times and Y up to k0 times, namely

j0∑

j=0

(tX)j

j ! ·k0∑

k=0

(sY)k

k!

≡k0+j0∑

n=0

1

n!(

sY + tX +∑

1≤k≤k0, 1≤j≤j0k+j≥2

sktj Ck,j (X, Y)

)n

{modulo terms with X more than j0 times or Y more than k0 times}. (15.37)


In particular, (15.37) holds true if X, Y are our smooth vector fields in Ω , as differ-ential operators in the (non-commutative) algebra T (Ω, R

N) o f the smooth vectorfields on Ω .

As a consequence, (15.36) will follow from (15.37) if we prove the followingfact. After carrying out the n-th power in the second line of (15.36), those termswhere X appears more than j0 times or where Y appears more than k0 times, whenapplied to I and evaluated at x, all together give O(sk0+1) + O(tj0+1), uniformly inx ∈ K .

This is straightforwardly seen. Indeed, in the second line of (15.36) we have onlya finite number of summands (depending on j0, k0) and any summand

((tX)α1(sY )β1 · · · (tX)αi (sY )βi )I (x)

with α1 + · · · + αi > j0 or β1 + · · · + βi > k0 is bounded by

tα1+···+αi sβ1+···+βi supx∈K

∣∣(Xα1Yβ1 · · · Xαi Y βi)I (x)

∣∣

≤{

c0 tj0+1 if α1 + · · · + αi > j0,

c0 sk0+1 if β1 + · · · + βi > k0.

Here c0 depends only on Q0, k0, j0,K and on

max{

supx∈K

|HI (x)| : H ∈ R[X, Y ] monomial with degree ≤ (k0 + j0)2}.

This completes the proof. �Bibliographical Notes. Classical references on the Campbell–Hausdorff formulaare Bourbaki [Bou89], M. Hausner and J.T. Scwartz [HS68], G. Hochschild [Hoc68],N. Jacobson [Jac62], V.S. Varadarajan [Var84].

Some applications of this tool to analysis are discussed, for example, in [Hor67,RS76,VSC92], see also the paper [GG90], already mentioned in Section 14.1.

We would also like to cite [DST91,Egg93,Oki95,Ote91,Str87,Tho82] for otherremarkable applications, referring the reader to the references therein for further de-tails.


Ex. 1) Given a smooth vector field X on RN and a point x ∈ R

N , we recall that byexp(X)(x) we mean the point reached at unit time (if this point exists) by theintegral curve of X starting at x. In (1.7) of Chapter 1 (page 7), we found theexponential-type expansion


exp(X)(x) = I (x) + XI (x) + 1

2! X2I (x) + · · · . (15.38)

Suppose that there exists a dilation δλ on RN of the form δλ(x) =

δλ(x1, . . . , xN) = (λσ1x1, . . . , λσN xN) with 0 < σ1 ≤ · · · ≤ σN such that X

is δλ-homogeneous of positive degree. Prove that any integral curve of X ex-ists on the whole R (that is to say that X is complete) and there exists n ∈ N

such that XnI ≡ 0 (here Xn denotes the n-th power of X as a differentialoperator). Deduce that formula (15.38) makes a precise sense, for the sum isfinite.

Ex. 2) By means of the above exercise, find the integral curves of the vector fieldson R

2 given by ∂x , x ∂y and (x − 12 ) ∂y + ∂x both by solving a suitable sys-

tem of ODE’s and by the above formula (15.38). Then check the Campbell–Hausdorff formula (15.3), which, in this case, gives

exp(∂x)(

exp(x ∂y)(x, y)) = exp

(x ∂y + ∂x + 1

2[x ∂y, ∂x]

)(x, y).

The reader is once again invited to appreciate the interchange of the order ofthe two vector fields from one side to the other of the formula

exp(X)(

exp(Y )(x, y)) = exp(Y �X)(x, y) = exp

(Y +X+ 1

2[Y,X]

)(x, y).

Ex. 3) Taking into account the actual expressions (15.12) (page 601) of the first fourterms in the Campbell–Hausdorff formula, verify directly for n = 1, 2, 3, 4the alternating skew-symmetry relation

Zn(A,B) = (−1)n+1 Zn(B,A).

Compare to (15.21) (page 608).Ex. 4) Find the Campbell–Hausdorff composition � up to step five.Ex. 5) Prove Theorem 15.4.1 when k0 = 1 = j0, with direct arguments and without

the aid of the Campbell–Hausdorff formula. The assertion to be proved is thefollowing one.Let X, Y be smooth vector fields on the open set Ω ⊆ R

N , and let K ⊂ Ω

be any fixed compact set. Then there exists a real number r0 > 0 (dependingon K, X, Y ) such that if |s|, |t | < r0, then, for every x ∈ K , we have

exp(sY )(

exp(tX)(x))

= exp

(sY + tX + st

2[X, Y ]

)(x) + O(s2) + O(t2), (15.39)

where the above O’s mean |O(r2)| ≤ c r2 for every |r| ≤ r0 and uniformlyin x ∈ K .


Hint: The Taylor expansion w.r.t. (s, t) up to order 2 of the left-hand sideof (15.39) is9 given by

I (x) + s Y I (x) + t XI (x) + st

2XYI (x) + O(s2) + O(t2).

Prove by a simple computation that this is also the expansion of

exp

(sY + tX + st

2[X, Y ]

)(x).

This can be derived by recalling that, by the Taylor formula with an integralremainder, we have

exp

(sY + tX + st

2[X, Y ]

)(x)

= exp

(r

(sY + tX + st

2[X, Y ]

))(x)

∣∣∣∣r=1

= I (x) +(

sY + tX + st

2[X, Y ]

)I (x)

+ 1

2

(sY + tX + st

2[X, Y ]

)2

I (x)

+ 1

2

∫ 1

0(1 − r)2

(sY + tX + st

2[X, Y ]

)3

I (γs,t,x(r)) dr,

where r �→ γs,t,x(r) is the integral curve of the vector field sY + tX +st2 [X, Y ] starting at x. Notice that the integral remainder in the above farright-hand side is O(s2) +O(t2), whereas the third line of the above expres-sion equals, modulo O(s2) + O(t2),

9 See (15.6); see also Ex. 8 at the end of Chapter 1 where it is proved that if X is a smoothvector field on an open set Ω ⊆ R

N , f ∈ C∞(Ω, RN) and k ∈ N, it holds

(d

d t

)k(f (γ (t))

)= (Xkf )(γ (t)),

whenever γ satisfies γ (t) = XI (γ (t)). Indeed, when k = 1, we have

d

d t

(f (γ (t))

)= 〈(∇f )(γ (t)), γ (t)〉 = 〈(∇f )(γ (t)),XI (γ (t))〉 = (Xf )(γ (t)).

Moreover, arguing by induction and by using the inductive step and the case n = 1 with f

replaced by Xkf , we have

(d

d t

)1+k(f (γ (t))

)=

(d

d t

)(d

d t

)k(f (γ (t))

)=

(d

d t

)(Xkf )(γ (t))

=(X(Xkf )

)(γ (t)) = (X1+kf )(γ (t)).


I (x) + sY I (x) + tXI (x) + st

2(XYI (x) − YXI (x)) + st

2YXI (x)

= I (x) + s Y I (x) + t XI (x) + st

2XYI (x).

This ends the proof.

16

Families of Diffeomorphic Sub-Laplacians

Let G be a homogeneous Carnot group, and let ΔG = ∑mi=1 X2

i be the canonicalsub-Laplacian on G. Suppose it is given a positive-definite symmetric matrix A =(ai,j )1≤i,j≤m. We consider the second order operator modeled on the matrix A andthe vector fields Xi’s

LA :=m∑

i,j=1

ai,j XiXj . (16.1)

For example, in the classical case G = (RN,+),

ΔA =N∑

i,j=1

ai,j ∂xi∂xj

is a constant coefficient second order operator of elliptic type, whereas Δ =∑Ni=1(∂xi

)2 is the usual Laplace operator. It is well known that a linear change ofcoordinates in R

N transforms the above operator ΔA into Δ. Thus, the operator ΔA

is “equivalent” to the operator Δ (in a new system of coordinates).Following the above naive idea, in order to study the operator LA in (16.1), it

is natural to ask the following question. Does there exist a diffeomorphism T =TA : G → G such that, in the new coordinate system defined by T , the operator(16.1) is turned1 into ΔG?

Unfortunately, when the Xi’s have not constant coefficients, classical changes ofbasis may fail to apply, and simple examples (provided in this chapter) show that T

may not exist at all. Then, to face the above question, it seems natural to make anadditional assumption on the group G. Indeed, we shall prove the existence of sucha diffeomorphism T whenever G is a free Carnot group.

Finally, in Section 16.4, we present an example of application to PDE’s of thetopics developed in this chapter.

1 We say that T turns LA into ΔG if LA(u ◦ T ) = (ΔGu) ◦ T for every u ∈ C∞(G).

622 16 Families of Diffeomorphic Sub-Laplacians

16.1 An Isomorphism Turning the Operator∑

i,j ai,j XiXj intothe Canonical Sub-Laplacian ΔGGG

Let G be a homogeneous Carnot group, and let

ΔG =m∑

i=1

X2i

be the canonical2 sub-Laplacian on G. Suppose it is given a positive-definite sym-metric matrix A = (ai,j )1≤i,j≤m. We then consider the following new second orderdifferential operator modeled on the matrix A and the vector fields Xi’s

LA :=m∑

i,j=1

ai,j XiXj . (16.2)

This is exactly what is done in the classical case G = (RN,+) when, starting fromthe classical Laplace operator

Δ :=N∑

i=1

(∂

∂ xi

)2

,

one considers the following constant coefficient second order operator of elliptic type

ΔA =N∑

i,j=1

ai,j

∂2

∂ xi ∂ xj

.

It is well known and easy to prove that a linear change of coordinates in RN (natu-

rally related to the matrix A−1/2) transforms the above operator ΔA into Δ. Indeed,consider the new coordinates in R

N given by

y = T (x) := C x, where C is a symmetric matrix such that C2 = A−1,

i.e. C = B−1 is the inverse of the symmetric (square root of A) matrix B such thatB2 = A. With this choice, for every u ∈ C∞(RN), u = u(y), we have

2 We recall that the Jacobian basis of g (=algebra of G) is the basis of vector fields in g

agreeing at the origin with the coordinate partial derivatives of RN ≡ G. Then, the canon-

ical sub-Laplacian on G is the second order differential operator ΔG = ∑mi=1 X2

i, where

X1, . . . , Xm are the first m vector fields of the Jacobian basis for g. Here m = N1 is the di-mension of the first layer of the stratification of g or, equivalently, m = N1 is the dimensionof the layer of coordinates where the dilation group δλ of G

δλ(x) = δλ(x(1), x(2), . . . , x(r)) = (λx(1), λ2x(2), . . . , λrx(r)), x(i) ∈ RNi ,

acts like the multiplication times λ.

16.1 An Isomorphism Turning∑

i,j ai,j XiXj into ΔG 623

ΔA

(u(T (x))

) =N∑

i,j=1

ai,j ∂xi

(N∑

h=1

ch,j (∂yhu)(T (x))

)

=N∑

h,k=1

(N∑

i,j=1

ai,j ch,j ck,i

)(∂2

yk,yhu)(T (x))

=N∑

h,k=1

(tC · A · C)k,h (∂2yk,yh

u)(T (x))

=N∑

h=1

(∂2yh,yh

u)(T (x)) = (Δu)(T (x)),

since C is symmetric and C · A · C = B−1 · B2 · B−1 = IN , the identity matrix oforder N . Thus, the operator ΔA is “equivalent” to the operator Δ (in a new systemof coordinates).

Following the above naive idea, in order to study the operator LA in (16.2), it isnatural to ask the following question.

Does there exist a diffeomorphism T = TA : G → G such that, in the newcoordinate system defined by T , the operator (16.2) is turned into ΔG?

Here, we said that T turns LA into ΔG if

LA(u ◦ T ) = (ΔGu) ◦ T

for every u ∈ C∞(G). A further justification for this natural question is given by thefollowing assertion.

Proposition 16.1.1 ({LA : A} and {L}). Any sub-Laplacian on G is of the form(16.2) for a suitable symmetric positive definite matrix A. Vice versa, any operatorLA in that form is a sub-Laplacian

LA =m∑

k=1

Y 2k with Yk = ∑m

j=1(A1/2)k,j Xj . (16.3)

Proof. First, suppose that L = ∑mk=1 Y 2

k is a sub-Laplacian on G. By the definitiongiven in Section 1.5 (page 62), this means that {Y1, . . . , Ym} and {X1, . . . , Xm} areboth bases for the first layer of the stratification of the algebra of G. In particular,there exists a non-singular m × m matrix D = (dk,j )k,j such that

Yk =m∑

j=1

dk,j Xj for every k = 1, . . . , m.

Then we havem∑

k=1

Y 2k =

m∑

k=1

(m∑

i=1

dk,iXi

)(m∑

j=1

dk,jXj

)

=m∑

i,j=1

(m∑

k=1

dk,i dk,j

)XiXj =:

m∑

i,j=1

ai,j XiXj = LA,


where we have set

A = (ai,j )i,j :=(

m∑

k=1

dk,i dk,j

)

i,j

= tD · D

(clearly, A is a symmetric positive-definite matrix).Vice versa, if A is a symmetric positive-definite matrix and LA is the opera-

tor (16.2), denoting by B = (bi,j )1≤i,j≤m the symmetric positive-definite matrixsuch that A = B2 (i.e. B = A1/2), one has

LA =m∑

i,j=1

ai,j XiXj =m∑

i,j=1

m∑

k=1

bi,kbk,j XiXj

=m∑

k=1

(m∑

i=1

bk,iXi

)(m∑

j=1

bk,jXj

)=

m∑

k=1

Y 2k ,

where Yk := ∑mj=1 bk,jXj for every k = 1, . . . , m. This proves that LA is a sub-

Laplacian. �Proposition 16.1.1 shows that, looking for a change of coordinates T turning LA

into ΔG, it is natural to look for a T turning

Yk =m∑

j=1

(A1/2)k,j Xj

into Xk , for k = 1, . . . , m. In the simple case when Xi = ∂/∂xi , we showed at thebeginning of the section that the problem always has a solution via a linear change ofcoordinates in R

N . However, when the Xi’s have not constant coefficients, classicalchanges of basis may fail to apply. Moreover, and this is the most striking fact, simpleexamples (see Remark 16.2.2 and Remark 16.2.3 in the next section) show that T

may not exist in the general case! Some further assumptions either on the coefficientmatrix A or on the structure of the group G must be made.

To face this question, rather than restricting the form of the matrix A, it seemsmore natural to make an additional assumption on the group G. Indeed, we shallprove the existence of such a diffeomorphism T , whenever G is a free Carnot group(we recall below the relevant definition; see also Chapter 14, page 577). The freegroup setting turns out to be the most natural one in order to generalize the classicalEuclidean case. Though, we stress that T is not necessarily a linear application, inthe general case (see Remark 16.2.1 in the next section).

We now recall the definition of fm,r , the free nilpotent Lie algebra of step r with m

(≥ 2) generators x1, . . . , xm. By definition, fm,r is the unique (up to isomorphism)nilpotent Lie algebra of step r generated by m of its elements x1, . . . , xm such that,for every nilpotent Lie algebra n of step r and for every map ϕ from {x1, . . . , xm}to n, there exists a (unique) Lie algebra morphism ϕ from fm,r to n extending ϕ.(See Chapter 14 for more details.) We say that the Carnot group G is a free Carnot



group if its Lie algebra g is isomorphic to fm,r , for some m and r . RN equipped with

the ordinary Abelian structure is an example of free Carnot group. The Heisenberggroup H

1 is also a free Carnot group, while Hn is not free for any n ≥ 2, as can be

seen by a dimensional argument.We are now in the position to prove the main result of this chapter.

Theorem 16.1.2 (Equivalence of sub-Laplacians on free groups). Let G be a freehomogeneous Carnot group, and let A be a given positive-definite symmetric matrix.Let X1, . . . , Xm denote the first vector fields in the Jacobian basis related to G, i.e.

ΔG =m∑

k=1

X2k

is the canonical sub-Laplacian on G. Finally, let the Yk’s be defined by

Yk =m∑

j=1

(A1/2)k,j Xj , k = 1, . . . , m. (16.4)

Consider the related (non-canonical) sub-Laplacian

LA =m∑

k=1

Y 2k =

m∑

i,j=1

ai,j XiXj .

Then there exists a Lie group automorphism TA of G such that

Yk(u ◦ TA) = (Xku) ◦ TA, k = 1, . . . , m, (16.5a)

LA(u ◦ TA) = (ΔGu) ◦ TA (16.5b)

for every smooth function u : G → R. Moreover, TA has polynomial componentfunctions and commutes with the dilations of G.

Proof. First of all, we remark that (16.5b) is an immediate consequence of (16.5a)since LA = ∑m

k=1 Y 2k whereas ΔG = ∑m

k=1 X2k .

On the other hand, by the very definition (16.4) of Yk , (16.5a) reads(

m∑

j=1

(A1/2)k,j Xj

)(u ◦ TA) = (Xku) ◦ TA, k = 1, . . . , m,

which, in turn, is trivially equivalent to

Xk(u ◦ TA) =m∑

j=1

(A−1/2)k,j (Xju) ◦ TA, k = 1, . . . , m. (16.6)

Moreover, if we already knew that TA (denoted henceforth simply by T ) is a Liegroup morphism, then it would be sufficient to prove that (16.6) holds at the origin.Indeed, suppose that


Xk(u ◦ T )(0) =m∑

j=1

bk,j (Xju)(0) ∀ u ∈ C∞(RN), k = 1, . . . , m, (16.7)

where we have set for brevity B = (bk,j )k,j = A−1/2. We fix y ∈ G, v ∈ C∞(RN)

and apply (16.7) tou(x) = v(T (y) ◦ x).

Since T is a Lie group morphism and Xi is left-invariant, we have

Xk

(v ◦ T

)(y) = Xk

(v(T (y ◦ ·)))(0) = Xk

(v(T (y) ◦ T (·)))(0)

=m∑

j=1

bk,j Xj

(v(T (y) ◦ ·))(0) =

m∑

j=1

bk,j (Xjv)(T (y)),

i.e. (16.6) holds.We now turn to show (16.7). We first observe that, by the chain rule and since B

is symmetric, (16.7) is equivalent to

JT (0) (XI)(0) = (XI)(0) B. (16.8)

Here we have denoted by I the identity map on G, by XI = (X1I · · ·XmI

)the

(N × m)-matrix whose i-th column is XiI and by JT the Jacobian matrix of T .We thus aim to construct a Lie group automorphism T of G satisfying (16.8).

This will complete the proof of the first statement of the theorem. Consider the linearmap defined as follows

ϕ : span{X1, . . . , Xm} → span{X1, . . . , Xm}, Xi �→m∑

j=1

bi,j Xj .

Since X1, . . . , Xm are linearly independent, ϕ is well posed. Moreover, B being anon-singular matrix, ϕ is a bijective linear map. Since G is a free Carnot group andits Lie algebra g is nilpotent of step r and generated by {X1, . . . , Xm}, clearly g isisomorphic to fm,r . Then, by Lemma 14.1.4 (page 578), there exists a unique Liealgebra automorphism ϕ : g → g extending ϕ. For the simplicity of notation, wealso denote ϕ by ϕ. We are now in the position to define T . We set

T : GLog−→ g

ϕ−→ gExp−→ G, T := Exp ◦ ϕ ◦ Log , (16.9)

where Exp denotes the exponential map and Log its inverse function. Clearly, T isbijective,

T −1 = Exp ◦ ϕ−1 ◦ Log ,

whence both T and T −1 are C∞ maps. Moreover, T is a Lie group automorphismof G: indeed, when g is equipped with the Campbell–Hausdorff3 group law , thenExp , Log and any Lie algebra morphism of g are Lie group morphisms. Indeed, the

3 See Section 14.2, page 584, for the definition and the properties of .



fact that Exp and Log are Lie-algebra morphisms between (g, ) and (G, ◦) is aconsequence of Theorem 14.2.1 (page 586). The fact that any Lie-algebra morphismof (g, [·, ·]) into itself is actually a Lie-group morphism of (g, ) into itself is easilyseen.4

Let us prove the matrix equality (16.8). We recall that, by definition,

(XI)(0) =(

Im

0

)

where Im is the identity matrix of order m and 0 is the null matrix of order (N −m)×m. If g is equipped with the well-known Jacobian basis, we have (see, for instance,the arguments in Remark 1.3.30, page 50)

JLog (0) = (JExp (0))−1 = IN,

whenceJT (0) = JExp (0)Jϕ(0)JLog (0) = Jϕ(0).

Thus, in order to prove (16.8), it is enough to prove that the (N × m)-matrix of thefirst m-columns of Jϕ(0) is equal to

(Im

0

)B =

(B

0

),

which straightforwardly follows from the definition of ϕ and from the fact thatX1, . . . , Xm are the first m vectors of the Jacobian basis.

We now turn to the proof of the last statement of the theorem. We recall thatExp and Log have polynomial components (see Theorem 1.3.28, 49). Hence T haspolynomial component functions since ϕ is linear. Finally, we prove that T commuteswith δλ, the dilations on G. Recalling that δλ has the form


Ni ,

let us accordingly denote by

Z(1)1 , . . . , Z

(1)N1

, . . . , Z(r)1 , . . . , Z

(r)Nr

the Jacobian basis of g, and let δλ also denote the map

4 Indeed, let ϕ : g → g be a Lie-algebra morphism, i.e. ϕ([X,Y ]) = [ϕ(X), ϕ(Y )] for everyX, Y ∈ g. Then, comparing to the definition (14.2) of the operation, we have

ϕ(X Y ) = ϕ

(X + Y + 1

2[X,Y ] + 1

12[X, [X,Y ]] + · · ·

)

= ϕ(X) + ϕ(Y ) + 1

2[ϕ(X), ϕ(Y )] + 1

12[ϕ(X), [ϕ(X), ϕ(Y )]] + · · ·

= ϕ(X) ϕ(Y ).

This amounts to say that ϕ is a Lie-group morphism of (g, ) into itself.


g �r∑

i=1

Ni∑

j=1

ξ(i)j Z

(i)j �→

r∑

i=1

Ni∑

j=1

λiξ(i)j Z

(i)j ∈ g.

With this notation, both Exp and Log commute with δλ (see (1.71), page 49, inTheorem 1.3.28). Thus, we only have to show

ϕ ◦ δλ = δλ ◦ ϕ.

Now, Z(k)j is a δλ-homogeneous vector field of degree k, hence in particular Z

(k)j

can be expressed as a homogeneous Lie polynomial of degree k in the generatorsX1, . . . , Xm. By the definition of the Lie algebra morphism ϕ, ϕ(Z

(k)j ) is a homoge-

neous Lie polynomial of degree k in X1, . . . , Xm as well. Hence, there exist scalarsc(k)i,j such that

ϕ(Z(k)j ) =

Nk∑

i=1

c(k)i,j Z

(k)i .

Consequently, for every k = 1, . . . , r and j = 1, . . . , Nk , we have

(ϕ ◦ δλ)(Z(k)j ) = ϕ(λk Z

(k)j ) = λkϕ(Z

(k)j ) = λk

Nk∑

i=1

c(k)i,j Z

(k)j

=Nk∑

i=1

c(k)i,j λkZ

(k)j =

Nk∑

i=1

c(k)i,j δλ(Z

(k)j ) = δλ

(Nk∑

i=1

c(k)i,j Z

(k)j

)

= (δλ ◦ ϕ)(Z(k)j ).

This completes the proof. �We explicitly remark that, in the proof of Theorem 16.1.2, we have proved that

the diffeomorphism T has the following form

T : GLog−→ g

ϕ−→ gExp−→ G,

T := Exp ◦ ϕ ◦ Log ,

where ϕ is a linear map which, w.r.t. the Jacobian basis, is related to a block-diagonalmatrix of the type

⎛

⎜⎜⎝

C(1) 0 · · · 00 C(2) · · · 0...

.... . .

...

0 0 · · · C(r)

⎞

⎟⎟⎠ , where C(k) = (c(k)i,j

)1≤i,j≤Nk

.

16.2 Examples and Counter-examples

In this section, we give some examples and counter-examples concerning with thetopics of the previous section. We follow all the notation of Section 16.1.

16.2 Examples and Counter-examples 629

Remark 16.2.1. The diffeomorphism TA constructed in Theorem 16.1.2 may fail tobe linear.

Proof. We consider the free homogeneous Carnot group G = (R3, ◦, δλ), where

x ◦ y = (x1 + y1, x2 + y2, x3 + y3 + x1 y2)

andδλ(x) = (λ x1, λ x2, λ

2 x3).

The Jacobian basis for g is given by

X1 = ∂1, X2 = ∂2 + x1 ∂3, X3 = [X1, X2] = ∂3.

Moreover, we have

Exp (ξ1 X1 + ξ2 X2 + ξ3 X3) =(

ξ1, ξ2, ξ3 + 1

2ξ1 ξ2

),

Log (x) = x1 X1 + x2 X2 +(

x3 − 1

2x1 x2

)X3.

Let now B = (bi,j )i,j≤2 be an assigned symmetric matrix. Since G is free, thereexists a unique Lie algebra morphism ϕ from g into itself which maps the generatorsX1, X2 respectively in

b1,1 X1 + b1,2 X2, b1,2 X1 + b2,2 X2.

In Jacobian coordinates, ϕ is represented by the block-diagonal matrix

(b1,1 b1,2 0b1,2 b2,2 0

0 0 det(B)

).

We now setT = Exp ◦ ϕ ◦ Log .

A direct computation gives

T (x) =⎛

⎝b1,1 x1 + b1,2 x2b1,2 x1 + b2,2 x2

det(B) x3 + 12 b1,1 b1,2 x2

1 + 12 b1,2 b2,2 x2

2 + b21,2 x1 x2

⎞

⎠ .

In particular, if we choose

b1,1 = 2, b1,2 = b2,1 = 1, b2,2 = 4,

we obtain

T (x) = (2 x1 + x2, x1 + 4 x2, 7 x3 + x21 + 2 x2

2 + x1 x2).


With the notation in the proof of Theorem 16.1.2, we see that, choosing

A = B−2 =( + 17

49 − 649

− 649 + 5

49

),

the diffeomorphism T = TA turns

Y1 = 47 X1 − 1

7 X2 and Y2 = − 17 X1 + 2

7 X2

respectively into X1 and X2. As a consequence, TA turns the sub-Laplacian

LA = 17

49X2

1 − 6

49X1X2 − 6

49X2X1 + 5

49X2

2

into the canonical sub-Laplacian

ΔG = X21 + X2

2.

We remark that TA is not linear. �We observe that the group G in the above remark is isomorphic to H

1. However,on H

1, TA is always linear. This follows from the fact that, in H1, the map Exp is

linear (with the obvious choices of basis in H1 and in its Lie algebra).

Remark 16.2.2. If G is not free, a diffeomorphism TA satisfying (16.5a) of Theo-rem 16.1.2 may not exist.

Proof. We consider the group G = H2, the Heisenberg group on R

5. If the points ofH

2 are denoted by (a, b, c), with a, b ∈ R2, c ∈ R, we have

(a, b, c) ◦ (α, β, γ ) = (a + α, b + β, c + γ + 2〈b, α〉 − 2〈a, β〉)and

δλ(a, b, c) = (λ a, λ b, λ2c),

and the canonical sub-Laplacian on H2 is given by

2∑

j=1

(A2j + B2

j ),

whereAj = ∂aj

+ 2 bj ∂c, Bj = ∂bj− 2 aj ∂c, j = 1, 2.

(H2, ◦, δλ) is then a homogeneous Carnot group of step 2 with 4 generators. Thedimension of the Lie algebra of H

2 is 5, whereas

dim(f4,2) = 10,

hence H2 is not free. We observe that there does not exist any diffeomorphism on H

2

turning the set of vector fields


Y = {A1 + B2, A2, B1, B2}into the set

X = {A1, A2, B1, B2}.Indeed, each vector field in X has exactly one non-vanishing commutator with any

other vector field in X ; on the contrary, A1 + B2 ∈ Y has two non-vanishing com-mutators with the other vector fields in Y . �

According to Remark 16.2.2 (we follow the notation therein), we may expect thatthe two sub-Laplacians on H

2

(A1 + B2)2 + A2

2 + B21 + B2

2 and A21 + A2

2 + B21 + B2

2

cannot be turned into each other, thus furnishing an example of two sub-Laplacianson the same group which are not “equivalent”. However, since (16.5a) is only suffi-cient but not necessary to deduce (16.5b), further comments must be made: these aregiven in the next theorem.

Theorem 16.2.3. If G is not free, a diffeomorphism TA satisfying (16.5b) of Theo-rem 16.1.2 may not exist.

Proof. The proof of this result makes use of some Liouville-type theorems for sub-Laplacians proved in Section 5.8, page 269. We shall exhibit a rather elaboratedcounter-example.

First of all, if A = (ai,j )i,j≤m is a fixed symmetric matrix and the Xi’s aresmooth vector fields, we provide a lemma characterizing the maps T turning theoperator

∑i,j ai,j XiXj into the operator

∑i X2

i .

Lemma 16.2.4. Suppose A = (ai,j )i,j≤m is a fixed symmetric matrix and

Xi =N∑

k=1

α(i)k (x) ∂k, i = 1, . . . , m,

are smooth vector fields on RN . Then a map T ∈ C2(RN, R

N) satisfies

m∑

i,j=1

ai,j XiXj (u ◦ T ) =(

m∑

i=1

X2i u

)◦ T ∀ u ∈ C∞(RN) (16.10)

(i.e. T turns the operator∑

i,j ai,j XiXj into the operator∑

i X2i ) if and only if the

following system of quasi-linear PDE’s is satisfied⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

m∑

i,j=1

ai,j XiXjTk =m∑

i=1

(Xiα(i)k ) ◦ T ∀ k = 1, . . . , N ,

m∑

i,j=1

ai,j XiTl XjTk =m∑

i=1

(α(i)k α

(i)l ) ◦ T ∀ k, l = 1, . . . , N .

(16.11)


Proof. The left-hand side of (16.10) is equal to

N∑

k,l=1

(∑

i,j

ai,j XiTl XjTk

)· (∂k,lu) ◦ T +

N∑

k=1

(∑

i,j

ai,j XiXjTk

)· (∂ku) ◦ T .

With the position Xi = ∑Nk=1 α

(i)k (x) ∂k , the right-hand side of (16.10) equals

N∑

k,l=1

(m∑

i=1

(α(i)k α

(i)l ) ◦ T

)· (∂k,lu) ◦ T +

N∑

k=1

(m∑

i=1

(Xiα(i)k ) ◦ T

)· (∂ku) ◦ T .

Now, if we put u(x) = xk in (16.10) and equate the left-hand and right-hand sidesof (16.10) expressed above, we obtain

m∑

i,j=1

ai,j XiXjTk =m∑

i=1

(Xiα(i)k ) ◦ T ∀ k = 1, . . . , N. (16.12)

Moreover, if we put u(x) = xk xl in (16.10), and (16.12) holds, we get (recall thatai,j = aj,i)

m∑

i,j=1

ai,j XiTl XjTk =m∑

i=1

(α(i)k α

(i)l ) ◦ T ∀ k, l = 1, . . . , N. (16.13)

This proves that (16.10) is equivalent to (16.11). �We explicitly remark that, if we set

J XT := (XjTi)1≤i≤N, 1≤j≤m (the “X-Jacobian” matrix of the map T )

and if we consider the N × m matrix X whose j -th column is given by the N -tupleof the coefficient functions of XjI , i.e.

X := (X1I · · · XmI) = (α

(j)i

)1≤i≤N, 1≤j≤m

, (16.14)

then the second equation in (16.11) turns out to be equivalent to

J XT · A · tJ X

T = (X · tX

) ◦ T . (16.15)

We now return to the proof of Theorem 16.2.3. We consider the special caseof the Heisenberg group H

2 (whose points are denoted by (x1, . . . , x5)), with thenatural choice of the fields X1, . . . , X4, i.e.

X1 := ∂1 + 2 x3 ∂5, X2 := ∂2 + 2 x4 ∂5,

X3 := ∂3 − 2 x1 ∂5, X4 := ∂4 − 2 x2 ∂5.

We aim to prove that there does not exist any map T ∈ C2(H2, H2) satisfying

(16.10), for a suitable matrix A to be specified in the sequel (henceforth, we write


LA := ∑mi,j=1 ai,j XiXj ). To this end, we suppose by contradiction that there exists

such a T .Obviously, it is not restrictive to suppose that T (0) = 0 (since X1, . . . , X4 are

left-translation invariant). With the above choice of X1, . . . , X4 (and following thenotation in (16.14)), one has

X · tX =

⎛

⎜⎜⎜⎝

1 0 0 0 2 x30 1 0 0 2 x40 0 1 0 −2 x10 0 0 1 −2 x2

2 x3 2 x4 −2 x1 −2 x2 4(x21 + x2

2 + x23 + x2

4)

⎞

⎟⎟⎟⎠ . (16.16)

Hence, the first equation in (16.11) becomes

LATk =4∑

i,j=1

ai,j XiXjTk = 0 ∀ k = 1, . . . , 5. (16.17)

The second equation in (16.11) gives, in particular,

4∑

i,j=1

ai,j XiTh XjTh = 1 ∀ h = 1, . . . , 4, (16.18a)

4∑

i,j=1

ai,j XiT5 XjT5 = 4 (T 21 + T 2

2 + T 23 + T 2

4 ). (16.18b)

Moreover, from (16.17) and (16.18a) we easily obtain

LA

((Tk)

2) = 2 Tk LATk + 24∑

i,j=1

ai,j XiTk XjTk = 2. (16.19)

Suppose now that A = (ai,j )i,j≤4 is a symmetric positive-definite matrix. Then,by Proposition 16.1.1, LA is a sub-Laplacian on H

2. In particular, we are in theposition to apply the Liouville-type theorems for sub-Laplacians in Section 5.8(page 269), Chapter 5.

By the hypoellipticity of LA, from (16.17) it follows that Tk ∈ C∞(RN) for everyk = 1, . . . , 5. Let now k ∈ {1, . . . , 4} be fixed. We note that (Tk)

2 is a non-negativefunction such that

LA((Tk)2)

is a polynomial of H2-degree5 zero (see (16.19)). As a consequence, by Liouville

Theorem 5.8.4 (page 270), (Tk)2 is a polynomial function of H

2-degree at most 2for every k = 1, . . . , 4. In particular, this proves that Tk is bounded from below bya polynomial function of H

2-degree at most 1. Hence, by Theorem 5.8.2, page 270

5 For the definition of G-degree, see Definition 1.3.3, page 33.


(being LATk = 0), it follows that, for any k = 1, . . . , 4, Tk is a polynomial of H2-

degree at most 1. Having supposed that T (0) = 0, there exists a constant matrixB = (bi,j )i,j≤4 such that ⎛

⎜⎝

T1T2T3T4

⎞

⎟⎠ = B ·⎛

⎜⎝

x1x2x3x4

⎞

⎟⎠ . (16.20)

From (16.17) and (16.18b) we obtain

LA

((T5)

2) = 2 T5 LAT5 + 24∑

i,j=1

ai,j XiT5 XjT5 = 8 (T 21 + T 2

2 + T 23 + T 2

4 ).

From (16.20) we then infer that LA((T5)2) is a polynomial function of H

2-degree atmost 2. Arguing as above, we derive that T5 is a polynomial function of H

2-degree atmost 2. Then there exist c5 ∈ R, a 1×4 constant matrix D = (di)i≤4 and a symmetric4 × 4 constant matrix C = (ci,j )i,j≤4 such that (setting x = t (x1, x2, x3, x4))

T5(x) = c5 x5 + D · x + t x · C · x.

First, we show that the entries of D are zero. From (16.20) we obtain

XiTl = ∂iTl = bl,i

for every l = 1, . . . , 4 and i = 1, . . . , 4. Hence, if we take 1 ≤ l ≤ 4 and 1 ≤ k ≤ 4in the second equation of (16.11), then we derive (see also (16.16))

B · A · tB = I4 (16.21)

(In denoting henceforth the m × n identity matrix). In particular, we notice that B isnon-singular. If in the second equation of (16.11) we take l ∈ {1, . . . , 4} and k = 5,we obtain

B · A ·⎛

⎜⎝

X1T5X2T5X3T5X4T5

⎞

⎟⎠ =⎛

⎜⎝

2 T32 T4

−2 T1−2 T2

⎞

⎟⎠ .

By evaluating both sides of this last identity at 0 (and recalling the form of T5), onehas

(XiT5)(0) = (∂iT5)(0) = di,

hence B · A · D = 0. Since A and B are non-singular, it follows D = 0.Consequently, T5 is a homogeneous polynomial of the form

T5(x) = c5 x5 + t x · C · x, where tC = C, x = t (x1, x2, x3, x4). (16.22)

By writing (16.15) in a more compact form and making use of (16.20) and (16.22),one gets


J XT =

(B

2 t x · t C

),

where we have set

C := C + c5 I and I :=(

0 I2−I2 0

).

Thus, with a block notation, we infer

J XT · A · tJ X

T =(

B · A · tB 2 B · A · C · x

2 t x · t C · A · tB 4 t x · t C · A · C · x

).

Again from (16.20) and (16.16), we also recognize that

(X · tX) ◦ T =(

I4 2 I · B · x

2 t x · tB · t I 4 t x · tB · t I · I · B · x

).

Collecting together all the above facts, we have proved that (16.15) is equivalent to⎧⎪⎪⎪⎨

⎪⎪⎪⎩

B · A · tB = I4,

2 B · A · C · x = 2 I · B · x,

2 t x · t C · A · tB = 2 t x · tB · t I ,

4 t x · t C · A · C · x = 4 t x · tB · t I · I · B · x.

The third equation follows from the second one by transposing, whereas the fourthone follows from the first and the second equations. As a consequence, (16.15) turnsout to be equivalent to {

B · A · tB = I4,

2 B · A · C = 2 I · B,

or equivalently, {B · A · tB = I4,

C = tB · I · B.(16.23)

We now explicitly observe that I is skew-symmetric, then the same is true of tB ·I ·B.From the second equation in (16.23) we infer that the matrix

C =⎛

⎜⎝

c11 c12 c13 + c5 c14c12 c22 c23 c24 + c5

c13 − c5 c23 c33 c34c14 c24 − c5 c34 c44

⎞

⎟⎠

must be skew-symmetric. This is possible if and only if C = 0. Thus, (16.23) isequivalent to {

B · A · tB = I4,

c5 I = tB · I · B.(16.24)

From the second equation of (16.24) it follows that c5 �= 0 (thus, T is bijective!).


Finally, we choose the symmetric positive-definite matrix A of the followingform: A = t S · S, where S is a suitable non-singular matrix. From the first equationin (16.24) we then have

(B · S) · t (B · S) = I4,

i.e. B = O · S−1, with an orthogonal matrix O. In particular, the second equation of(16.24) becomes

t S · I · S = 1

c5

tO · I · O, where O is an orthogonal matrix. (16.25)

We choose

S =⎛

⎜⎝

1 0 0 10 1 0 00 0 1 00 0 0 1

⎞

⎟⎠ .

Thus, (16.25) is equivalent to⎛

⎜⎝

0 0 1 00 0 0 1

−1 0 0 −10 −1 1 0

⎞

⎟⎠ = 1

c5

tO · I · O, where O is an orthogonal matrix.

(16.26)In particular, the eigenvalues of these matrices should be equal. But the eigenvaluesof 1

c5

tO · I ·O (being O orthogonal) are ± 1c5

i, whereas the eigenvalues of the matrix

in the left-hand side of (16.26) are ±i

√3±√

52 . This gives a contradiction, proving

that, with the above choice of S and consequently of A, it does not exist any mapT ∈ C2(H2, H

2) satisfying (16.10). This ends the proof of Theorem 16.2.3. �Remark 16.2.5. In the proof of Theorem 16.2.3, we explicitly proved that the opera-tor LA = ∑

i,j ai,j XiXj on H2, where

A = t S · S =⎛

⎜⎝

1 0 0 00 1 0 00 0 1 01 0 0 1

⎞

⎟⎠ ·⎛

⎜⎝

1 0 0 10 1 0 00 0 1 00 0 0 1

⎞

⎟⎠ =⎛

⎜⎝

1 0 0 10 1 0 00 0 1 01 0 0 2

⎞

⎟⎠ ,

cannot be turned into the canonical sub-Laplacian ΔH2 = ∑i X2

i by any C2-map T .However, by Proposition 16.1.1, we know that LA is (exactly as ΔH2 ) a sub-Laplacian on H

2. Since the symmetric square root B of A is

B =

⎛

⎜⎜⎝

2√5

0 0 1√5

0 1 0 00 0 1 01√5

0 0 3√5

⎞

⎟⎟⎠ ,

again from the result in Proposition 16.1.1 we know that LA is equal to the followingsub-Laplacian

16.3 Canonical or Non-canonical? 637

LA =m∑

k=1

Y 2k , with Yk =

m∑

j=1

(A1/2)k,j Xj ,

i.e.

Y1 = 2√5

X1 + 1√5

X4, Y2 = X2, Y3 = X3, Y4 = 1√5

X1 + 3√5

X4.

This proves that the following two sub-Laplacians on H2 cannot be turned into each

other by any C2-map:

ΔH2 = (∂1 + 2 x3 ∂5)2 + (∂2 + 2 x4 ∂5)

2 + (∂3 − 2 x1 ∂5)2 + (∂4 − 2 x2 ∂5)

2,

LA =(

2√5

∂1 + 1√5

∂4 +(

4√5

x3 − 2√5

x2

)∂5

)2

+ (∂2 + 2 x4 ∂5)2 + (∂3 − 2 x1 ∂5)

2

+(

1√5

∂1 + 3√5

∂4 +(

2√5

x3 − 6√5

x2

)∂5

)2

= X21 + X2

2 + X23 + 2 X2

4 + X1 X4 + X4 X1

= (∂1 + 2 x3 ∂5)2 + (∂2 + 2 x4 ∂5)

2 + (∂3 − 2 x1 ∂5)2 + 2 (∂4 − 2 x2 ∂5)

2

+ 2 (∂1 + 2 x3 ∂5) (∂4 − 2 x2 ∂5).

However, another natural question still arises: is it possible to create a new Carnotgroup G on R

5, (different but) diffeomorphic to H2, such that its canonical sub-

Laplacian ΔG is turned into (i.e. equivalent to) the non-canonical LA? We answer tothis question in the next section.

16.3 Canonical or Non-canonical?

In this section, we deal with the following problem. Let G = (RN, ◦, δλ) be a homo-geneous Carnot group with an assigned (not necessarily canonical) sub-Laplacian L.We know that there may not exist any C2-map on G itself turning L into the canoni-cal sub-Laplacian ΔG of G (see, for instance, Theorem 16.2.3). We aim to show thatthere exists a different homogeneous Carnot group H = (RN, ∗, δλ) (with the sameunderlying manifold R

N and the same group of dilations δλ) and a Lie-group isomor-phism from G to H turning L into the canonical sub-Laplacian ΔH. Roughly speak-ing, even if we are given a non-canonical sub-Laplacian L, up to an isomorphism ofLie groups, we can consider L to be equivalent to a canonical sub-Laplacian, moduloa change of the Lie group.

Let us denote by L = ∑mj=1 Y 2

j a given sub-Laplacian on G and by ΔG =∑m

j=1 X2j the canonical sub-Laplacian on G. By definition, there exists a non-

singular matrixB = (bi,j )i,j≤m


such that

Yj =m∑

i=1

bi,j Xi for every j = 1, . . . , m. (16.27)

With reference to the particular form

δλ(x) = δλ(x(1), x(2), . . . , x(r)) = (λx(1), λ2x(1), . . . , λrx(r)), x(i) ∈ R

Ni ,

of the dilations on G, we introduce a block-diagonal matrix

C =

⎛

⎜⎜⎝

C(1) 0 · · · 0

0 C(2) . . ....

.... . .

. . . 00 · · · 0 C(r)

⎞

⎟⎟⎠ , (16.28)

where C(i) is a matrix of order Ni × Ni , for every i = 1, . . . , r , and we choose C(i)

in such a way that

C(1) = B−1 and C(2), . . . , C(r) are non-singular. (16.29)

In the sequel, we also denote by C the linear map

C : RN → R

N, x �→ y = C · x.

We now consider a new Lie group H = (RN, ∗), where ∗ is defined by

H × H � (x, y) �→ x ∗ y := C(C−1(x) ◦ C−1(y)

) ∈ H. (16.30)

We claim that:

1) C : (G, ◦) → (H, ∗) is a Lie group isomorphism;2) H = (RN, ∗, δλ) is a homogeneous Carnot group (this is not obvious, since we

know that the homogeneity is not an invariant of equivalence classes of abstractCarnot groups);

3) C turns L on G into the canonical sub-Laplacian ΔH of H.

Now, 1) is trivial (see also Example 2.1.48, page 112): for every ξ, η ∈ G wehave, by (16.30),

C(ξ) ∗ C(η) = C(C−1(C(ξ)) ◦ C−1(C(η))

) = C(ξ ◦ η).

We now prove 2). We must show that:

2’) the first m vector fields of the Jacobian basis of H are Lie-generators of the Liealgebra h of H; and

2”) δλ is a Lie group morphism of (H, ∗).


First of all, we show that

C turns Yj into the j -th vector field of the Jacobian basis of H. (16.31)

We denote by H1, . . . , HN the Jacobian basis of h, the Lie algebra of H. In order toprove (16.31), we have to show that

C turns Yj into Hj for every j = 1, . . . , m.

Now, from Example 2.1.48 (page 112) we know that the map C turns the vectorfield Yj into a ∗-left-invariant vector field Yj on H such that (see (2.17a) in Exam-ple 2.1.25, page 101)

Yj I (y) = JC(C−1(y)) · (Yj I )(C−1(y)).

In particular, we have

Yj I (0) = JC(C−1(0)) · (Yj I )(C−1(0)) = C · (Yj I )(0)

(see (16.27)) = C ·(

m∑

i=1

bi,j XiI (0)

)

= C · (b1,j , . . . , bm,j , 0, . . . , 0)T

= (C(1) · b(1)j , 0(2), . . . , 0(r)),

where b(1)j denotes the j -th column of B. By the choice (16.29) of C(1), this proves

that Yj I (0) is the j -th vector of the canonical basis of RN . Since the same is true of

HjI (0) (by the definition of Jacobian basis) and the left-invariant vector fields aredetermined by their value at the origin, this proves (16.31).

From Theorem 2.1.50 (page 114) we know that Yj = dC(Yj ). Hence, the abovecomputation proves that

dC(Yj ) = Hj . (16.32)

Since X1, . . . , Xm is a system of Lie-generators for the Lie-algebra g of G and theXj ’s are linear combinations of the Yj ’s, then every element of g is a linear combi-nation of commutators of the type

U = [U1[U2[U3[· · · [Un−1, Un] · · ·]]]],with U1, . . . , Un ∈ {Y1, . . . , Ym}. Now, take any H ∈ h; since dC : g → h isa Lie-algebra isomorphism (see Theorem 2.1.50, page 114), then (dC)−1(H) ∈ g

is a linear combination of brackets as the above U . Consequently, H is a linearcombination of elements in h of the type

(dC)(U) = (dC)([U1[U2[U3[· · · [Un−1, Un] · · ·]]]])= [(dC)U1[(dC)U2[(dC)U3[· · · [(dC)Un−1, (dC)Un] · · ·]]]]= [V1[V2[V3[· · · [Vn−1, Vn] · · ·]]]],


where V1, . . . , Vn ∈ {(dC)Y1, . . . , (dC)Ym} = {H1, . . . , Hm} (see (16.32)). Thisproves (2’).

In order to prove (2”), we first remark that δλ commutes with C, i.e.

δλ(C(x)) = C(δλ(x)) ∀ x ∈ RN, ∀ λ > 0. (16.33)

Indeed, following our usual notation, we have

C(δλ(x)) =

⎛

⎜⎜⎝

C(1) 0 · · · 0

0 C(2) . . ....

.... . .

. . . 00 · · · 0 C(r)

⎞

⎟⎟⎠ ·

⎛

⎜⎜⎝

λ x(1)

λ2 x(2)

...

λr x(r)

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

C(1)λ x(1)

C(2)λ2 x(2)

...

C(r)λr x(r)

⎞

⎟⎟⎠

= δλ

⎛

⎜⎜⎝

C(1) x(1)

C(2) x(2)

...

C(r) x(r)

⎞

⎟⎟⎠ = δλ(C(x)).

Obviously, (16.33) implies that δλ commutes with C−1 too. Now, (2”) is equivalentto

δλ(x ∗ y) = (δλ(x)) ∗ (δλ(y)) ∀ x, y ∈ H, ∀ λ > 0,

which follows from the simple argument below

(δλ(x)) ∗ (δλ(y)) = (see (16.30)) C(C−1(δλ(x)) ◦ C−1(δλ(y))

)

= (see (16.33)) C((δλ(C

−1(x))) ◦ (δλ(C−1(y)))

)

(δλ is a Lie group morphism of (G, ◦))

= C(δλ

((C−1(x)) ◦ (C−1(y))

))

= (see (16.33)) δλ

(C

((C−1(x)) ◦ (C−1(y))

))

= (see (16.30)) δλ(x ∗ y).

Finally, (3) is a straightforward consequence of (16.31). �In the next subsection, we shall see an explicit example of non-canonical sub-

Laplacian turning into a canonical one.

Remark 16.3.1. The above computations also prove the following fact. Let (RN,

◦, δλ) be a homogeneous Carnot group. As usual, we denote the points of G by

x = (x(1), . . . , x(r)) with x(i) ∈ RNi

and the dilation group by δλ(x) = (λx(1), . . . , λrx(r)). Let

C(1), . . . , C(r)

be fixed non-singular matrices, with C(i) of dimension Ni × Ni for every i =1, . . . , r . We denote by C the matrix as in (16.28). Then C defines a linear change ofcoordinates on R

N . The expression of ◦ in the new coordinates defined by ξ = C(x)


is precisely the composition ∗ defined in (16.30). Then, H = (RN, ∗, δλ) is a homo-geneous Carnot group isomorphic to G = (RN, ◦, δλ), and C turns a left-invariantvector field X on (G, ◦) into the left-invariant vector field dC(X) on H = (RN, ∗);in particular, dC(X) coincides with C · XI (0) at the origin, i.e.

(dC(X)

)I (0) = C · XI (0). (16.34)

This last fact can be proved as follows. From Example 2.1.48 (page 112) we have(dC(X)

)I (y) = JC(C−1(y)) · (XI)(C−1(y)) = C · (XI)(C−1(y)).

When y = 0, we have (16.34). �

16.3.1 Example of a Non-canonical Sub-Laplacian Turning into a CanonicalOne

In Theorem 16.2.3, we proved that the following sub-Laplacian on H2

L =(

2√5

∂1 + 1√5

∂4 +(

4√5

x3 − 2√5

x2

)∂5

)2

+ (∂2 + 2 x4 ∂5)2 + (∂3 − 2 x1 ∂5)

2

+(

1√5

∂1 + 3√5

∂4 +(

2√5

x3 − 6√5

x2

)∂5

)2

cannot be turned into the canonical sub-Laplacian ΔH2 of H2 itself by any C2-map.

Nonetheless, we have just showed a constructive method which allows us to turn theabove L into the canonical sub-Laplacian on another homogeneous Carnot group.

Indeed (following the notation in Section 16.3), we have G = (R5, ◦, δλ), where

x ◦ y =

⎛

⎜⎜⎜⎝

x1 + y1x2 + y2x3 + y3x4 + y4

x5 + y5 + 2(x3 y1 + x4 y2 − x1 y3 − x2 y4)

⎞

⎟⎟⎟⎠

andδλ(x1, x2, x3, x4, x5) = (λx1, λx2, λx3, λx4, λ

2x5).

Moreover,

L =4∑

j=1

Y 2j , where Yj =

4∑

i=1

bi,j Xi

with

B = (bi,j )i,j≤4 =

⎛

⎜⎜⎝

2√5

0 0 1√5

0 1 0 00 0 1 01√5

0 0 3√5

⎞

⎟⎟⎠ .


More explicitly, since

X1 = ∂1 + 2 x3 ∂5, X2 = ∂2 + 2 x4 ∂5,

X3 = ∂3 − 2 x1 ∂5, X4 = ∂4 − 2 x2 ∂5,

the Yj ’s are given by

Y1 = 2√5

X1 + 1√5

X4, Y2 = X2, Y3 = X3, Y4 = 1√5

X1 + 3√5

X4.

We have

C(1) = B−1 =

⎛

⎜⎜⎝

3√5

0 0 − 1√5

0 1 0 00 0 1 0

− 1√5

0 0 2√5

⎞

⎟⎟⎠ .

Consequently, we define

C :=(

C(1) 00 1

)=

⎛

⎜⎜⎜⎜⎝

3√5

0 0 − 1√5

00 1 0 0 00 0 1 0 0

− 1√5

0 0 2√5

00 0 0 0 1

⎞

⎟⎟⎟⎟⎠

and the relevant linear map

C : R5 → R

5, x �→ y = C x =

⎛

⎜⎜⎜⎝

3√5

x1 − 1√5

x4x2x3

− 1√5x1 + 2√

5x4

x5

⎞

⎟⎟⎟⎠ .

Following (16.30), we equip R5 with a new Lie group law ∗ defined by

y ∗ η := C(C−1(y) ◦ C−1(η)

)

=(

y1 + η1, y2 + η2, y3 + η3, y4 + η4,

y5 + η5 + 2√5

(y3(2η1 + η4) + (y1 + 3y4)η2 − (2y1 + y4)η3

− y2(η1 + 3η4)))

.

We explicitly remark that this Lie group composition law can be written as in (3.6)(page 158) of Section 3.2, i.e.

y ∗ η = (y1 + η1, y2 + η2, y3 + η3, y4 + η4,

y5 + η5 + 〈H(y1, y2, y3, y4), (η1, η2, η3, η4)〉),


where

H = 1√5

⎛

⎜⎝

0 1 −2 0−1 0 0 −32 0 0 10 3 −1 0

⎞

⎟⎠ .

Finally, we observe that the first four vector fields of the Jacobian basis of (R5, ∗)

are given by6

H1 = ∂y1 +(

4√5y3 − 2√

5y2

)∂y5 ,

H2 = ∂y2 +(

2√5y1 + 6√

5y4

)∂y5 ,

H3 = ∂y3 +(

− 4√5y1 − 2√

5y4

)∂y5 ,

H4 = ∂y4 +(

2√5y3 − 6√

5y2

)∂y5 .

We obviously recognize that the Hj ’s are the expression of the Yj ’s w.r.t. the newsystem of coordinates y = C x, i.e.

(Hjf )(C(x)) = Yj (f ◦ C)(x) ∀ f ∈ C∞(RN, R), ∀ x ∈ RN.

As a consequence, the sub-Laplacian L on H2 is turned by C into the canonical

sub-Laplacian of H∗ = (R5, ∗)

ΔH∗ =4∑

j=1

H 2j

=(

∂y1 +(

4√5y3 − 2√

5y2

)∂y5

)2

+(

∂y2 +(

2√5y1 + 6√

5y4

)∂y5

)2

+(

∂y3 +(

− 4√5y1 − 2√

5y4

)∂y5

)2

+(

∂y4 +(

2√5y3 − 6√

5y2

)∂y5

)2

.

6 Here, we use the formula

(Hjϕ)(y) = (∂/∂ηj )|η=0ϕ(y ◦ η).


16.4 Further Reading: An Example of Application to PDE’s

We close the chapter, by giving an example of application to PDE’s of the topicstreated in Section 16.1. To this end, let us introduce the parabolic-type operator innon-divergence form ∑

i,j

ai,j (x)XiXj − ∂t , (16.35)

being {Xi}i the generators of the Lie algebra of a Carnot group G and ai,j Höldercontinuous functions.

The fundamental solution for (16.35) can be constructed via the Levi-parametrixmethod7, provided suitable uniform estimates of the fundamental solutions for thefrozen operators are established (see [BLU03]). By the frozen operators, we meanthe constant coefficient operators

HA = LA − ∂t =m∑

i,j=1

ai,j XiXj − ∂t , A = (ai,j )i,j ∈ MΛ,

where MΛ is the set of m × m symmetric matrices A such that

Λ−1|ξ |2 ≤ 〈Aξ, ξ 〉 ≤ Λ|ξ |2 ∀ξ ∈ Rm

(Λ ≥ 1 being a fixed constant). The Levi-parametrix method for the operator (16.35)requires the following Gaussian uniform estimates

∣∣Xi1 · · · Xip(∂t )q ΓA(x, t) − Xi1 · · ·Xip(∂t )

q ΓB(x, t)∣∣

≤ cΛ,p,q ‖A − B‖1/r t−(Q+p+2q)/2 exp

(−d2

G(x)

cΛ t

)

for every x ∈ G, t > 0, A,B ∈ MΛ, (16.36)

where we have denoted by ΓA the fundamental solution for HA and by dG a fixedsymmetric homogeneous norm8 on G.

A natural question in approaching (16.36) (whose proof can be found in [BLU02])is to ask whether the sub-Laplacians LA’s are all diffeomorphic to the canonical op-erator ΔG, via a change of variables. This naive idea stands at the basis of the Levi-parametrix method in the classical case. As we showed in Section 16.1, in the generalcase of a Carnot group, this problem is not trivial.

Theorem 16.1.2 allows us to obtain the fundamental solution ΓA for HA as thecomposition of TA with the fundamental solution ΓG for HG, where

HG = ΔG − ∂t .

7 See, e.g. [Kal92] for a reference to the Levi-parametrix method in the classical case.8 I.e. a continuous function dG : R

N → [0, ∞[, smooth away from the origin, such thatdG(δλ(x)) = λ dG(x), dG(x−1) = dG(x) and dG(x) = 0 iff x = 0; see Section 5.1,page 229.


Indeed, if G is free, it turns out that

ΓA(x, t; ξ, τ ) = | detJTA(x)| ΓG(TA(x), t; TA(ξ), τ ) (16.37)

for every x, ξ ∈ RN and t, τ ∈ R. A crucial step in order to obtain the uniform

estimates in (16.36) is then to establish ad hoc uniform estimates for TA (these canbe found in [BU04b, Theorem 2.7]).

In order to handle the case of an arbitrary Carnot group G, a possible approachis to lift G to a free Carnot group G in such a way that

ΔG is lifted to ΔG

(the lifting technique is treated in Chapter 17). Indeed, the following result holds (seeTheorem 17.1.5).

There exists a free homogeneous Carnot group G on RH (with H ≥ N ) such

that, denoting by π : RH → R

N the projection on the first N coordinates, we have

Xi(u ◦ π) = (Xiu) ◦ π ∀ u ∈ C∞(RN),

where∑m

i=1 X2i and

∑mi=1 X2

i are the canonical sub-Laplacians ΔG and ΔG

, re-spectively.

This result, together with Theorem 16.1.2, allows to prove (16.36) in the generalcase. We briefly describe how (we refer to [BLU02] for the complete proofs).

A family of sub-Laplacians {LA} on an arbitrary Carnot group G is lifted to afamily of sub-Laplacians {LA} on a free G. The related family {ΓA} of fundamentalsolutions fulfills identity (16.37). By means of the uniform estimates of TA, thisallows to derive (16.36) for {ΓA}. Finally, the fundamental solutions ΓA are explicitlyrepresented by integrating ΓA over the added variables. As a consequence, estimate(16.36) can be obtained on G.

Bibliographical Notes. For the details of the topics presented in Section 16.4, werefer the reader to [BLU02,BLU03]. Some of the topics presented in this chapter alsoappear in [BU04b].


Ex. 1) Prove that RN equipped with the ordinary Abelian structure is a free Carnot

group. Then prove that the Heisenberg group H1 is also a free Carnot group.

Ex. 2) Consider the Heisenberg group H1. Let B be a 2 × 2 symmetric non-singular

matrix. Arguing as in Theorem 16.1.2, write down explicitly a diffeomor-phism T which turns the sub-Laplacian LB−2 into ΔH1 . Recognize that T soconstructed turns out to be linear. Compare with Remark 16.2.1.

Ex. 3) Consider the following sub-Laplacian on the Heisenberg group H2

L = X21 + X2

2 + X24 + (X2 + X3)

2,


where∑4

j=1 X2j is the canonical sub-Laplacian of H

2. Following the argu-

ments of Section 16.3, find a new group law ∗ on R5 such that H

∗ := (R5, ∗)

is a homogeneous Carnot group isomorphic to H2, via a Lie group isomor-

phism which turns L into the canonical sub-Laplacian ΔH∗ of H∗. Write

down explicitly ΔH∗ .Ex. 4) Let (G, ◦) be a free homogeneous Carnot group of step r , and let dG be a

symmetric homogeneous norm on G. Given a fixed constant Λ ≥ 1, let MΛ

denote the set of symmetric m × m constant matrices A such that

Λ−1|ξ |2 ≤ 〈Aξ, ξ 〉 ≤ Λ|ξ |2 ∀ ξ ∈ Rm.

In what follows we shall denote by cΛ any positive constant depending onlyon Λ and the structure of G. Finally, ‖A‖ will stand for the matrix normmax|ξ |=1 |Aξ |.Consider the diffeomorphism TA constructed in Theorem 16.1.2 and set, forx ∈ G,

JA(x) = | detJTA(x)|.

Prove that the following properties hold for any A, A1, A2 ∈ MΛ andx ∈ G:a) JA is constant in x,b) (cΛ)−1 ≤ JA ≤ cΛ,c) |JA1 − JA2 | ≤ cΛ ‖A1 − A2‖,d) (cΛ)−1 dG(x) ≤ dG(TA(x)) ≤ cΛ dG(x),e) dG((TA2(x))−1 ◦ TA1(x)) ≤ cΛ ‖A1 − A2‖1/r dG(x).Hint: First prove Ex. 5 below. Then recall that TA is defined in (16.9), whereϕ = ϕA is a linear map represented in Jacobian coordinates by a block-diagonal matrix whose entries are polynomials in the entries of A−1/2. Usethe following facts

detJExp = detJLog ≡ 1

to prove (a). Then use (b) of Ex. 5 to prove (b). Now, observing that JA is inthe form

JA = |Ψ (A−1/2)|,being Ψ a polynomial function in the entries of A−1/2, estimate |JA1 − JA2 |by the mean value theorem. Use then (b) and (c) of Exercise 5 to prove (c).Exploiting the fact that TA commutes with the dilations of G (see Theo-rem 16.1.2) and that dG is δλ-homogeneous of degree 1, reduce the proofsof (d) and (e) to the case

x ∈ SG := {ξ ∈ G | dG(ξ) = 1}.Then prove that the map MΛ ×SG � (A, ξ) �→ TA(ξ) is continuous, observ-ing that TA(ξ) is a polynomial function both in ξ and in the entries of A−1/2,and using (c) of Ex. 5. Observe also that dG(TA(ξ)) > 0 for all ξ ∈ SG, andthen prove (d). Finally, apply (5.5) to dG and K := {TA(ξ)|A ∈ MΛ, ξ ∈SG} in order to obtain (e).


Ex. 5) With the notation of Ex. 4, prove that:a) If A ∈ MΛ, then A−1 ∈ MΛ;b) If A ∈ MΛ, then

Λ−1 ≤ ‖A‖, ‖A−1‖ ≤ Λ and Λ−1/2 ≤ ‖A1/2‖, ‖A−1/2‖ ≤ Λ1/2;c) For every A, B ∈ MΛ, we have

‖A1/2 − B1/2‖, ‖A−1 − B−1‖, ‖A−1/2 − B−1/2‖ ≤ cΛ ‖A − B‖.Ex. 6) In the setting and with the notation of Ex. 4 (and using the very results of

Ex. 4, besides Theorem 16.1.2), prove the following facts:a) For every A ∈ MΛ, we have

ΓA(x) = JA · ΓG(TA(x)), x ∈ RN,

where ΓG and ΓA denote, respectively, the fundamental solutions for thecanonical sub-Laplacian

ΔG =m∑

k=1

X2k

and for the sub-Laplacian

LA =m∑

i,j=1

ai,j XiXj ;

b) The fundamental solution ΓA satisfy the following uniform estimates

c−1Λ (dG(x))2−Q ≤ ΓA(x) ≤ cΛ (dG(x))2−Q, x ∈ R

N \ {0},for every A ∈ MΛ (as usual, Q denotes the homogeneous dimensionof G);

c) The derivatives of the fundamental solution ΓA satisfy the following uni-form estimates

∣∣Xi1 · · ·Xip ΓA(x)∣∣ ≤ cΛ,p (dG(x))2−Q−p, x ∈ R

N \ {0},for every A ∈ MΛ and for every i1, . . . , ip ∈ {1, . . . , m}.

17

Lifting of Carnot Groups

As we discussed in the Preface of this book, L.P. Rothschild and E.M. Stein [RS76]obtained sharp regularity results for the sum of squares of Hörmander vector fields∑m

j=1 X2j by using analysis on nilpotent Lie groups. A crucial step in [RS76] is the

construction of new vector fields X1, . . . , Xm (on a larger manifold) which “lift”the Xj ’s and which can be locally approximated by left-invariant vector fields on astratified group.

The aim of this chapter is to study the lifting procedure in the special case whenthe Xj ’s generate the Lie algebra of a Carnot group G. In this case, we prove that G

can be directly lifted to a free group G which preserves the homogeneous structureof G, besides being itself a homogeneous Carnot group.

We also give an example of application of the lifting technique to PDE’s. Indeed,in Section 17.3 we write an explicit formula for the fundamental solutions for all thesub-Laplacians on Carnot groups of step two. This formula is given in terms onlyof the fundamental solution for the canonical sub-Laplacian on a fixed free Carnotgroup.

17.1 Lifting to Free Carnot Groups

We first recall the basic notation for Carnot groups that we are going to use through-out the chapter. We denote, as usual, by G = (RN, ◦, δλ) a homogeneous Carnotgroup. The dilations on G have the usual form

δλ(x) = δλ(x(1), . . . , x(r)) = (λx(1), . . . , λrx(r)). (17.1)

Here x(i) ∈ RNi for i = 1, . . . , r and N1 + · · · + Nr = N . We denote by g the Lie

algebra of G. For i = 1, . . . , N , let Zi be the vector field in g that agrees at the originwith ∂/∂xi , i.e. the Zi’s form the Jacobian basis for g.

We know that the crucial hypothesis on G is that the Lie algebra generated byZ1, . . . , ZN1 is the whole g. We also say that G is of step r and has m := N1 gen-erators. Let us also explicitly recall that g is thus an N -dimensional nilpotent Lie

650 17 Lifting of Carnot Groups

algebra of step r generated by Z1, . . . , Zm. Moreover, if Z(i)j ∈ g agrees at the origin

with ∂/∂x(i)j , then Z

(i)j is δλ-homogeneous of degree i (see Remark 1.4.5, page 58).

As usual, we denote by Q = ∑rj=1 j Nj the homogeneous dimension of G. The

canonical sub-Laplacian on G is the second order differential operator

ΔG =m∑

i=1

Z2i .

If Y1, . . . , Ym is any basis for span{Z1, . . . , Zm}, the second order differential oper-ator

L =m∑

i=1

Y 2i

is called a sub-Laplacian on G.We now recall the definition of fm,r , the free nilpotent Lie algebra of step r with m

(≥ 2) generators x1, . . . , xm (see also Chapter 14). By definition, fm,r is the unique(up to isomorphism) nilpotent Lie algebra of step r generated by m of its elementsF1, . . . , Fm such that, for every nilpotent Lie algebra n of step r and for every mapϕ from {F1, . . . , Fm} to n, there exists a (unique) Lie algebra morphism ϕ from fm,r

to n extending ϕ. We say that the Carnot group G is a free Carnot group if its Liealgebra g is isomorphic to fm,r for some m and r .

Following the above notation, by the definition of fm,r , there exists a unique Liealgebra morphism Π from fm,r to g such that

Π(Fi) = Zi for every i = 1, . . . , m.

Clearly, Π is surjective, whence

dim fm,r = dim ker(Π) + dim g.

We set H = dim fm,r . The following result will be relevant in the sequel.

Proposition 17.1.1 (A lifting basis F ). There exists a basis

F = {F1, . . . , FH }for fm,r such that

{Π(Fj ) = Zj for every j = 1, . . . , N ,

Π(Fj ) = 0 for every j = N + 1, . . . , H ,(17.2)

and each Fj (j = 1, . . . , H ) is a homogeneous Lie polynomial in F1, . . . , Fm.

Proof. Let αj denote the δλ-homogeneity degree of Zj . In particular, α1 = · · · =αm = 1. For a multi-index I = (i1, . . . , ik) with i1, . . . , ik ∈ {1, . . . , m}, we set|I | = k (the height of I ) and define

17.1 Lifting to Free Carnot Groups 651

ZI := [Zi1, [Zi2 . . . [Zik−1 , Zik ] . . . ]].Then, by simple homogeneity arguments, we have (see Proposition 1.3.9, page 36,and Proposition 1.1.7, page 12)

Zj =∑

I∈Ij

c(j)I ZI , c

(j)I ∈ R, for every j = 1, . . . , N,

where Ij is a set of multi-indices all with height αj . If we analogously set

FI := [Fi1, [Fi2 . . . [Fik−1 , Fik ] . . . ]],the first N elements of the basis F can be chosen as

Fj =∑

I∈Ij

c(j)I FI , j = 1, . . . , N.

Indeed Π(Fj ) = Zj and F1, . . . , FN are linearly independent homogeneous Liepolynomials in the generators F1, . . . , Fm. Let now FN+1, . . . , FH be a basis forker(Π). We can write

Fj =∑

I∈Aj

q(j)I FI , j = N + 1, . . . , H,

for a certain set of multi-indices Aj and scalars q(j)I ’s. For every k = 1, . . . , r , we

set A(k)j := {I ∈ Aj : |I | = k }. We evidently have

Aj = A(1)j ∪ · · · ∪ A(r)

j .

ThenFj =

∑

I∈A(1)j

q(j)I FI + · · · +

∑

I∈A(r)j

q(j)I FI =: F

(1)j + · · · + F

(r)j .

The system of vectors

F := {F (k)j : j = N + 1, . . . , H, k = 1, . . . , r}

has the following properties:

1) each of its vectors is a homogeneous Lie polynomial in the generatorsF1, . . . , Fm;

2) the system spans ker(Π);3) for every j = N + 1, . . . , H and k = 1, . . . , r , we have F

(k)j ∈ ker(Π).

Let us prove this last assertion: for j = N + 1, . . . , H , we have

0 = Π(Fj ) = Π(F(1)j ) + · · · + Π(F

(r)j ).


We notice that Π(F(k)j ) is a δλ-homogeneous vector field of degree k, unless it van-

ishes. Consequently, since the cited δλ-degrees are all different, each Π(F(k)j ) must

vanish identically (see Proposition 1.3.9, page 36).Finally, the proposition is proved by extracting from the system of vectors F a

basis FN+1, . . . , FH for ker(Π). ��In the sequel, we shall consider some abstract finite-dimensional algebras; we

hence introduce a useful notation, which will allow us to deal with ordinary Rn

spaces.Let h be a finite-dimensional nilpotent real Lie algebra. It is possible to prove

(see Theorem 14.2.3, page 586) that h is equipped with a Lie group structure by theso-called Campbell–Hausdorff composition law defined by (14.2), page 585. For X,Y ∈ h, the first few terms in the sum (14.2) (which is a finite sum since h is nilpotent)are given by

X � Y = X + Y + 1

2[X, Y ] + 1

12[X, [X, Y ]] − 1

12[Y, [X, Y ]] + · · · .

We explicitly remark that � is defined in a universal way (independent of h) as a Liepolynomial in X and Y . We now fix a basis

E = (E1, . . . , EN)

for h (N := dim h), and we identify h with RN via the map

πE : h → RN,

N∑

i=1

ξi Ei → (ξ1, . . . , ξN ).

The group law � is then turned into a group law �E on RN in the natural way

a �E b := πE(π−1E (a) � π−1

E (b)), a, b ∈ R

N.

In other words, if a, b ∈ RN , then a �E b is the only element of R

N such that(

N∑

i=1

ai Ei

)�

(N∑

i=1

bi Ei

)=:

N∑

i=1

(a �E b)i Ei.

The Lie groups (RN,�E ) and (h,�) are clearly isomorphic1 via πE . We stress thatthe Lie group morphism πE is also a linear map.

1 It is evident that the particular form of the operation �E on RN does depend on the choice

of the basis E on h. However, if E1 and E2 are two different bases of h, obviously the twoLie groups (RN, �E1

) and (RN, �E2) are canonically Lie-isomorphic via the natural linear

change of basis from E1 to E2:

(h, �)

πE1

(h, �)

πE2

(RN, �E1)

change of basis(RN, �E2

).


With the above notation, we now consider the Lie groups (RN,�Z ) and(RH ,�F ), where Z is the Jacobian basis of g, whereas F is the basis of fm,r in-troduced in Proposition 17.1.1. First of all, we remark that the map

π := πZ ◦ Π ◦ π−1F : (RH ,�F ) → (RN,�Z )

is a surjective Lie group morphism coinciding with the usual projection of RH

onto RN

RH � (ξ1, . . . , ξN , ξN+1, . . . , ξH ) → (ξ1, . . . , ξN ) ∈ R

N.

The following diagram may be considered

(fm,r ,�)Π

(g,�)

πZ

(RH ,�F )

π−1F

canonical projection(RN,�Z ).

Indeed, if ξ ∈ RH , we have

(πZ ◦ Π ◦ π−1F )(ξ) = (πZ ◦ Π)

(H∑

i=1

ξi Fi

)

= πZ

(H∑

i=1

ξi Π(Fi)

)

(see (17.2)) = πZ

(N∑

i=1

ξi Zi

)= (ξ1, . . . , ξN ).

We have thus proved a first kind of lifting result (we notice that the Lie algebra ofvector fields g may be replaced by any finite-dimensional nilpotent Lie algebra h

generated by m of its elements).

Proposition 17.1.2 (Lifting via a lifting basis F ). Let h be a finite-dimensional Liealgebra generated by m of its elements, nilpotent of step r . Let N := dim (h). Let Ebe a fixed arbitrary linear basis of h. Then there exists a basis F of fm,r such that theprojection

π : (RH ,�F ) −→ (RN,�E ),

(ξ1, . . . , ξN , ξN+1, . . . , ξH ) → (ξ1, . . . , ξN )

is a surjective Lie group homomorphism.

We now turn to the Lie algebras of the Lie groups (RN,�Z ) and (RH ,�F ) con-sidered above: let rN and rH denote these Lie algebras, respectively. Since π is a Liegroup morphism, then its differential

dπ : rH → r

N


is a Lie algebra morphism (see Theorem 2.1.50, page 114) with the following prop-erty (

dπ(E))π(ξ)

= dπ(Eξ ) ∀ E ∈ rH , ∀ ξ ∈ R

H . (17.3)

Roughly speaking, the differential of a Lie group morphism which coincides with aprojection gives a lifting of vector fields.

More precisely, we have the following lemma.

Lemma 17.1.3. With the above notation, if E ∈ rH , then E is a lifting of dπ(E) inthe following sense:

if f ∈ C∞(RH ) depends only on ξ1, . . . , ξN , i.e. there exists g ∈ C∞(RN) suchthat

f (ξ1, . . . , ξH ) = g(ξ1, . . . , ξN )

for every ξ ∈ RH , then we have

(Ef

)(ξ1, . . . , ξH ) = (

dπ(E)g)(ξ1, . . . , ξN) ∀ ξ ∈ R

H .

Proof. Let g ∈ C∞(RN) and f ∈ C∞(RH ) be such that f = g ◦ π . We have toprove

Eξ(f ) = (dπ(E)

)π(ξ)

(g) for every ξ ∈ RH .

From (17.3) we immediately obtain

(dπ(E)

)π(ξ)

(g) = (dπ(Eξ )

)(g) = Eξ(g ◦ π) = Eξ(f ).

This ends the proof. ��The rest of the lifting method consists in the transferring of this result to the group G,after a suitable definition of the larger group which projects onto G. We refer thereader to the following diagram of Lie group morphisms:

(fm,r ,�)Π−→ (g,�)

Exp←→ (G, ◦)

πF � � πZ(G × R

H−N, •)Φ←→ (RH ,�F )

π−→ (RN,�Z ).

We recall that (RN,�Z ) is isomorphic to (g,�) via πZ . On the other hand, the ex-ponential map

Exp : (g,�) → (G, ◦)

is a Lie group isomorphism (see Theorem 2.2.13, page 129). As a consequence, themap

Ψ := Exp ◦ π−1Z

is a Lie group isomorphism from (RN,�Z ) to (G, ◦).We then look for a suitable group structure on G × R

H−N and a Lie group iso-morphism

Φ : (RH ,�F ) → (G × RH−N, •)


such thatϑ := Ψ ◦ π ◦ Φ−1 = Exp ◦ π−1

Z ◦ π ◦ Φ−1

is the projection of G × RH−N onto G. To this end, we set

Φ(ξ1, . . . , ξH ) := (Ψ (ξ1, . . . , ξN ), ξN+1, . . . , ξH ).

Clearly, Φ is an invertible map of class C∞ and the same is true of its inverse map.We then define on G × R

H−N the composition law “induced” by Φ

(g1, a1) • (g2, a2) := Φ(Φ−1(g1, a1) �F Φ−1(g2, a2)

),

so that Φ becomes a Lie group isomorphism between the groups (RH ,�F ) and (G×R

H−N, •). Finally, the Lie group morphism

ϑ := Ψ ◦ π ◦ Φ−1 : (G × RH−N, •) → (G, ◦)

is, by construction, the natural projection. Indeed,

ϑ(x1, . . . , xH ) = (Ψ ◦ π)(Ψ −1(x1, . . . , xN), xN+1, . . . , xH )

= Ψ (Ψ −1(x1, . . . , xN)) = (x1, . . . , xN).

We denote by g rH−N the Lie algebra of (G × RH−N, •).

The proof of the following result is simply a restatement of the proof of Lem-ma 17.1.3.

Lemma 17.1.4. If W ∈ g rH−N , then W is a lifting of dϑ(W) in the following sense:if f ∈ C∞(G × R

H−N) depends only on x1, . . . , xN , i.e. there exists g ∈ C∞(G)

such that

f (x1, . . . , xH ) = g(x1, . . . , xN) for every x ∈ G × RH−N,

then we have(Wf

)(x1, . . . , xH ) = (

dϑ(W)g)(x1, . . . , xN) ∀ x ∈ G × R

H−N.

Proof. Let g ∈ C∞(G) and f ∈ C∞(G × RH−N) be such that f = g ◦ ϑ , i.e.

f (x1, . . . , xH ) = g(x1, . . . , xN) ∀ x ∈ G × RH−N.

We have to prove that

Wx(f ) = (dϑ(W))ϑ(x)(g) ∀ x ∈ G × RH−N.

To this end, it suffices to notice that

(dϑ(W))ϑ(x)(g) = (dϑ(Wx))(g) = Wx(g ◦ ϑ) = Wx(f ).

This ends the proof. ��


We claim that the first N vector fields of the Jacobian basis W1, . . . ,WH for g rH−N

lift orderly the Jacobian basis Z1, . . . , ZN for g. Indeed, by Lemma 17.1.4, it sufficesto show that

dϑ(Wk) = Zk for every k = 1, . . . , N .

To this end, we remark that, for all f ∈ C∞(G), we have(dϑ(Wk)

)0(f ) = (Wk)0(f ◦ ϑ) = (∂xk

f )(0) = (Zk

)0f,

which proves the assertion, since two left-invariant vector fields are equal if and onlyif they coincide at the origin.

We are now in the position to prove the main result of this section.

Theorem 17.1.5 (Lifting). Let G be a homogeneous Carnot group on RN of step r

and m (= N1) generators. Then there exists a free homogeneous Carnot group G

on RH (H = dim fm,r ) with the properties (i) and (ii) stated below.

We fix the following notation:

δλ(x) = δλ(x(1), x(2), . . . , x(r)) = (λx(1), λ2x(2), . . . , λrx(r))

andδλ(x) = δλ(x

(1), x(2), . . . , x(r)) = (λx(1), λ2x(2), . . . , λr x(r))

denote the dilations on G and G, respectively, with the usual notation

x(i) ∈ RNi , i = 1, . . . , r , N1 + · · · + Nr = N ,

and analogously

x(i) ∈ RNi , i = 1, . . . , r , N1 + · · · + Nr = H ;

the vector fieldsZ

(i)j , i = 1, . . . , r, j = 1, . . . , Ni,

denote the Jacobian basis of the Lie algebra g of G, and analogously

Z(i)j , i = 1, . . . , r, j = 1, . . . , Ni ,

denote the Jacobian basis of the Lie algebra g of G. Then:

(i) G has step r and m generators, and its Lie algebra is isomorphic to fm,r .(ii) For a certain i0 ∈ {1, . . . , r}, we have

Ni = Ni, i = 1, . . . , i0 and Ni > Ni, i = i0 + 1, . . . , r;moreover, if �(i) : R

Ni → RNi denotes the projection on the first Ni coordinates

and � : RH → R

N is defined by

�(x) = (�(1)(x(1)), . . . , �(r)(x(r))),

then

Z(i)j (u ◦ �) = (

Z(i)j u

) ◦ � ∀ u ∈ C∞(RN), i ≤ r, j ≤ Ni, (17.4)

i.e. Z(i)j lifts Z

(i)j . Moreover, � is a Lie group morphism.


Proof. Let (G × RH−N, •) be the previously defined Lie group on R

H . We showthat G can be constructed from G × R

H−N by a permutation of the coordinates.Let F = (F1, . . . , FH ) be the basis for fm,r as in Proposition 17.1.1. Then Fj is ahomogeneous Lie polynomial of degree αj in the generators F1, . . . , Fm. We stressthat, by Proposition 17.1.1, αj is also the δλ-homogeneity degree of Zj for j =1, . . . , N , i.e. the dilation δλ on R

N can be written as

δλ(x1, . . . , xN) = (λα1x1, . . . , λαN xN).

We also observe that only F1, . . . , Fm have degree 1. We define the dilations on fm,r

as follows

δλ

(H∑

i=1

ξi Fi

):=

H∑

i=1

λαi ξi Fi .

First, we prove that δλ is a Lie algebra automorphism of fm,r . Indeed, for all i, j ∈{1, . . . , H }, one has

δλ

([Fi, Fj ]) = λαi+αj [Fi, Fj ] = [λαi Fi, λ

αj Fj ] = [δλ(Fi), δλ(Fj )].The first equality holds since [Fi, Fj ] is a homogeneous Lie polynomial in F1,

. . . , Fm of degree αi + αj . Then δλ is also a Lie group automorphism of (fm,r ,�) byRemark 2.2.16, page 130. As a consequence, the map

δ∗λ := πF ◦ δλ ◦ π−1

F

is a Lie group automorphism of (RH ,�F ). Analogously, the map

δλ := Φ ◦ δ∗λ ◦ Φ−1

is a Lie group automorphism of (G × RH−N, •). If x ∈ G × R

H−N , we have

δλ(x) = ((Ψ ◦ δλ ◦ Ψ −1)(x1, . . . , xN), λαN+1xN+1, . . . , λαH xH )

= (δλ(x1, . . . , xN), λαN+1xN+1, . . . , λαH xH ).

Indeed, Ψ commutes with the dilations of G (see Theorem 1.3.28, page 49).We then reorder the coordinates of G×R

H−N in the following way. Let (x, y) ∈G × R

H−N , where

x = (x(1), . . . , x(r)) ∈ G and y = (yN+1, . . . , yH ) ∈ RH−N.

We can suppose that the coordinates of y are ordered in such a way that αN+1 ≤· · · ≤ αH . Setting i0 := αN+1 − 1, we can write

y = (y(i0+1), . . . , y(r)),

where to each coordinate ys of y(k) (i0 + 1 ≤ k ≤ r) corresponds a degree ofhomogeneity αs equal to k. We now set


P : G × RH−N → R

H ,

(x, y) → (x(1); . . . ; x(i0); x(i0+1), y(i0+1); . . . ; x(r), y(r)).

We define G as RH with the group structure naturally induced by P from (G ×

RH−N, •).

With respect to the coordinates of G, δλ assumes the usual form(λ x(1); . . . ; λi0x(i0); λi0+1x(i0+1), λi0+1y(i0+1); . . . ; λrx(r), λry(r)

).

We now set � := ϑ ◦ P −1. It is then easy to recognize that (17.4) follows fromLemma 17.1.4.

In order to complete the proof, we have to show that G is a homogeneous Carnotgroup with Lie algebra g isomorphic to fm,r . To this purpose, let (Gm,r , ∗) be a freehomogeneous Carnot group2 on R

H whose Lie algebra is isomorphic to fm,r . It isnot restrictive to consider fm,r to be the Lie algebra of Gm,r itself. If we denote byExp ∗ the exponential map from (fm,r ,�) to (Gm,r , ∗), then the map

Exp ∗ ◦ π−1F : (RH ,�F ) → (Gm,r , ∗)

is a Lie group isomorphism (see Theorem 2.2.13, page 129). As a consequence, byTheorem 2.1.50, page 114, the differential

d(Exp ∗ ◦ π−1F )

is a Lie algebra isomorphism from rH to fm,r . Let E1, . . . , EH be the Jacobian basisfor rH . We now prove that

d(Exp ∗ ◦ π−1F )(Ei) = Fi, i = 1, . . . , m. (17.5)

In particular, since F1, . . . , Fm are generators for fm,r , this will prove that E1, . . . ,

Em are generators for rH . If f ∈ C∞(Gm,r ), we have

(d (Exp ∗ ◦ π−1

F )(Ei))

0(f ) = (Ei)0(f ◦ Exp ∗ ◦ π−1

F)

= (∂ξi|ξ=0)f

(Exp ∗

(H∑

j=1

ξjFj

))= d

dt

∣∣∣∣t=0

f(Exp ∗(t Fi)

)

= d

dt

∣∣∣∣t=0

f(

expFi(t)

) = (Fi)0(f ).

Since a left invariant vector field is determined by its value at the origin, this proves(17.5). An analogous argument shows that

dΦ : rH → g r

H−N

2 The existence of such a group can be easily derived by the third fundamental theorem ofLie (see Theorem 2.2.14, page 130) and by Theorem 2.2.18, page 131.

17.2 An Example of Lifting 659

maps the first m fields of the Jacobian basis for rH into the first m fields of the Ja-cobian basis for g rH−N . Finally, since G is obtained from G × R

H−N by the Liegroup isomorphism P (which permutes the coordinates, leaving the first m coordi-nates unaltered), we can assert that G is a homogeneous Carnot group. Moreover, g

is isomorphic to fm,r via the Lie algebra isomorphism

d(Exp ∗ ◦ π−1F ◦ Φ−1 ◦ P −1).

This ends the proof. ��In Section 17.2 below, we give an example of the lifting method just described.

The reader is referred to this example for a straightforward comprehension of thetopics presented so far.

17.2 An Example of Lifting

We here give an example of a homogeneous Carnot group, and we lift it to a freehomogeneous Carnot group. We shall follow the notation introduced in Section 17.1.We consider the homogeneous Carnot group G on R

5 with the group law and dilationdefined as follows:

x ◦ y = (x1 + y1, x2 + y2, x3 + y3, x4 + y4, x5 + y5 + x1 y3 + x2 y4)

and

δλ(x) = (λ x1, λ x2, λ x3, λ x4, λ2x5).

The Jacobian basis Z for g is given by

Z1 = ∂1, Z2 = ∂2, Z3 = ∂3 + x1 ∂5, Z4 = ∂4 + x2 ∂5, Z5 = ∂5.

The Campbell–Hausdorff formula on a Lie algebra nilpotent of step 2 is

X � Y = X + Y + 1

2[X, Y ].

Hence the group (R5,�Z ) has the composition law given by

ξ �Z η =(

ξ1 + η1, ξ2 + η2, ξ3 + η3, ξ4 + η4,

ξ5 + η5 + 1

2(ξ1 η3 − ξ3 η1 + ξ2 η4 − ξ4 η2)

).

With few modifications, we recognize that �Z is the usual group law on H2, the

Heisenberg group on R5. The Jacobian basis X for r5 is given by


X1 = ∂1 − ξ3

2∂5, X2 = ∂2 − ξ4

2∂5,

X3 = ∂3 + ξ1

2∂5, X4 = ∂4 + ξ2

2∂5, X5 = ∂5.

We now turn to consider f4,2. We have dim f4,2 = 10. Let Π : f4,2 → g be the Liealgebra morphism such that Π(Fi) = Zi for every i = 1, . . . , 4. The following is abasis F for f4,2 as in Proposition 17.1.1:

F1, F2, F3, F4, [F1, F3];[F1, F2], [F1, F4], [F2, F3], [F3, F4], [F2, F4] − [F1, F3].

One can easily recognize that the last 5 vectors in F form a basis for ker(Π). TheLie group (R10,�F ) has the composition law given by

ξ �F η =(

ξ1 + η1, ξ2 + η2, ξ3 + η3, ξ4 + η4,

ξ5 + η5 + 1

2(ξ1 η3 − ξ3 η1 + ξ2 η4 − ξ4 η2),

ξ6 + η6 + 1

2(ξ1 η2 − ξ2 η1),

ξ7 + η7 + 1

2(ξ1 η4 − ξ4 η1), ξ8 + η8 + 1

2(ξ2 η3 − ξ3 η2),

ξ9 + η9 + 1

2(ξ3 η4 − ξ4 η3), ξ10 + η10 + 1

2(ξ2 η4 − ξ4 η2)

).

The first 5 vector fields of the Jacobian basis for r10 are then given by

E1 = ∂1 − ξ3

2∂5 − ξ2

2∂6 − ξ4

2∂7,

E2 = ∂2 − ξ4

2∂5 + ξ1

2∂6 − ξ3

2∂8 − ξ4

2∂10,

E3 = ∂3 + ξ1

2∂5 + ξ2

2∂8 − ξ4

2∂9,

E4 = ∂4 + ξ2

2∂5 + ξ1

2∂7 + ξ3

2∂9 + ξ2

2∂10,

E5 = ∂5.

It is evident that Ei lifts dπ(Ei) = Xi for every i = 1, . . . , 5, as stated inLemma 17.1.3. By a straightforward calculation of the exponential map from g to G,we have

Ψ = Exp ◦ π−1Z : (R5,�Z ) → (G, ◦),

Ψ (ξ) =(

ξ1, ξ2, ξ3, ξ4, ξ5 + 1

2(ξ1 ξ3 + ξ2 ξ4)

).

In particular, the Lie group isomorphism Φ : (R10,�F ) → (G × R5, •) is

17.3 An Example of Application to PDE’s 661

Φ(ξ) =(

ξ1, ξ2, ξ3, ξ4, ξ5 + 1

2(ξ1 ξ3 + ξ2 ξ4); ξ6, ξ7, ξ8, ξ9, ξ10

).

As a consequence, the group law on G × R5 is given by

x • y =(

x1 + y1, x2 + y2, x3 + y3, x4 + y4,

x5 + y5 + x1 y3 + x2 y4, x6 + y6 + 1

2(x1 y2 − x2 y1),

x7 + y7 + 1

2(x1 y4 − x4 y1), x8 + y8 + 1

2(x2 y3 − x3 y2),

x9 + y9 + 1

2(x3 y4 − x4 y3), x10 + y10 + 1

2(x2 y4 − x4 y2)

).

The first 5 vector fields of the Jacobian basis for g r5 are then given by

W1 = ∂1 − x2

2∂6 − x4

2∂7,

W2 = ∂2 + x1

2∂6 − x3

2∂8 − x4

2∂10,

W3 = ∂3 + x1 ∂5 + x2

2∂8 − x4

2∂9,

W4 = ∂4 + x2 ∂5 + x1

2∂7 + x3

2∂9 + x2

2∂10,

W5 = ∂5.

As stated in Lemma 17.1.4, Wi lifts dϑ(Wi) = Zi for every i = 1, . . . , 5. Finally, inthis case, the free homogeneous Carnot group G which lifts G has the same grouplaw as (G × R

5, •) since the permutation of the coordinates P can be chosen as theidentity map. We remark that the dilations of G are given by

δλ(x) = (λ x1, λ x2, λ x3, λ x4, λ

2x5, λ2x6, λ

2x7, λ2x8, λ

2x9, λ2x10

).

17.3 An Example of Application to PDE’s

In this section, we focus our attention on Carnot groups of step two. We recall a cha-racterization of the composition law on such groups, and we explicitly exhibit theirlifting. As a consequence, we derive a direct formula for the fundamental solutionsfor all the sub-Laplacians on groups of step two. This formula is given in terms onlyof the fundamental solution for the canonical sub-Laplacian on a fixed free Carnotgroup. The latter fundamental solution can be written in a somewhat explicit formby means of a result by Beals, Gaveau and Greiner [BGG96].

Let us start by recalling the result below which easily follows from Theo-rem 1.3.15, page 39 (see also Theorem 3.2.2, page 160).


Remark 17.3.1. The N -dimensional homogeneous Carnot groups of step two andm generators are characterized by being (RN, ◦) with the following Lie group law(N = m + n, x ∈ R

m, t ∈ Rn)

(x, t) ◦ (ξ, τ ) =(

xj + ξj , j = 1, . . . , m

tj + τj + 12 〈x, B(j)ξ 〉, j = 1, . . . , n

),

where the B(j)’s are m × m matrices whose skew-symmetric parts

1

2(B(j) − (B(j))T )

are linearly independent.

We fix G = (RN, ◦), a Carnot group of step two as above, i.e. we fix matricesB(1), . . . , B(n) as in Remark 17.3.1. Let L = ∑m

j=1 Y 2j be a fixed sub-Laplacian

on G. Then there exists a non-singular m × m matrix A = (ak,j )k,j such that

Yj =m∑

k=1

ak,j Zk,

where Z1, . . . , Zm denote the first m vector fields of the Jacobian basis of g.Our aim here is to write the fundamental solution for L in terms only of the

matricesA, B(1), . . . , B(n)

and the fundamental solution for the canonical sub-Laplacian of the prototype freeCarnot group (Fm,2, �), which we introduced in Section 3.3, page 163.

For the reader’s convenience, we recall the notation. We set

N = m(m + 1)/2,

and we denote the points of RN = R

m × Rm(m−1)/2 by (x, γ ), where x ∈ R

m andγ ∈ R

m(m−1)/2, and we agree to write the coordinates of γ by γi,j , where (i, j)

varies in the setI = {(i, j) | 1 ≤ j < i ≤ m}.

Then the composition law � on RN is given by

(x, γ ) � (x′, γ ′) =(

xh + x′h, h = 1, . . . , m

γi,j + γ ′i,j + 1

2 (xi x′j − xj x′

i ), (i, j) ∈ I

).

It is easily proved that the Lie algebra of (Fm,2, �) is (isomorphic to) fm,2. Then Fm,2is a free homogeneous Carnot group of step two on R

N , with m generators.Since, by the assumption, the skew-symmetric parts of B(1), . . . , B(n) are linearly

independent and the matrix A is non-singular, then the matrices

AT B(r) − (B(r))T

2A, r = 1, . . . , n,


are also linearly independent. Hence, there exist indices (i1, j1), . . . , (in, jn) ∈ Isuch that the following n × n matrix

((AT B(r) − (B(r))T

2A

)

is , js

)

1≤r,s≤n

is non-singular. We denote by K = (kr,s)r,s the inverse of the above matrix. We alsodefine the subset of indices

C = I \ {(i1, j1), . . . , (in, jn)}.With the above notation, we shall also denote the points of Fm,2 by (x, t, β), wherex ∈ R

m, t ∈ Rn, and

β = (βh,k)(h,k)∈C ∈ Rm(m−1)/2−n.

We are now in the position to state the main result of this section.

Proposition 17.3.2. With the above notation, let ΓL and Γm,2 denote respectivelythe fundamental solutions of L and of the canonical sub-Laplacian ΔFm,2 . Then thefollowing formula holds

ΓL(x, t) = | det K|| det A|

∫

β∈Rm(m−1)/2−n

dβ

× Γm,2

(A−1x,

[ ∑

s≤n

kr,s

{ts − 1

4〈x, B(s)x〉

−∑

(h,k)∈Cβh,k

(AT B(s) − (B(s))T

2A

)

h,k

}]

r≤n

, β

).

The above result can be applied in order to manage, in a uniform way, familiesof sub-Laplacians (letting A vary) and families of step two Carnot groups with thesame number of generators (letting B(1), . . . , B(n) vary). Indeed, an example of ap-plication is given in [BLU02,BLU03] where uniform estimates for some familiesof sub-Laplacians are derived and, as a consequence, the fundamental solutions fornon-divergence form operators with Hölder-continuous coefficients are constructed.

Remark 17.3.3. We explicitly remark that a rather explicit formula is given by Beals,Gaveau and Greiner in [BGG96] for the fundamental solution of the canonical sub-Laplacian on any step two Carnot group G:

ΓΔG(x, t) =

∫

Rn

V (ρ) f (x, t, ρ)(2−Q)/2 dρ,

where f is the action associated to a complex Hamiltonian problem and V solvesa transport equation. Collecting together this formula in the case of the free groupFm,2 and our Proposition 17.3.2, we obtain the following formula



∫

β∈Rm(m−1)/2−n

dβ

∫

ρ∈Rm(m−1)/2dρ

×{

det

(S(ρ)

sinh S(ρ)

)}1/2{

1

2

⟨S(ρ) coth S(ρ)A−1x, A−1x

⟩

− ι

(n∑

r=1

ρir ,jr

[ ∑

s≤n

kr,s

(ts − 1

4〈x, B(s)x〉

−∑

(h,k)∈Cβh,k

(AT B(s) − (B(s))T

2A

)

h,k

)]+

∑

i,j∈Cρi,j βi,j

)}(2−m2)/2

.

Here, S(ρ) is the matrix S(ρ) = ι∑

i,j∈I ρi,j S(i,j), where the matrices S(i,j) havebeen introduced in Section 3.3, page 163, and ι is the imaginary unit. So, we alsohave an even more explicit representation


∫

β∈Rm(m−1)/2−n

dβ

∫

ρ∈Rm(m−1)/2dρ

×{

det

[ ∞∑

p=0

(−1)p

(2p + 1)!(

−∑

i,j∈Iρi,j S(i,j)

)2p]}−1/2

×{

1

2

⟨[ ∞∑

p=0

(−1)p2(2p)B2p

(2p)!(

−∑

i,j∈Iρi,j S(i,j)

)2p]A−1x, A−1x

⟩

− ι

(n∑

r=1

ρir ,jr

[ ∑

s≤n

kr,s

(ts − 1

4〈x, B(s)x〉

−∑

(h,k)∈Cβh,k

(AT B(s) − (B(s))T

2A

)

h,k

)]

+∑

i,j∈Cρi,j βi,j

)}(2−m−2n)/2

.

As an example, if m = 3, Γ3,2 has been explicitly written in [BGG96, p. 322]. Hence,the above formula writes

ΓL(x, t) = −Gamma(7/2)

(2π)7/2

| det K|| det A|

∫

β∈R3−n

dβ

∫

ρ∈R3dρ

2|ρ|sinh(2|ρ|)

×{

〈A−1x, ρ〉2

2|ρ|2 + |ρ| coth(2|ρ|)[|A−1x|2 − 〈A−1x, ρ〉2

|ρ|2

− ι

(n∑

r=1

ρr

{ ∑

s≤n

kr,s

(ts − 1

4〈x, B(s)x〉


−∑

(h,k)∈Cβh,k

(AT B(s) − (B(s))T

2A

)

h,k

)}

+3∑

r=n+1

ρr βr−n

)]}−7/2

.

Proof (of Proposition 17.3.2). As the first step, we turn the arbitrary sub-LaplacianL on G into the canonical sub-Laplacian on a new group H. This can be done by theisomorphism of Lie groups

H � (ξ, τ ) → (Aξ, τ ) ∈ G

which turns the composition ◦ on G into

(ξ, τ ) • (ξ ′, τ ′) =(

ξj + ξ ′j , j = 1, . . . , m,

τj + τ ′j + 1

2 〈ξ,AT B(j) Aξ ′〉, j = 1, . . . , n

).

Then, it is easy to see that we have the relation

ΓL(x, t) = | det A|−1ΓΔH(A−1x, t). (17.6)

As the second step, we turn the composition law • into a composition law ∗ whoseassociated matrices (according to Remark 17.3.1) are skew-symmetric. This can bedone by identifying H with its Lie algebra via the exponential map. Then, the iso-morphism of Lie groups

P � (x, t) →(

x, tj + 1

4〈x, AT B(j) Ax〉

)∈ H

turns the composition • on H into

(x, t) ∗ (x′, t′) =( xj + x′

j , j = 1, . . . , m,

tj + t′j + 12 〈x, AT (

B(j)−(B(j))T

2 )Ax′〉, j = 1, . . . , n

),

where the associated matrices are skew-symmetric. It is easy to see that we have therelation

ΓΔH(ξ, τ ) = ΓΔP

(ξ, τj − 1

4〈ξ,AT B(j) Aξ 〉). (17.7)

As the third step, we lift P to a free Carnot group V, as in Theorem 17.1.5. Thefundamental solution of the canonical sub-Laplacian on P can be obtained by inte-grating the fundamental solution of the canonical sub-Laplacian on V with respectto the added variables, i.e.

ΓΔP(x, t) =

∫

β∈Rm(m−1)/2−n

ΓΔV(x, t, β) dβ. (17.8)

We need the explicit form of the lifting for an arbitrary group of step two (withassociated skew-symmetric matrices). This is given in the following lemma, whichcan be proved by retracing the proof of Theorem 17.1.5.


Lemma 17.3.4. Let (RN, ◦) be a Carnot group of step two and m generators as inRemark 17.3.1, where its associated matrices B(j)’s are linearly independent andskew-symmetric. Then (RN, ◦) is lifted to the group

(RN × Rm(m−1)/2−n, ◦)

(according to Theorem 17.1.5), where (using the notation x ∈ Rm, t ∈ R

n, β ∈R

m(m−1)/2−n)

(x, t, β) ◦ (x′, t ′, β ′) =⎛

⎜⎝

xj + x′j , j = 1, . . . , m

tj + t ′j + 12 〈x, B(j)x′〉, j = 1, . . . , n

βi,j + β ′i,j + 1

2 (xix′j − xjx

′i ), (i, j) ∈ C

⎞

⎟⎠ .

The next step is to explicitly write a Lie group isomorphism between V and Fm,2.This can be obtained by the composition

Exp ◦ ϕ ◦ Log =: T ,

where Exp is the exponential map on Fm,2, Log is the logarithmic map on V, and ϕ

is the unique Lie algebra morphism mapping the first m vector fields of the Jacobianbasis related to V into the first m vector fields of the Jacobian basis related to Fm,2.A laborious computation shows that the map

T : (V, ∗) → (Fm,2, �)

is defined by

T (x, t, β) :=(

x,

[n∑

s=1

kr,s

(ts −

∑

(h,k)∈Cβh,k

(AT B(s) − (B(s))T

2A

)

h,k

)]

r≤n

, β

).

It is easy to see that we have the relation

ΓΔV(x, t, β) = | det K|Γm,2(T (x, t, β)). (17.9)

Finally, collecting together equations (17.6) to (17.9), we obtain the desired formulain Proposition 17.3.2. ��

17.4 Folland’s Lifting of Homogeneous Vector Fields

The aim of this section is to provide another lifting theorem for vector fields moregeneral than those considered in Section 17.1. An example will clarify the new situ-ation. Let us consider on R

2 the vector fields X = ∂x and Y = x ∂y . As we saw inExample 1.2.14 (page 18), they are not left-invariant with respect to any group lawin R

2, so that the lifting Theorem 17.1.5 cannot be applied. In this section, we shallprovide a lifting result which can be applied to X, Y . It will give a constructive way

17.4 Folland’s Lifting of Homogeneous Vector Fields 667

to build new vector fields (generating a homogeneous Carnot group!) lifting X and Y

(namely, X = ∂x and Y = ∂z + x ∂y on R3, whose points are denoted by (x, z, y)).

This lifting result (due to Folland, see [Fol77]) only relies on the homogeneity prop-erties of X and Y (namely, they are homogeneous of degree one, w.r.t. the dilationδλ(x, y) = (λx, λ2y) on R

2).Before entering into the details of this new lifting result, it is useful to focus on

the example above and to see how this result works.

Example 17.4.1. Let us denote the points of R2 by (x1, x2), and let us consider the

following vector fields on R2: X1 = ∂x1 and X2 = x1 ∂x2 . They are δλ-homogeneous

of degree one w.r.t. the dilation δλ(x1, x2) = (λx1, λ2x2) on R

2. As a consequence,the Lie algebra a generated by them in T (R2) (i.e. the set of smooth vector fieldson R

2 equipped with the usual bracket operation) is nilpotent (of step two), namely,since [X1, X2] = ∂x2 and all other commutators vanish, we have

a = Lie{X1, X2} = span{X1, X2, [X1, X2]}, whence dim(a) = 3.

As we saw in Theorem 2.2.13 (page 129; see also the relevant Definition 2.2.11), theCampbell–Hausdorff operation � (truncated of step two)

X � Y = X + Y + 1

2[X, Y ]

defines on a a Lie group structure (a,�). We first realize this group as a Lie group onR

3 in the usual way. We fix the basis X = (X1, X2, [X1, X2]) for a, and we identifya with R

3 via the map

φ : R3 → a, (ξ1, ξ2, ξ3) → ξ1 X1 + ξ2 X2 + ξ3 [X1, X2].

The group law � is then turned into a group law3 ◦ on R3 in the natural way

a ◦ b := φ−1(φ(a) � φ(b)

), a, b ∈ R

3.

The Lie group G = (R3, ◦) is obviously isomorphic to (a,�), and the Lie algebra g

of G is isomorphic (as a Lie algebra) to the Lie algebra a (see Ex. 3 of Chapter 2,page 148). A direct computation shows that in this case we have

a ◦ b =(

a1 + b1, a2 + b2, a3 + b3 + 1

2(a1b2 − a2b1)

).

Denoting the points of R3 by ξ = (ξ1, ξ2, ξ3), we have g = span{Z1, Z2, Z3}, where

Z1 = ∂ξ1 − 1

2ξ2 ∂ξ3 , Z2 := ∂ξ2 + 1

2ξ1 ∂ξ3 , Z3 := [Z1, Z2] = ∂ξ3 .

The Lie algebra isomorphism between g and a is the linear map α : g → a such that

3 In the previous sections, we denoted it by �X , and we also wrote φ−1 = πX .


α(Z1) = X1, α(Z2) = X2, α(Z3) = [X1, X2].Let us denote by Log the logarithmic map related to the Lie group (G, ◦). Moreover,we write I for the identity map on R

2. We then introduce the following map

π : R3 −→ R

2,(17.10)

π(ξ) := eα(Log (ξ))I (0, 0) :=∑

k≥0

(α(Log (ξ))

)k

k! I (0, 0).

We explicitly remark that the map π is well defined. Indeed, if ξ ∈ R3 ≡ G, then

Log (ξ) ∈ g, so that α(Log ξ) ∈ a. As a consequence, by the properties of δλ-

homogeneity of the vector fields in a, any power(α(Log (ξ))

)k with k ≥ 3 annihilatesthe identity map I , so that the sum in (17.10) is finite. We compute π explicitly,

π(ξ) =(

I + (α(ξ1Z1 + ξ2Z2 + ξ3Z3))I + 1

2(α(ξ1Z1 + ξ2Z2 + ξ3Z3))

2I

)

x=0(0)

=(

I + (ξ1∂x1 + ξ2x1∂x2 + ξ3∂x2)I + 1

2(ξ1∂x1 + ξ2x1∂x2 + ξ3∂x2)

2I

)

x=0(0)

=(

I +(

ξ1∂x1 + ξ3∂x2 + 1

2ξ1ξ2∂x2

)I

)

x=0(0)

=(

ξ1, ξ3 + 1

2ξ1ξ2

).

We now verify that Z1 and Z2 (the vector fields in g corresponding to X1 and X2 ina via α) lift X1 and X2 via π in the sense that

Xi |π(ξ) = dξπ(Zi) for all ξ ∈ R3 and i = 1, 2.

Indeed, since the component functions of Xi |π(ξ) are given by XiI (π(ξ)) whereasthose of dξπ(Zi) are given by Jπ (ξ) · ZiI (ξ), we only have to check the matrixidentity

(1 00 ξ1

)=

(1 0 0

12ξ2

12ξ1 1

)·( 1 0

0 1− 1

2ξ212ξ1

),

which actually holds.The ideas presented so far appear in the paper [Fol77] by G.B. Folland. We now

aim to add a new feature: we are interested in a “change of coordinates” on R3

turning Zi into a vector field on R3 lifting Xi . To this aim, we define the following

map

T : R3 → R

3, T (ξ) = (π(ξ), ξ2) =(

ξ1, ξ3 + 1

2ξ1ξ2, ξ2

).

We obtained T by completing π with the only coordinate function (i.e. ξ2) not ap-pearing in the first order expansion of π(ξ). We remark that T is a smooth dif-feomorphism of R

3 onto itself (with polynomial coordinates and the same holds


for T −1). We perform a change of coordinates on R3 by introducing new variables

x = (x1, x2, x3) as follows

x = T (ξ), x, ξ ∈ R3.

We express Z1 and Z2 (the vector fields in g corresponding to X1 and X2 in a via α)in these new coordinates and obtain Z1 and Z2. We claim that Z1 and Z2 lift X1and X2. Indeed, for every x ∈ R

3 and i = 1, 2, we have

Zi = ⟨ZiI (x),∇x

⟩, where ZiI (x) = JT (T −1(x)) · (ZiI )(T −1(x)),

so that (after a simple computation)

Z1 = ∂x1 and Z2 = ∂x3 + x1 ∂x2 .

It is now immediately seen that Z1 and Z2 lift X1 and X2, respectively. ��The aim of the rest of the section is to generalize the previous example to sets ofvector fields satisfying suitable hypotheses.

17.4.1 The Hypotheses on the Vector Fields

Assume that we are given on Rn (whose points will be denoted by x = (x1, . . . , xn))

a set of m smooth vector fields X1, . . . , Xm satisfying the following conditions:

(F1) X1, . . . , Xm are linearly independent and dλ-homogeneous of degree one withrespect to a suitable family of dilations {dλ}λ>0 of the following type

dλ : Rn → R

n, dλ(x) = (λσ1x1, . . . , λσnxn),

where 1 = σ1 ≤ · · · ≤ σn.(F2) dim(Lie{X1, . . . , Xm}I (0)) = n.

The reader should notice that these conditions are resemblant to conditions (H0)and (H2) on page 191, but condition (H1) on that page has no analogue. For example,the vector fields in Example 17.4.1 fulfill hypotheses (F1) and (F2) but not (H1).

We henceforth use the following notation: a denotes the Lie algebra generated bythe Xj ’s (sub-algebra of T (Rn), the set of the smooth vector fields on R

n equippedwith the usual bracket of differential operators, not necessarily left-invariant on aLie-group), i.e.

a := Lie{X1, . . . , Xm}. (17.11)

The first task of the section is to prove the following theorem (see also [Fol77, The-orem 1]).

Theorem 17.4.2 (Folland’s lifting, [Fol77]). Let X1, . . . , Xm satisfy the above con-ditions (F1) and (F2). Then there exist a homogeneous Carnot group G on R

N withm generators and nilpotent of step r (where N ≥ m is the dimension of a and r thestep of nilpotence of a), a polynomial surjective map


π : RN → R

n,

and a system of Lie-generators Z1, . . . , Zm for the algebra of G such that Zi lift Xi

via π , i.e.dξπ(Zi) = (Xi)π(ξ) for every ξ ∈ R

N . (17.12)

More precisely, Zi is the i-th vector of the Jacobian basis for G.

We give a constructive proof of Theorem 17.4.2 consisting of six steps.STEP 1 (The properties of a). We leave as an exercise the verification of the

following results. If X1, . . . , Xm satisfy the above conditions (F1) and (F2), then thefollowing facts hold:

(i) Each Xj has the following form

Xj =n∑

i=1

ai,j (x) ∂xi,

where ai,j is a dλ-homogeneous polynomial of degree σi − 1 (so that ai,j de-pends only on the xk’s such that σk ≤ σi − 1).

(ii) a is nilpotent (of step at most σn, say r). As a consequence, since a is spannedby high-order brackets of the form

[Xik · · · [Xi1, Xi2] · · ·], i1, . . . , ik ∈ {1, . . . , m}, k ∈ N,

and only a finite number of them are not vanishing (due to the nilpotence of a),then a is finite-dimensional. We set

N := dim(a). (17.13)

(iii) The Lie algebra a is stratified, i.e. (see also Definition 2.2.3, page 122)

a = a1 ⊕ a2 ⊕ · · · ⊕ ar with

{ [a1, ai−1] = ai if 2 ≤ i ≤ r ,

[a1, ar ] = {0}. (17.14)

Here a1 = span{X1, . . . , Xm} and r is the step of nilpotence of a. We explicitlyremark that

any operator in ai is dλ-homogeneous of degree i, (17.15)

for it is a linear combination of elements of the type

Xj1Xj2 · · · Xjs with j1 + j2 + · · · + js = i

(and any Xj is dλ-homogeneous of degree 1 by hypothesis (F1)). The abovedecomposition also allows us to define a group of dilations {δλ}λ>0 on a in thefollowing way:

δλ : a → a, δλ

(r∑

i=1

Ai

):=

r∑

i=1

λi Ai, where Ai ∈ ai for i = 1, . . . , r .

(17.16)


STEP 2 (The choice of a basis for a). Recalling hypothesis (F1) and (17.13),we can complete X1, . . . , Xm to a basis of a

X = (X1, . . . , Xm,Xm+1, . . . , XN)

satisfying the following two conditions:

(1) we have (by making use of hypothesis (F2))

span{X1I (0), . . . , XmI (0),Xm+1I (0), . . . , XNI (0)

} = Rn, (17.17)

(2) X is adapted to the stratification of a, i.e. we have

X = (X

(1)1 , . . . , X(1)

m1; X

(2)1 , . . . , X(2)

m2; · · · ; X

(r)1 , . . . , X(r)

mr

), (17.18)

where m1 = m, X(1)i = Xi for every 1 ≤ i ≤ m and, with reference to (17.14),

ai = span{X

(i)1 , . . . , X(i)

mi

}for every i = 2, . . . , r.

STEP 3 (The group operations). It is well known that (see Theorem 2.2.13,page 129; see also the relevant Definition 2.2.11), the Campbell–Hausdorff operation� defines on a a Lie group structure (a,�). We can realize this group as a Lie groupon R

N in the usual way. Fixed a basis X for a as in Step 2, we identify a with RN

via the map

φ : RN → a, ξ = (ξ1, . . . , ξN ) →

N∑

j=1

ξj Xj .

The group law � is then turned into a group law4 ∗ on RN in the natural way

a ∗ b := φ−1(φ(a) � φ(b)

), a, b ∈ R

N.

The Lie group G := (RN, ∗) is obviously isomorphic to (a,�), for the mapφ : (G, ∗) → (a,�) is a Lie group isomorphism (which is also linear, when a andG ≡ R

N are equipped with their vector space structures).Analogously, we transfer the dilations {δλ}λ>0 on a defined in (17.16) to a group

of dilations {Dλ}λ>0 on G by setting

Dλ : G → G, Dλ(ξ) := φ−1(δλ(φ(ξ))

). (17.19)

Since δλ is an automorphism of the Lie group (a,�) (see Ex. 5 at the end of theChapter), it immediately follows that Dλ is an automorphism of the Lie group (G, ∗).

We claim that G = (RN, ∗,Dλ) is a homogeneous Carnot group. Let us provethis fact. With reference to the choice of the basis X for a in (17.18), for any ξ ∈ R

N ,we write

ξ = (ξ (1), . . . , ξ (r)), ξ (i) ∈ Rmi , i = 1, . . . , r.

4 In the previous sections, we denoted it by �X , and we also wrote φ−1 = πX .


First, we aim to show that, with this notation,

Dλ(ξ(1), ξ (2), . . . , ξ (r)) = (λξ (1), λ2ξ (2), . . . , λrξ (r)). (17.20)

We have

Dλ(ξ) = φ−1(δλ(φ(ξ))

) = φ−1

(δλ

(N∑

j=1

ξj Xj

))

= φ−1

(δλ

(r∑

i=1

mi∑

j=1

ξ(i)j X

(i)j

))= φ−1

(r∑

i=1

mi∑

j=1

ξ(i)j λi X

(i)j

)

= (λξ (1), λ2ξ (2), . . . , λrξ (r)).

Denote by Y(1)1 , . . . , Y

(1)m the first m vector fields of the Jacobian basis of g. In order

to prove that G is a homogeneous Carnot group, we have to show

Lie{Y (1)1 , . . . , Y (1)

m } = g. (17.21)

Let us prove this fact. Let Y(i)j denote the vector field in g coinciding at 0 with

∂ξ

(i)j

|0

(i.e. Y(i)j is the vector field in the Jacobian basis of g relative to the coordinate ξ

(i)j ).

Then, Y(i)j is Dλ-homogeneous of degree i: indeed, it holds

Y(i)j |ξ (u ◦ Dλ) = d0τξ (Y

(i)j |0)(u ◦ Dλ) = (Y

(i)j |0)(u ◦ Dλ ◦ τξ )

= (Y(i)j |0)(u ◦ τDλ(ξ) ◦ Dλ)

= (∂x

(i)j

∣∣0

)(u ◦ τDλ(ξ))(λx(1), . . . , λix(i), . . . , λrx(r))

= λi(∂y

(i)j

∣∣0

)(u ◦ τDλ(ξ))(y(1), . . . , y(i), . . . , y(r))

= λi (Y(i)j |0)(u ◦ τDλ(ξ)) = λiY

(i)j

∣∣Dλ(ξ)

u.

Hence, g is split into the direct sum g = H(1) ⊕ · · · ⊕ H(r), where H(i) is theset of the left-invariant vector fields, Dλ-homogeneous of degree i, i.e. H(i) =span{H(i)

1 , . . . , H(i)mi

}. In order to prove (17.21), it is sufficient to find another set{Z1, . . . , Zm} of independent vector fields in g which are Dλ-homogeneous of de-gree 1 and such that Lie{Z1, . . . , Zm} = g. This is done in the next step.

STEP 4 (Relations between g and a). Consider the following commutative (seeTheorem 2.1.59, page 119) diagram:

(G, ∗)φ

(a,�)

g

Exp G

dφLie(a).

Exp a


It is proved in Ex. 4 at the end of the chapter that Exp a is linear and it is a Liealgebra isomorphism, when a is equipped with its former structure of Lie algebra(and Lie(a) with its obvious one). Hence (recall that dφ is a Lie algebra isomorphism,see Theorem 2.1.50, page 114)

α : g → a, α := φ ◦ Exp G = Exp a ◦ dφ (17.22)

is a Lie algebra isomorphism. The reader should notice that the elements of a, whichare vector fields on R

n, have now been put into a bijective correspondence withvector fields on R

N (the elements of g). We consider the vector fields Z1, . . . , Zm ing corresponding to X1, . . . , Xm, respectively, i.e.

Zi ∈ g : α(Zi) = Xi for every i = 1, . . . , m. (17.23)

We claim that Zi is the i-th vector of the Jacobian basis for g. Indeed,

Zi |0u = (α−1(Xi))|0u = (dφ−1(Log a(Xi)))0u = (d0φ−1((Log a(Xi))0))u

= (Log a(Xi))0(u ◦ φ−1) = d

dt

∣∣∣∣t=0

(u ◦ φ−1)(tXi)

= d

dt

∣∣∣∣t=0

u(tei) = ∂

∂ξi

∣∣∣∣ξ=0

(u(ξ)).

For the fifth equality, we used the results of Ex. 6 at the end of the Chapter (recallthat (tX) � (sX) = (t + s)X for every X ∈ a and s, t ∈ R). Finally, we noticethat (17.23), the fact that α is a Lie algebra isomorphism, and Lie{X1, . . . , Xm} = a

imply Lie{Z1, . . . , Zm} = g. This completes the proof of Step 3.The aim of the following step is to find a “lifting” map π : R

N → Rn relating

the Zi’s to the Xi’s as in (17.12).STEP 5 (The lifting map π). Let us denote by Log the logarithmic map related

to the Lie group (G, ◦). Moreover, we write I for the identity map on Rn. We then

introduce the following map

π : RN −→ R

n,(17.24)

π(ξ) := eα(Log (ξ))I (0) :=∑

k≥0

(α(Log (ξ))

)k

k! I (0).

We explicitly remark that the map π is well defined. Indeed, if ξ ∈ RN ≡ G, then

Log (ξ) ∈ g, so that α(Log ξ) ∈ a. As a consequence, by the properties of δλ-homogeneity of the vector fields in a (see hypothesis (F1)), any power (α(Log (ξ)))k

with k ≥ n+1 annihilates the identity map I of Rn, so that the sum in (17.24) is finite.

By the very definition of α in (17.22), we have α(Log (ξ)) = φ(ξ) = ∑Nj=1 ξj Xj ,

so that

π(ξ) :=∑

k≥0

( ∑Nj=1 ξj Xj

)k

k! I (0) for all ξ ∈ RN . (17.25)

STEP 6 (The properties of π ). The map π has the following properties:


1. (HOMOGENEITY PROPERTY.) We have

π(Dλ(ξ)) = dλ(π(ξ)) ∀ ξ ∈ RN, λ > 0. (17.26)

2. (SURJECTIVE PROPERTY.) The map π : RN → R

n is onto.3. (EXPONENTIAL PROPERTY.) For every X ∈ a, we have

π(φ−1(X)) = exp(X)(0), (17.27)

where the right-hand side is meant as a formal exponential.5

4. (LIFTING PROPERTY.) It holds

dξπ(Zi) = (Xi)π(ξ) for every ξ ∈ RN . (17.28)

Let us begin with the homogeneity property. If we shortly write

Dλ(ξ) = (λβ1ξ1, . . . , λβN ξN), (17.29)

then from (17.25) we have

π(Dλ(ξ)) :=∑

k≥0

( ∑Nj=1 ξj λβj Xj

)k

k! I (0).

Now, let us fix i0 ∈ {1, . . . , n}. The claimed (17.26) will follow if we show that

∑

k≥0

( ∑Nj=1 ξj λβj Xj

)k

k! Ii0(0) = λσi0∑

k≥0

( ∑Nj=1 ξj Xj

)k

k! Ii0(0). (17.30)

Let us prove (17.30). From (17.15), the choice (17.18) of the basis X , the very de-finition (17.16) of δλ, (17.20) and the notation in (17.29) it follows that Xj is dλ-homogeneous of degree βj . Moreover, any non-vanishing summand in the left-handside of (17.30) has the following form (for some c ∈ R, i1, . . . , is ∈ {1, . . . , N})

c λβi1 +···+βis ξi1 · · · ξis Xi1 · · ·Xis Ii0(0) with βi1 + · · · + βis = σi0 ,

because the function Ii0 is dλ-homogeneous of degree σi0 . This proves (17.30).Let us turn to the surjective property. From (17.25) it is evident that

π(ξ) :=N∑

j=1

ξj Xj I (0) + O(|ξ |2), as ξ → 0.

5 Id est, the formal exponential of a high-order differential operator on Rn

exp(X)(0) :=∑

k≥0

Xk

k! I (0),

which is well defined for the sum is finite (recall the dλ-homogeneity properties of theoperators in a).


This givesJπ (0) = (

X1I (0) · · · XNI (0)),

so that, recalling (17.17),rank

(Jπ (0)

) = n.

By the rank theorem from elementary calculus, this proves the existence of an openneighborhood W ⊆ R

n of π(0) = 0 such that π : RN → W is surjective. We next

demonstrate that π is also onto Rn. Indeed, if x ∈ R

n, there exists λ = λ(x) > 0such that dλ(x) ∈ W . Let ξ ∈ R

N be such that π(ξ) = dλ(x). Then, thanks to(17.26),

π(D1/λ(ξ)) = d1/λ(π(ξ)) = d1/λ(dλ(x)) = x.

This shows that π is surjective. Obviously, π has polynomial entries, for the sum in(17.25) is finite.

Let us prove (17.27). This is just the definition (17.24) of π jointly with (α ◦Log ◦ φ−1)(X) = (φ ◦ φ)(X) = X.

Finally, Theorem 17.4.2 will be completely proved if we show that (17.28) holds(the lifting property of π). Let us fix ξ ∈ R

N . Since in (17.12) we are testing theequality of two first order differential operators on R

n, (17.12) follows if we demon-strate that

dξπ(Zi)Is = (Xi)π(ξ)Is for every s = 1, . . . , n, (17.31)

where Is(x) = xs is the s-th coordinate projection of Rn. Recalling that Zi is the i-th

vector of the Jacobian basis for g, we have

dξπ(Zi)Is = Zi |ξ (Is ◦ π) = d

dt

∣∣∣∣t=0

((Is ◦ π)(ξ ∗ (tei))

)

= d

dt

∣∣∣∣t=0

(Is ◦ π)(φ−1(

φ(ξ) � φ(tei)))

= d

dt

∣∣∣∣t=0

(Is ◦ π ◦ φ−1)

((N∑

j=1

ξjXj

)� (tXi)

)

= d

dt

∣∣∣∣t=0

Is ◦ exp

((N∑

j=1

ξjXj

)� (tXi)

)(0)

= d

dt

∣∣∣∣t=0

Is ◦(

exp(tXi)

(exp

(N∑

j=1

ξjXj

)(0)

))g

= (XiIs)

(exp

(N∑

j=1

ξjXj

)(0)

)= (XiIs)(π(ξ)),

which gives (17.31). Here we used the following facts: The first equality is the defi-nition of the differential; the second equality follows from the left-invariance of Zi ;the third equality is the definition of ∗; the fourth equality is the definition of φ; the


fifth equality follows from (17.27); the sixth equality follows from the Campbell–Hausdorff formula (applied to the vector fields of a which have dλ-homogeneityproperties; see (15.3) page 595); the seventh equality is trivial; the last equality fol-lows again from (17.27). This ends the proof. ��Bibliographical Notes. Different proofs of the lifting procedure have been pro-vided. Besides the pioneering paper by L.P. Rothschild and E.M. Stein [RS76],we refer the reader to L. Hörmander and A. Melin [HM78], G.B. Folland [Fol77],R.W. Goodman [Goo78]. In [Fol77], the case described in Section 17.4 is considered.Some of the topics presented in this chapter also appear in [BU05a].


Ex. 1) Let α ∈ R be fixed. Lift the Carnot group on R4 with the composition law

⎛

⎜⎜⎜⎜⎝

x1 + y1,

x2 + y2,

x3 + y3 + 12 (x1 y2 − x2 y1),

x4 + y4 + 12 (x1 y3 − x3 y1) + α

2 (x2 y3 − x3 y2)

+ 112 (x1 − y1) (x1 y2 − x2 y1) + α

12 (x2 − y2) (x1 y2 − x2 y1)

⎞

⎟⎟⎟⎟⎠.

Verify that, following the notation and definitions in Section 17.1, the follow-ing facts hold:a) The Jacobian basis Z for g is

Z1 = ∂1 − 1

2x2 ∂3 −

(1

2x3 + 1

12x2(x1 + α x2)

)∂4,

Z2 = ∂2 + 1

2x1 ∂3 +

(−1

2α x3 + 1

12x1(x1 + α x2)

)∂4,

Z3 = ∂3 +(

1

2x1 + 1

2α x2

)∂4,

Z4 = ∂4.

b) The Campbell–Hausdorff law related to Z on R4 coincides with ◦.

Hence, the Jacobian basis X for r4 is given by

X1 = ∂1 − 1

2ξ2 ∂3 −

(1

2ξ3 + 1

12ξ2(ξ1 + α ξ2)

)∂4,

X2 = ∂2 + 1

2ξ1 ∂3 +

(−1

2α ξ3 + 1

12ξ1(ξ1 + α ξ2)

)∂4,


X3 = ∂3 +(

1

2ξ1 + 1

2α ξ2

)∂4,

X4 = ∂4.

c) Consider f2,3. We have dim f2,3 = 5. Let Π : f2,3 → g be the Lie algebramorphism such that Π(Fi) = Zi for every i = 1, 2. The Hall basis forf2,3 is given by

F1, F2, F3 = [F2, F1], F4 = [[F2, F1], F1],F5 = [[F2, F1], F2].

The following is a basis F for f2,3, as in Proposition 17.1.1,

F1, F2, −F3, F4; F5 − α F4.

d) The Lie group (R5,�F ) is such that ξ �F η equals

⎛

⎜⎜⎜⎜⎜⎜⎝

ξ1 + η1,

ξ2 + η2,

ξ3 + η3 + 12 (ξ1 η2 − ξ2 η1),

ξ4 + η4 + 12 (ξ1 η3 − ξ3 η1) + α

2 (ξ2 η3 − ξ3 η2)

+ 112 (ξ1 − η1) (ξ1 η2 − ξ2 η1) + α

12 (ξ2 − η2) (ξ1 η2 − ξ2 η1),

ξ5 + η5 + 12 (ξ2 η3 − ξ3 η2) + 1

12 (ξ2 − η2) (ξ1 η2 − ξ2 η1)

⎞

⎟⎟⎟⎟⎟⎟⎠.

Verify that the first 4 vector fields of the Jacobian basis for r5 lift Xi , forevery i = 1, . . . , 4, as stated in Lemma 17.1.3.

e) Verify thatψ = Exp ◦ π−1

Z : (R4,�Z ) −→ (G, ◦)

satisfies ψ(ξ1, ξ2, ξ3, ξ4) = (ξ1, ξ2, ξ3, ξ4) ∈ G. In particular, the Liegroup isomorphism Φ : (R5,�F ) → (G × R, •) is defined by

Φ(ξ1, ξ2, ξ3, ξ4, ξ5) = (ξ1, ξ2, ξ3, ξ4; ξ5) ∈ G × R.

As a consequence, for the group law • on G × R, we have that x • y

equals

⎛

⎜⎜⎜⎜⎜⎜⎝

x1 + y1,

x2 + y2,

x3 + y3 + 12 (x1 y2 − x2 y1),

x4 + y4 + 12 (x1 y3 − x3 y1) + α

2 (x2 y3 − x3 y2)

+ 112 (x1 − y1) (x1 y2 − x2 y1) + α

12 (x2 − y2) (x1 y2 − x2 y1),

x5 + y5 + 12 (x2 y3 − x3 y2) + 1

12 (x2 − y2) (x1 y2 − x2 y1)

⎞

⎟⎟⎟⎟⎟⎟⎠.

Find the Jacobian basis W1, . . . ,W5 for g r1. Verify that, in particular, itholds


W1 = ∂1 − 1

2x2 ∂3 −

(1

2x3 + 1

12x2(x1 + α x2)

)∂4− 1

12x2

2 ∂5,

W2 = ∂2 + 1

2x1 ∂3 +

(−1

2α x3 + 1

12x1(x1 + α x2)

)∂4

+(

−12x3 + 1

12x1x2

)∂5.

Verify that, as stated in Lemma 17.1.4, Wi lifts dϑ(Wi) = Zi , for everyi = 1, . . . , 4.

Ex. 2) Give a detailed proof of Lemma 17.3.4, page 666.Ex. 3) Prove properties (i)–(ii)–(iii) in Step 1, page 670.Ex. 4) Prove the following result.

Proposition 17.5.1. Let a be a (finite-dimensional) nilpotent Lie algebra.Consider the Lie group (a,�), where � is the Campbell–Hausdorff opera-tion6 on a. Denote by Lie(a) the Lie algebra of the Lie group (a,�) and byExp a : Lie(a) → a the relevant exponential map. Then, Exp a is linear andit is a Lie algebra isomorphism, when a is equipped with its former structureof Lie algebra.

(Hint: By the third fundamental theorem of Lie (see Theorem 2.2.14, page 130)there exists a connected, simply connected Lie group (G, ·) such that g :=Lie(G) is isomorphic to a as Lie algebras. Let ϕ : a → g be the relevant Liealgebra isomorphism. Then ϕ : (a,�) → (g,�) is a Lie group isomorphism(see Remark 2.2.16, page 130). Consider the following commutative diagram

(a,�)ϕ

(g,�)Exp G

(G, ·)

Lie(a)d ϕ

Exp a

Lie(g)d Exp G

g.

Exp GA

We have

Exp a = ϕ−1 ◦ (Exp G)−1 ◦ Exp G ◦ dExp G ◦ dϕ = ϕ−1 ◦ dExp G ◦ dϕ,

which is evidently a Lie algebra isomorphism.)Ex. 5) Let a be a (finite-dimensional) nilpotent Lie algebra. Consider the Lie group

(a,�), where � is the Campbell–Hausdorff operation on a. Suppose a is strat-ified, i.e. it admits a decomposition of the form

a = a1 ⊕ a2 ⊕ · · · ⊕ ar with

{ [a1, ai−1] = ai if 2 ≤ i ≤ r ,

[a1, ar ] = {0}.Define a group of dilations {δλ}λ>0 on a in the following way:

6 See Theorem 2.2.13, page 129; see also the relevant Definition 2.2.11.


δλ : a → a, δλ

(r∑

i=1

Ai

):=

r∑

i=1

λi Ai, where Ai ∈ ai for i = 1, . . . , r .

Prove that δλ is an automorphism of the Lie group (a,�). (Hint: First showthat if X ∈ ai and Y ∈ aj , then [X, Y ] ∈ ai+j (where ai := {0} for i > r).Then prove that δλ

([X, Y ]) = [δλ(X), δλ(Y )] for every X, Y ∈ a. End byusing Remark 2.2.16.)

Ex. 6) Let V be a real (finite-dimensional) vector space equipped with a Lie groupstructure by the operation ∗. Denote by v the relevant Lie algebra, by Exp andLog the relevant exponential and logarithmic maps. Furthermore, supposethat

(�) (tv) ∗ (sv) = (t + s)v for every v ∈ V and s, t ∈ R.

Let us introduce the following notation: for every v ∈ V , Ξ(v) denotes thetangent vector at 0 ∈ V such that

Ξ(v)f = d

dt

∣∣∣∣t=0

(f (tv)

) ∀ V ∈ C∞(V , R).

Prove the following facts:a) Show that Ξ(v) is indeed an element of T0(V );b) Show that

T0(V ) = {Ξ(v) : v ∈ V },whence the natural identification between T0(V ) and V ;

c) Denote by α : v → T0(V ) the usual identification between v and T0(V ),i.e.

α(X) = X0 for every X ∈ v.

Prove that the flow of X ∈ v is t → t v, where v = Ξ−1(X0).d) Derive that Exp (X) = Ξ−1(X0), i.e.

(Exp ◦ α−1 ◦ Ξ

)(v) = v for every v ∈ V ,

or analogously, α ◦ Log = Ξ , that is, for every v ∈ V , it holds

(Log v)0f = d

dt

∣∣∣∣t=0

f (tv) for every f ∈ C∞(V ).

(Hint: The main task is to prove (c). We must show that γ : R → V ,γ (t) = tv solves the Cauchy problem γ (t) = Xγ(t), γ (0) = 0. Note that

Xγ(t) = d0τγ (t)(X0) = d0τγ (t)(Ξ(v)).

We thus have to prove that, for every f ∈ C∞(V ),

d

dt

∣∣∣∣t

(f (t v)

) ?= d0τγ (t)(Ξ(v))f = [· · ·] = d

ds

∣∣∣∣s=0

f ((tv) ∗ (sv)),

which follows from (�).)


Ex. 7) Prove that the vector fields on R3 (whose points are denoted by (x, y, t))

defined byX := ∂x + y2 ∂t , Y := ∂y

are dλ-homogeneous of degree 1, where

dλ : R3 −→ R

3, dλ(x, y, t) := (λx, λy, λ3t),

and satisfy hypotheses (F1)–(F2) of page 669 (these are the fields consideredin Section 4.4.2, page 212, as an example of fields not satisfying hypothesis(H1) of page 191). Lift them by the method illustrated in Section 17.4.

18

Groups of Heisenberg Type

In this chapter, we treat the so-called Heisenberg-type groups (also referred to asH-type groups), a significant class of Carnot groups of step two, generalizing theclassical Heisenberg–Weyl groups.

In Section 18.2, we give a direct characterization of Heisenberg-type groups, viaa suitable choice of a coordinate system, with respect to which the canonical sub-Laplacian has a simple and suggestive representation. We compare the definitionof (abstract) H-type group given in the present chapter to our previous definition ofprototype H-type group (see Definition 3.6.1): roughly speaking, any abstract H-typegroup is naturally isomorphic to a prototype H-type group.

In Section 18.3, we prove in a direct way the existence of a fundamental so-lution for some sub-Laplacians on Heisenberg-type groups, following the proof byA. Kaplan in [Kap80]. In Sections 18.4 and 18.5 we define an inversion and a Kelvin-transform on H-type groups, generalizing those from the classical theory of Laplace’soperator. Unfortunately, this Kelvin transform has remarkable properties only on asub-class of the H-type groups: the so-called Iwasawa-type groups (the relevant def-inition is given in Section 18.4). Indeed given a H-type group H, the following resultholds: The H-Kelvin transform of a ΔH-harmonic function is ΔH-harmonic if andonly if H is of Iwasawa-type.

18.1 Heisenberg-type Groups

Let us begin with the definition of Heisenberg-type algebra and Heisenberg-typegroup.

Definition 18.1.1 (H-type algebra, H-type group). A Heisenberg-type algebra(H-type algebra, in short) is a finite-dimensional real Lie algebra g which can beendowed with an inner product 〈 , 〉 such that

[z⊥, z⊥] = z,

where z is the center of g and moreover, for every fixed z ∈ z, the map

682 18 Groups of Heisenberg Type

Jz : z⊥ → z

⊥

defined by1

〈Jz(v), w〉 = 〈z, [v,w]〉 ∀ w ∈ z⊥ (18.1)

is an orthogonal map whenever 〈z, z〉 = 1.A Heisenberg-type group (H-type group, in short) is a connected and simply

connected Lie group whose Lie algebra is an H-type algebra.

We recall that the center of g is, by definition,

z = {z ∈ g | [z, g] = 0 ∀ g ∈ g}.In the following we shall always suppose that z is not the null subspace. We shalloften use the notation

b := z⊥.

Let us consider some examples.

Example 18.1.2 (The Heisenberg–Weyl group is an H-type group). The classicalHeisenberg–Weyl group H

N is an H-type group. As usual, we shall denote by(x, y, t) the points of H

N ≡ R2N+1, x, y ∈ R

N , t ∈ R. Consider the vector fields

Xj := ∂xj+ 2yj ∂t , Yj := ∂yj

− 2xj ∂t , j = 1, . . . , N.

Then we have

g = Lie{Xj , Yj | j = 1, . . . , N}, z = span{∂t }.Let us now fix on g the coordinate system associated to the basis

{X1, . . . , XN, Y1, . . . , YN ,−4 ∂t },and let us consider on g the standard inner product generated by this system. In otherwords, we identify g with R

2N+1 in the following way

g N∑

j=1

(aj Xj + bj Yj ) − 4c ∂t ←→ (a, b, c) ∈ R2N+1,

and we set⟨(a, b, c), (a′, b′, c′)

⟩ :=N∑

j=1

(aj a′j + bj b′

j ) + c c′.

In this coordinate system, the Lie bracket has the following expression

[(a, b, c), (a′, b′, c′)] =(

0, 0,

N∑

j=1

(aj b′j − bj a′

j )

).

1 Jz is well-posed, see (18.3) below.

18.1 Heisenberg-type Groups 683

Then we haveb := z

⊥ = span{Xj , Yj | j = 1, . . . , N},which gives [b, b] = z. Let now z ∈ z be such that |z| = 1, i.e.

z = ∓4 ∂t ≡ (0, 0,±1).

With the notation of Definition 18.1.1, we have

Jz

(N∑

j=1

(aj Xj + bj Yj )

)= ±

N∑

j=1

(−bj Xj + aj Yj ),

and, clearly, Jz is an orthogonal endomorphism of b. �Example 18.1.3. It can be proved (and it will be proved in Section 18.2) that thefollowing group law on R

6 defines an H-type group:

x ◦y :=(

x(1) +y(1), x(2)1 +y

(2)1 + 1

2〈P1 x(1), y(1)〉, x

(2)2 +y

(2)2 + 1

2〈P2 x(1), y(1)〉

),

where x = (x(1), x(2)1 , x

(2)2 ) ∈ R

6, x(1) ∈ R4, x

(2)1 , x

(2)2 ∈ R and

P1 :=√

2

2

⎛

⎜⎝

0 1 1 0−1 0 0 −1−1 0 0 10 1 −1 0

⎞

⎟⎠ , P2 :=√

2

2

⎛

⎜⎝

0 −1 1 01 0 0 −1

−1 0 0 −10 1 1 0

⎞

⎟⎠ .

Example 18.1.4. The Lie group obtained as the direct product of the group (R,+)

with the Heisenberg group (H1, ◦) is not an H-type group. More precisely, if weconsider the algebra g generated by the vector fields on R

4

∂x1 , ∂x2 + 2x3∂x4 , ∂x3 − 2x2∂x4 ,

we observe that the center z of g is given by span{∂x1 , ∂x4}, but ∂x1 cannot be ob-tained as linear combinations of brackets in g. Thus the hypothesis [b, b] = z inDefinition 18.1.1 cannot be satisfied by any b ⊆ g. �Remark 18.1.5. If a Lie algebra g has a center z such that dim(g) − dim(z) is odd,then g is not an H-type algebra. This fact will be proved in Section 18.2 (see alsoRemark 18.1.6 below). �Remark 18.1.6. We have the following general result (see A. Kaplan [Kap80, Corol-lary 1]). Let N1, N2 ∈ N \ {0}. Then there exists an H-type algebra of dimensionN1 + N2 whose center has dimension N2 (thus N1 is the dimension of the orthog-onal completion of the center) if and only if N2 < ρ(N1), where ρ is the so-calledHurwitz–Radon function, i.e.

ρ : N → N, ρ(n) := 8p + q, where n = (odd) · 24p+q, 0 ≤ q ≤ 3.

We observe that ρ(N1) = 0 if N1 is odd. Thus there cannot be H-type algebras wherethe dimension of the orthogonal completion of the center is odd.


We explicitly remark that the relation

〈Jz(v), v′〉 = 〈z, [v, v′]〉 ∀ v′ ∈ b (18.2)

in Definition 18.1.1 actually defines an endomorphism of b. Indeed, for fixed v ∈ b

and z ∈ z, the map

Ψ : b → R, v′ �→ Ψ (v′) := 〈z, [v, v′]〉is linear. Hence, there exists exactly one w ∈ b (depending only on v and z) suchthat Ψ (v′) = 〈w, v′〉 for every v′ ∈ b. Then we set Jz(v) := w. We finally show that,for fixed z ∈ z, Jz(·) is linear: if u, v ∈ b, α, β ∈ R, we have

〈Jz(αu + βv), v′〉 = 〈z, [αu + βv, v′]〉 = α〈z, [u, v′]〉 + β〈z, [v, v′]〉= α〈Jz(u), v′〉 + β〈Jz(v), v′〉= 〈αJz(u) + βJz(v), v′〉 ∀ v′ ∈ b.

Then we have

Jz(αu + βv) − αJz(u) − βJz(v) ∈ b⊥ ∩ b = {0}.

Moreover, for fixed v ∈ b, the map J(·)(v) : z → b, z �→ Jz(v), is a linear map.Indeed, if z, z′ ∈ z and α, β ∈ R, we have

〈J(αz+βz′)(v), v′〉 = 〈αz + βz′, [v, v′]〉 = α〈z, [v, v′]〉 + β〈z′, [v, v′]〉= α〈Jz(v), v′〉 + β〈Jz′(v), v′〉= 〈αJz(v) + βJz′(v), v′〉 ∀ v′ ∈ b.

Thus J(αz+βz′)(v) = αJz(v) + βJz′(v) as above. This shows that the map

J(·) : z → End(b), z �→ Jz, defined by (18.2) is well-posed and linear. (18.3)

Let us make some other simple but useful remarks on H-type algebras.

Remark 18.1.7. Let g be an H-type algebra. With the above notation, we have:

1) is a nilpotent Lie algebra of step two;2) g = b ⊕ z, [b, b] = z, [b, z] = {0}; hence, if G is an H-type group, then G is a

step two Carnot group;3) the notion of H-type algebra (respectively, of H-type group) is invariant under Lie

algebra (respectively, Lie group) isomorphisms.

Proof. 1) Since b is the orthogonal completion of z �= {0} with respect to 〈 , 〉, wehave g = b ⊕ z. Let now g1, g2, g3 ∈ g. We have gi = vi + zi , with vi ∈ b andzi ∈ z, for every i = 1, 2, 3. Since z is the center of g, every zi commutes with anyelement of g. As a consequence,

[g1, [g2, g3]] = [v1 + z1, [v2 + z2, v3 + z3]] = [v1, [v2, v3]] = 0,

18.1 Heisenberg-type Groups 685

since we have [v2, v3] ∈ [b, b] = z.2) It directly follows from the above argument.3) Let g be an H-type algebra, let g∗ be a Lie algebra isomorphic to g and let

ϕ : g∗ → g be a Lie algebra isomorphism. Since ϕ is a vector space isomorphism,the bilinear form

〈g∗1 , g∗

2〉∗ := 〈ϕ(g∗1), ϕ(g∗

2)〉, g∗1 , g∗

2 ∈ g∗,

is an inner product in g∗. Let z∗ := ϕ−1(z). It is immediate to recognize that z∗ isthe center of g∗. Moreover, setting b∗ := (z∗)⊥ (orthogonal completion of z∗ withrespect to 〈 , 〉∗), we have b∗ = ϕ−1(z⊥) = ϕ−1(b). It follows that

[b∗, b∗] = [ϕ−1(b), ϕ−1(b)] = ϕ−1([b, b]) = ϕ−1(z) = z∗.

For z∗ ∈ z∗ fixed, we consider the linear map J ∗z∗ : b∗ → b∗ defined as follows: if

v∗ ∈ b, J ∗z∗(v∗) is defined by the identity

〈J ∗z∗(v∗), (v′)∗〉∗ = 〈z∗, [v∗, (v′)∗]〉∗ ∀ (v′)∗ ∈ b

∗.

It is immediate to verify that

J ∗z∗ = ϕ−1 ◦ Jϕ(z∗) ◦ ϕ.

Let now z∗ ∈ z∗ be such that 〈 z∗, z∗ 〉∗ = 1. Then 〈ϕ(z∗), ϕ(z∗) 〉 = 1. Thus J ∗z∗ is

an orthogonal map. �Proposition 18.1.8. Let g be an H-type algebra. With the above notation,

〈Jz(v), v〉 = 0, (18.4a)

〈Jz(v), v′〉 = −〈v, Jz(v′)〉, (18.4b)

|Jz(v)| = |z| · |v|, (18.4c)

〈Jz(v), Jz′(v)〉 = 〈z, z′〉 · |v|2, (18.4d)

[v, Jz(v)] = |v|2 · z (18.4e)

for every z, z′ ∈ z and for every v, v′ ∈ b.

Proof. Let z, z′ ∈ z and v, v′ ∈ b be fixed. From (18.2) we have

〈Jz(v), v〉 = 〈z, [v, v]〉 = 0,

which gives (18.4a). It follows that

0 = 〈Jz(v + v′), v + v′〉 = 〈Jz(v), v〉 + 〈Jz(v), v′〉 + 〈Jz(v′), v〉 + 〈Jz(v

′), v′〉= 〈Jz(v), v′〉 + 〈Jz(v

′), v〉,which gives (18.4b). If z = 0, we have Jz ≡ 0 and (18.4c) trivially follows. If z �= 0,from (18.3) it follows that Jz = |z| Jz/|z|. In this case, by the definition of H-typealgebra, we have that Jz/|z| is an orthogonal map. As a consequence,


|Jz(v)| = |z| · |Jz/|z|(v)| = |z| · |v|.This proves (18.4c). Using (18.3), it is immediate to verify that we have

〈Jz+z′(v), Jz+z′(v)〉 = 〈Jz(v), Jz(v)〉 + 〈Jz′(v), Jz′(v)〉 + 2 〈Jz(v), Jz′(v)〉.Thus, from (18.4c) it follows that

〈Jz(v), Jz′(v)〉 = 1

2(〈Jz+z′(v), Jz+z′(v)〉 − 〈Jz(v), Jz(v)〉 − 〈Jz′(v), Jz′(v)〉)

= 1

2(|z + z′|2 · |v|2 − |z|2 · |v|2 − |z′|2 · |v|2) = 〈z, z′〉 · |v|2,

which gives (18.4d). Finally, we have

〈z′, [v, Jz(v)]〉 = 〈Jz′(v), Jz(v)〉 = 〈z′, z〉 · |v|2 = 〈z′, |v|2 · z〉.The first equality follows from (18.2), the second equality follows from (18.4d). Asa consequence, we have

〈z′, [v, Jz(v)] − |v|2 · z〉 = 0 ∀ z′ ∈ z.

Since |v|2 · z ∈ z and [v, Jz(v)] ∈ [b, b] = z, we immediately get (18.4e). �

18.2 A Direct Characterization of H-type Groups

The main aim of this section is to prove that (under a suitable system of coordinates)the group law of any H-type group has a somewhat explicit form (see (18.6) below).Roughly speaking, following our definitions from Section 3.6 (page 169), any ab-stract H-type group is naturally isomorphic to a prototype H-type group. Moreover,we furnish several useful details on H-type groups.

By means of the natural identification of a Carnot group with its Lie algebra (seeSection 2.2 on page 121 for details), we already proved that (see Theorem 3.2.2,page 160, for the precise statement) any Carnot group of step two is canonicallyisomorphic to R

N equipped with the following group law (N = m + n, x(1) ∈ Rm,

x(2) ∈ Rn)

(x(1), x(2)) ◦ (y(1), y(2)) =(

x(1)j + y

(1)j , j = 1, . . . , m

x(2)j + y

(2)j + 1

2 〈B(j)x(1), y(1)〉, j = 1, . . . , n

),

(18.5)where the B(j)’s are m × m linearly independent skew-symmetric matrices. Let nowG be a general H-type group according to Definition 18.1.1. We set

m := dim(z⊥) and n := dim(z).

Since G has step two and since the stratification of the Lie algebra g is evidentlyz⊥ ⊕ z (see Remark 18.1.7), in the sequel we shall fix on G a system of coordinates

18.2 A Direct Characterization of H-type Groups 687

(x, t), and we shall suppose that the dilations on G are δλ(x, t) = (λx, λ2t) and thatthe group law has the form

(x, t) ◦ (ξ, τ ) =(

xj + ξj , j = 1, . . . , m

tj + τj + 12 〈U(j)x, ξ 〉, j = 1, . . . , n

),

x, ξ ∈ Rm, t, τ ∈ R

n, (18.6)

for suitable skew-symmetric matrices U(j)’s. Our main aim is to give a necessaryand sufficient condition on the U(j)’s such that the composition law (18.6) defineson R

m+n an H-type group. A complete answer is given by the following result.

Theorem 18.2.1 (Characterization). G is an H-type group if and only if G is (iso-morphic to) R

m+n with the group law (18.6) and the matrices U(1), . . . , U(n) havethe following properties:

1) U(j) is an m × m skew-symmetric and orthogonal matrix for every j ≤ n;2) U(i) U(j) + U(j) U(i) = 0 for every i, j ∈ {1, . . . , n} with i �= j .

Remark 18.2.2. Rm+n equipped with the group law (18.6) (the matrices U(j)’s be-

ing as in Theorem 18.2.1) will be referred to as a prototype H-type group. Theo-rem 18.2.1 states that any H-type group is naturally isomorphic to a prototype H-typegroup.

Remark 18.2.3. Conditions 1) and 2) imply that U(1), . . . , U(n) are linearly indepen-dent. Namely, as it will appear from the proof, if 1) and 2) hold, then

∑ns=1 zsU

(s) isthe product of |z| times an orthogonal matrix.

For example, the following three matrices

U(1) =⎛

⎜⎝

0 −1 0 01 0 0 00 0 0 −10 0 1 0

⎞

⎟⎠ , U(2) =⎛

⎜⎝

0 0 1 00 0 0 −1

−1 0 0 00 1 0 0

⎞

⎟⎠ ,

U(3) =⎛

⎜⎝

0 0 0 10 0 1 00 −1 0 0

−1 0 0 0

⎞

⎟⎠

satisfy conditions 1)–2) and, together with (18.6), define on R7 an H-type group

whose center has dimension 3.

Proof (of Theorem 18.2.1). First of all, we prove the “only if” part of the theorem.Let G be an H-type group: we make use of all the notation in Definition 18.1.1. LetB1, . . . , Bm and Z1, . . . , Zn be orthonormal bases for b := z⊥ and z, respectively.The hypothesis [b, b] = z ensures that there exist scalars U

(s)i,j such that

[Bi, Bj ] =n∑

s=1

U(s)j,i Zs ∀ i, j ∈ {1, . . . , m}. (18.7)


With the notation in (18.7), we define the following square matrices

U(s) := (U

(s)i,j

)i,j≤m

, s = 1, . . . , n. (18.8)

Since [Bi, Bj ] = −[Bj , Bi], and recalling that Z1, . . . , Zn is a basis of z, U(s) isskew-symmetric. We now recall that the group (G, ◦) is canonically isomorphic (viathe exponential map) to its Lie algebra g (endowed with the Campbell–Hausdorffoperation (X, Y ) �→ X + Y + 1

2 [X, Y ]). As a consequence, by means of the fixedbasis B1, . . . , Bm,Z1, . . . , Zn on g, we can identify G to g ≡ R

m+n with the grouplaw (18.6), where the matrices U(j)’s are as in (18.8).

We fix v = ∑mi=1 viBi and z = ∑n

i=1 ziZi . We look for x1, . . . , xm ∈ R suchthat Jz(v) = ∑m

i=1 xiBi satisfies condition (18.1) of Definition 18.1.1 for everyw = ∑m

i=1 wiBi . Precisely,

m∑

j=1

xjwj =⟨

m∑

i=1

xiBi,

m∑

i=1

wiBi

⟩= 〈Jz(v), w〉 = 〈z, [v,w]〉

=⟨

n∑

i=1

zi Zi,

[m∑

i=1

vi Bi,

m∑

j=1

wj Bj

]⟩

=⟨

n∑

i=1

ziZi,

m∑

i,j=1

viwj [Bi, Bj ]⟩

=⟨

n∑

i=1

ziZi,

m∑

i,j=1

viwj

n∑

s=1

U(s)j,i Zs

⟩

=m∑

j=1

wj

(n∑

r,s=1

m∑

i=1

vi U(r)j,i zs

⟨Zr,Zs

⟩)

=m∑

j=1

wj

(n∑

s=1

m∑

i=1

viU(s)j,i zs

). (18.9)

Since w is arbitrary, this gives

xj =∑

s≤n

∑

i≤m

viU(s)j,i zs,

whence

Jz : b → b,

m∑

j=1

vjBj �→m∑

j=1

(∑

s≤n

∑

i≤m

viU(s)j,i zs

)Bj .

As a consequence, w.r.t. the orthonormal basis B1, . . . , Bm for b, the endomorphismJz is represented by the following matrix


n∑

s=1

zs

⎛

⎜⎝U

(s)1,1 · · · U

(s)1,m

.... . .

...

U(s)m,1 · · · U

(s)m,m

⎞

⎟⎠ =n∑

s=1

zsU(s). (18.10)

By Definition 18.1.1, this matrix has to be orthogonal, whenever∑n

s=1(zs)2 = 1.

In particular, this implies that every U(s) is orthogonal (whence property 1) of theassertion is proved). We explicitly remark that since U(s) is both skew-symmetric andorthogonal, U(s) has no real eigenvalues, whence m is necessarily even. Moreover,

−U(s)U(s) = Im

(Im denotes the unit matrix of order m). If∑n

s=1(zs)2 = 1, the matrix

∑ns=1 zsU

(s)

is orthogonal if and only if

Im =(

n∑

s=1

zsU(s)

)·(

n∑

s=1

zsU(s)

)T

= −∑

r,s≤n

zrzs U(r)U(s)

= −∑

r≤n

z2r U(r)U(r) −

∑

r,s≤n, r �=s

zrzs U(r)U(s)

= Im −∑

r,s≤n, r �=s

zrzs U(r)U(s).

We have here used the fact that (U(r))T = −U(r) and

−(U(r))2 = U(r) · (−U(r)) = U(r) · (U(r))T = Im,

since U(r) is skew-symmetric and orthogonal. Therefore, we have

n∑

r,s=1, r �=s

zrzs U(r)U(s) = 0 ∀ z1, . . . , zn :n∑

s=1

z2s = 1. (18.11)

If in (18.11) we take z = (0, . . . , 1/√

2, . . . , 1/√

2, . . . , 0), we obtain

U(i)U(j) + U(j)U(i) = 0 for every i, j ∈ {1, . . . , n} with i �= j , (18.12)

which is property 2) of the assertion. Since we also have∑

r,s≤n, r �=s

zrzs U(r)U(s) =∑

r,s≤n, r<s

zrzs (U(r)U(s) + U(s)U(r)),

(18.11) turns out to be equivalent to (18.12).We now prove the “if” part of the theorem. Let U(1), . . . , U(n) be matrices having

properties 1)–2) of the assertion. Suppose Rm+n is endowed with the composition

law (18.6). It is immediately verified that ◦ defines a Lie group, nilpotent of step atmost two, in which the identity is the origin and the inverse of (x, t) is (−x,−t).Moreover, δλ(x, t) = (λx, λ2t) is a group of automorphisms. An easy computation


shows that the vector field in the algebra g of G = (Rm+n, ◦) that agrees at the originwith ∂/∂xj (j = 1, . . . , m) is given by

Xj = (∂/∂xj ) + 1

2

n∑

s=1

(m∑

i=1

U(s)j,i xi

)(∂/∂ts), (18.13)

and that g is spanned by

X1, . . . , Xm, ∂/∂t1, . . . , ∂/∂tn.

From (18.13) and the skew-symmetry of U(s) we obtain

[Xi,Xj ] =n∑

s=1

U(s)j,i (∂/∂ts)

for every i, j ∈ {1, . . . , m}. Now, since U(1), . . . , U(n) are linearly independent (seeRemark 18.2.3), the dimension of the vector space spanned by (U

(1)j,i , . . . , U

(n)j,i ) as

i, j ∈ {1, . . . , m} equals n. As a consequence, G is a homogeneous Carnot group.For every s = 1, . . . , n, we set Zs = ∂/∂ts . We claim that z, the center of g,

is spanned by Z1, . . . , Zn. Indeed, we suppose by contradiction that (for suitablescalars αi’s)

∑mi=1 αiXi ∈ z, i.e.

0 =[

m∑

i=1

αiXi,Xj

]=

n∑

s=1

m∑

i=1

αiU(s)j,i Zs for every j ∈ {1, . . . , m}.

Since the Zs’s are linearly independent, this means that

m∑

i=1

αiU(s)j,i = 0

for every s ≤ n and every j ≤ m, i.e. α = (α1, . . . , αm) belongs to the kernel ofthe transpose matrix of U(s) for every s ≤ n. Since every U(s) is orthogonal, this ispossible only if α = 0, which proves the claim.

Finally, let 〈 , 〉 be the standard inner product on g w.r.t. the basis

X1, . . . , Xm, Z1, . . . , Zn.

For what has been proved above, we have z⊥ = span{X1, . . . , Xm} and

[z⊥, z⊥] = span{Z1, . . . , Zn} = z.

By means of a computation analogous to (18.9), it is easy to recognize that prop-erties 1) and 2) of the assertion ensure that, with the above choice of 〈 , 〉, G is anH-type group according to Definition 18.1.1. Indeed, for a fixed z = ∑n

j=1 zj Zj ,setting


v :=m∑

i=1

vi Xi, w :=m∑

i=1

wi Xi,

we have

〈z, [v,w]〉 =⟨

n∑

i=1

zi Zi,

[m∑

i=1

vi Xi,

m∑

j=1

wj Xj

]⟩

=⟨

n∑

i=1

zi Zi,

m∑

i,j=1

vi wj [Xi,Xj ]⟩

=⟨

n∑

i=1

zi Zi,

m∑

i,j=1

vi wj

n∑

s=1

U(s)j,i Zs

⟩

=m∑

j=1

wj

(n∑

r,s=1

m∑

i=1

vi U(s)j,i zr 〈Zr,Zs〉

)

(〈Zr,Zs〉 = δr,s) =m∑

j=1

wj

(n∑

r=1

m∑

i=1

vi U(r)j,i zr

).

As a consequence, by definition of 〈, 〉, setting

Jz(v) :=m∑

j=1

xj Xj

with

xj :=n∑

r=1

m∑

i=1

vi U(r)j,i zr ,

we have〈Jz(v), w〉 = 〈z, [v,w]〉

for every w = ∑mi=1 wi Xi . Repeating exactly the same computations previously

done, we see that if U(1), . . . , U(n) are orthogonal matrices satisfying

U(r) · U(s) + U(s) · U(r) = 0

for every r, s ∈ {1, . . . , n} with r �= s, then the map

m∑

j=1

vj Xj �→ Jz(v)

defines an orthogonal endomorphism on span{X1, . . . , Xm} for every choice of z =∑nj=1 zj Zj such that

∑nj=1 |zj |2 = 1. This completes the proof. �

From the explicitness of the operation (18.6) we can derive some more propertiesof H-type groups. We first give the explicit form of the canonical sub-Laplacian.


Proposition 18.2.4. With the notation in the proof of Theorem 18.2.1, the canonicalsub-Laplacian on the H-type group G is given by

ΔG = Δx + 1

4|x|2 Δt +

n∑

s=1

〈U(s) x,∇x〉 ∂

∂ts, (18.14)

where the U(s) are as in Theorem 18.2.1. Here we used the notation

Δx =m∑

j=1

(∂

∂xj

)2

, Δt =n∑

s=1

(∂

∂ts

)2

, ∇x =(

∂

∂x1, . . . ,

∂

∂xm

).

Moreover, on functions u(x, t) = u(|x|, t), ΔG has the form

ΔG = Δx + 1

4|x|2 Δt =

(∂

∂r

)2

+ m − 1

r

∂

∂r+ 1

4r2 Δt, r = |x| �= 0. (18.15)

Proof. Since the canonical sub-Laplacian is ΔG = ∑mj=1 X2

j , where Xj is givenby (18.13), one has

ΔG = Δx + 1

4

m∑

j=1

n∑

s=1

(m∑

i=1

xiU(s)j,i

)2(∂

∂ts

)2

+ 1

2

m∑

j=1

n∑

s=1

U(s)j,j

∂

∂ts

+ 1

4

m∑

j,h,k=1

n∑

r,s=1, r �=s

xhxk U(r)j,hU

(s)j,k

∂2

∂tr∂ts

+m∑

i,j=1

n∑

s=1

xiU(s)j,i

∂2

∂xj ∂ts

= Δx + 1

4

n∑

s=1

|U(s)x|2(

∂

∂ts

)2

+m∑

i,j=1

n∑

s=1

xiU(s)j,i

∂2

∂xj ∂ts

+ 1

2

n∑

r,s=1, r<s

〈U(r)x, U(s)x〉 ∂2

∂tr∂ts

= Δx + 1

4|x|2 Δt +

m∑

i,j=1

n∑

s=1

xiU(s)j,i

∂2

∂xj ∂ts

+ 1

2

n∑

r,s=1, r<s

〈U(r)U(s) x, x〉 ∂2

∂tr∂ts

= Δx + 1

4|x|2 Δt +

m∑

i,j=1

n∑

s=1

xiU(s)j,i

∂2

∂xj ∂ts.

Here we used the following facts: U(s)j,j = 0 since U(s) is skew-symmetric; |U(s) x| =

|x| since U(s) is orthogonal; from (18.12) we have


〈U(r)x, U(s)x〉 = 〈−U(s)U(r)x, x〉 = 〈U(r)U(s) x, x〉for every r �= s. Again from (18.12) it follows that U(r)U(s) is skew-symmetric (forr �= s) since

(U(r)U(s))T = (−U(s))(−U(r)) = U(s)U(r) = −U(r)U(s),

whence 〈U(r)U(s)x, x〉 = 0 for every r �= s, since for every skew-symmetric matrixA we have 〈Ax, x〉 = 0. This proves the first part of Proposition 18.2.4.

We now prove the second part. The third differential summand in the right-handside of (18.14) vanishes on functions u(|x|, t). Indeed, we have

(m∑

i,j=1

n∑

s=1

xiU(s)j,i

∂2

∂xj ∂ts

)u(|x|, t)

= 1

r

n∑

s=1

(m∑

i,j=1

xixjU(s)j,i

)(∂2u

∂r∂ts

)(|x|, t) = 0,

since for every s ≤ n,∑m

i,j=1 xixjU(s)j,i = 〈U(s) x, x〉 = 0, being U(s) skew-

symmetric. �Remark 18.2.5. Starting from (18.13), another direct computation shows

|∇Gu|2 = |∇xu|2 + 1

4|x|2 · |∇t u|2 +

n∑

s=1

(m∑

i,j=1

xi U(s)j,i

∂ u

∂xj

)∂ u

∂ts

= |∇xu|2 + 1

4|x|2 · |∇t u|2 +

n∑

s=1

⟨U(s)x,∇xu

⟩ ∂u

∂ts.

Remark 18.2.6. Let (G, ◦) be an H-type group. Suppose that the center of the algebraof G has dimension 1. Then G is isomorphic to a Heisenberg group H

k (here k =12 (dim(G) − 1)).

Indeed, by Theorem 18.2.1 (and by the hypothesis n = 1) it is not restrictive tosuppose that G is R

m+1 equipped with the group law

(x, t) ◦ (ξ, τ ) =(

x + ξ, t + τ + 1

2〈U(1) x, ξ 〉

).

Let M be an m × m non-singular matrix and consider the following bijection

M : G → Rm+1, (x, t) �→ (M x, t).

Clearly, if G is Rm+1 equipped with the composition

(M x, t) ∗ (M ξ, τ) := M((x, t) ◦ (ξ, τ )),


then M : (G, ◦) → (G, ∗) is a Lie group isomorphism. It is then sufficient to showthat there exists M such that (G, ∗) is isomorphic to a Heisenberg group. In order toprove it, we notice that

(x, t) ∗ (ξ , τ ) = M((M−1x, t ) ◦ (M−1ξ , τ )

)

= M

(M−1x + M−1ξ , t + τ + 1

2〈U(1) M−1x,M−1 ξ〉

)

=(

x + ξ , t + τ + 1

2

⟨M U(1)M−1x, ξ

⟩)

if (x, t ), (ξ , τ ) ∈ G. It is known that (see, e.g. [HJ85, Corollary 2.5.14]) every skew-symmetric orthogonal matrix is congruent to a block diagonal matrix of the type

J = diag

{(0 −11 0

), . . . ,

(0 −11 0

)}.

Hence, if P is a non-singular matrix such that P T U(1) P = J and if we take M =P −1, Remark 18.2.6 is proved. �Remark 18.2.7. We end this section with the following natural question. Let R

m+n

be equipped with a homogeneous Carnot group structure G = (Rm+n, δλ, ◦) by theusual dilations δλ(x, t) = (λx, λ2t) and the usual operation

(x, t) ◦ (ξ, τ ) =(

xj + ξj , j = 1, . . . , m

tj + τj + 12 〈B(j)x, ξ 〉, j = 1, . . . , n

),

x, ξ ∈ Rm, t, τ ∈ R

n, (18.16)

for suitable skew-symmetric linearly independent matrices B(j).We aim to answer to the following question. When is G an H-type group? Obvi-

ously, we cannot say that this happens if and only if 2 the B(j)’s satisfy conditions 1)and 2) of Theorem 18.2.1, for this is only the characterization of prototype H-typegroups, but not the characterization of all H-type groups. (Consider the Heisenberg–Weyl group as a counterexample.)

The proof of Theorem 18.2.1 helps us finding the answer to this question. Indeed,let G be an H-type group. Suppose

B = {B1, . . . , Bm,Z1, . . . , Zn}is any fixed basis for the algebra g of G, which is an orthonormal basis w.r.t. aninner product endowing g with a structure of H-type algebra (see Definition 18.1.1,page 681). Moreover, identify G to g (via the exponential map) and then identify g

to Rm+n (via coordinates w.r.t. B). Denote by H this last group, which is obviously

isomorphic to G. We demonstrated in due course of the proof of Theorem 18.2.1 thatH is a prototype H-type group.

2 The “if” part is true, the “only if” is not.

18.3 The Fundamental Solution on H-type Groups 695

Roughly speaking, if G is an H-type group, then a suitable choice of a new basisof g turns G into a prototype H-type group. As we already studied the action of achange of basis in the algebra of a homogeneous Carnot group in Remark 2.2.20(precisely, see page 153), we can infer that3 there exist two non-singular matrices U

(of order m × m) and V (of order n × n) such that the matrices

U(i) := UT ·(

n∑

j=1

wi,j B(j)

)· U,

where V −1 = (wi,j )i,j≤n, fulfill the requirements 1)–2) of Theorem 18.2.1. We thushave proved the following result.

Corollary 18.2.8. The homogeneous Carnot group G = (Rm+n, δλ, ◦) with ◦ asin (18.16) is an H-type group if and only if there exist two non-singular matrices U

(of order m × m) and V (of order n × n) such that the matrices

U(i) := UT ·(

n∑

j=1

wi,j B(j)

)· U,

where V −1 = (wi,j )i,j≤n fulfill the requirements 1)–2) of Theorem 18.2.1.In particular, if the second layer of the stratification of G is one-dimensional (i.e.

if n = 1), then G is an H-type group if and only if there exists w ∈ R such thatthe matrices w B(j)’s are simultaneously congruent to a set of matrices U(j)’s as in1)–2) of Theorem 18.2.1.

Unfortunately, this result does not seem very operative in order to answer to theproposed question.

18.3 The Fundamental Solution for Sub-Laplacians on H-typeGroups

Throughout this section, (G, ◦) will denote a fixed H-type group with the Lie al-gebra g. Moreover, we shall denote by 〈 , 〉 the given inner product on g as in De-finition 18.1.1. It is not restrictive to suppose that, if z is the center of g, b is theorthogonal completion of z and N1 := dim(b), N2 := dim(z), then G is a homoge-neous Carnot group on R

N1+N2 with the dilations

δλ(x) = δλ(x(1), x(2)) = (λx(1), λ2x(2)), x(1) ∈ R

N1, x(2) ∈ RN2 .

We finally set N := N1+N2, and we observe that Q = N1+2N2 is the homogeneousdimension of G. Under our hypotheses, we have N2 ≥ 1 and thus Q ≥ 4.

3 This fact is proved in details at the cited page 153.


We fix an orthonormal basis X1, . . . , XN1 of b and an orthonormal basis Z1, . . . ,

ZN2 of z. Then X1, . . . , XN1, Z1, . . . , ZN2 is an orthonormal basis of g, and wehave

v =N1∑

j=1

〈v,Xj 〉Xj , |v|2 =N1∑

j=1

〈v,Xj 〉2 ∀ v ∈ b, (18.17a)

z =N2∑

j=1

〈z, Zj 〉Zj , |z|2 =N2∑

j=1

〈z, Zj 〉2 ∀ z ∈ z, (18.17b)

|v + z|2 = |v|2 + |z|2 ∀ v ∈ b, ∀ z ∈ z, (18.17c)N1∑

j=1

|Xj |2 = N1,

N2∑

j=1

|Zj |2 = N2. (18.17d)

The aim of this section is to find the fundamental solution for the sub-Laplacian

L :=N1∑

j=1

X2j .

To this end, we start by introducing the following functions on G:

v : G → b, v(x) :=N1∑

j=1

〈Log (x),Xj 〉Xj ,

z : G → z, z(x) :=N2∑

j=1

〈Log (x), Zj 〉Zj .

From the definition it immediately follows that v and z are characterized by thefollowing property:

∀ x ∈ G, x = Exp (v(x) + z(x)), v(x) ∈ b, z(x) ∈ z. (18.18)

We want to prove the following result, due to A. Kaplan [Kap80].

Theorem 18.3.1 (A. Kaplan [Kap80]). Let X1, . . . , XN1 be an orthonormal basisof b, and let L be the sub-Laplacian

L =N1∑

j=1

X2j .

Then, with the above notation, there exists a positive constant c such that the function

Φ(x) := c · {|v(x)|4 + 16|z(x)|2}(2−Q)/4

is the fundamental solution for L.


Remark 18.3.2. We stress that Theorem 18.3.1 gives a fundamental solution for L =∑N1j=1 X2

j only when X1, . . . , XN1 is an orthonormal basis of b and not for any sub-Laplacian L on the H-type group G.

We observe thatd(x) := (|v(x)|4 + 16|z(x)|2)1/4

is a symmetric homogeneous norm on G. Indeed, from the definition of v(x) andz(x) it immediately follows d ∈ C∞(G \ {0}) ∩ C(G) and

v(δλ(x)) =N1∑

j=1

〈Log (δλ(x)),Xj 〉Xj =N1∑

j=1

〈λLog (x),Xj 〉Xj = λv(x),

z(δλ(x)) =N2∑

j=1

〈Log (δλ(x)), Zj 〉Zj =N2∑

j=1

〈λ2Log (x), Zj 〉Zj = λ2z(x).

Therefore, d(δλ(x)) = λd(x) for every λ > 0. Arguing analogously, we prove thatv(x−1) = −v(x) and z(x−1) = −z(x), which give d(x−1) = d(x).

Remark 18.3.3. When G is a prototype H-type group as in Theorem 18.2.1, fromTheorem 18.3.1 we get the following more explicit formula for the fundamental so-lution Φ of the canonical sub-Laplacian ΔG in (18.14). Namely, with the notation ofTheorem 18.2.1, the fundamental solution of the operator

ΔG = Δx + 1

4|x|2 Δt +

n∑

s=1

U(s)〈x,∇x〉 ∂

∂ts,

is given byΦ(x) := c · (|x|4 + 16|t |2)(2−Q)/4,

for some positive constant c.

Proof (of Remark 18.3.3). Let X1, . . . , Xm be defined as in (18.13), so that ΔG =∑mj=1 X2

j , and let Zi = ∂ti , i = 1, . . . , n. Then {X1, . . . , Xm} is an orthonormalbasis of b and {X1, . . . , Xm,Z1, . . . , Zn} is an orthonormal basis of g (see the proofof Theorem 18.2.1). By means of Theorem 18.3.1, we are only left to prove that

|v(x, t)| = |x|, |z(x, t)| = |t |. (18.19)

To this end, we need to investigate the exponential map Exp : g → G. For a fixed(a, b) ∈ R

m+n, we have (by the definition of Exp )

Exp

(m∑

j=1

ajXj +n∑

i=1

biZi

)= γ (1),

where γ solves

698 18 Groups of Heisenberg Type{

γ (s) = ∑mj=1 ajXj (γ (s)) + ∑n

i=1 biZi(γ (s)),

γ (0) = 0.

Setting γ (s) = (x(s), t (s)), we get

⎧⎨

⎩

xj (s) = aj ,

ti (s) = bi + 12 〈U(i)x(s), a〉,

x(0) = 0, t (0) = 0,

so that xj (s) = s aj and

ti (s) = bi + s

2〈U(i)a, a〉 = bi,

since U(i) is skew-symmetric. Therefore, ti (s) = s bi , and we finally get

Exp

(m∑

j=1

ajXj +n∑

i=1

biZi

)= γ (1) = (x(1), t (1)) = (a, b).

As a consequence, we get

Log : G → g, (x, t) �→m∑

j=1

xjXj +n∑

i=1

tiZi

and then

v(x, t) =m∑

j=1

xjXj , z(x, t) =n∑

i=1

tiZi

which finally give (18.19). �Remark 18.3.4. We refer to Example 5.4.7 on page 250 for another, more direct proofof Remark 18.3.3 above.

Proof (of Theorem 18.3.1). We introduce a family of regular functions Φε approx-imating Φ, and we compute L(Φε). If ε > 0 and x ∈ G, we set

Φε(x) := c · {(|v(x)|2 + ε2)2 + 16|z(x)|2}(2−Q)/4.

It is clear (being Exp and Log analytic functions) that Φε is an analytic function forevery fixed ε > 0, and Φ is analytic on G\{0}. From the definition of the exponentialmap we have

L(Φε)(x) =N1∑

j=1

X2j (Φε)(x) =

N1∑

j=1

(d

dt

)2∣∣∣∣t=0

Φε(x ◦ Exp (tXj )).


Let us fix ε > 0 and x ∈ G. In order to compute L(Φε)(x), we need to compute thefirst and the second derivatives of

φj (t) := (|v(x ◦ Exp (tXj ))|2 + ε2)2 + 16|z(x ◦ Exp (tXj ))|2, j = 1, . . . , N1.

We observe thatφj (0) = (|v(x)|2 + ε2)2 + 16|z(x)|2

does not depend on j ; then we set φ(0) := φj (0). In the above notation, setting alsok := (Q − 2)/4, we then have

L(Φε)(x) =N1∑

j=1

(d

dt

)2∣∣∣∣t=0

c (φj (t))−k = c

N1∑

j=1

d

dt

∣∣∣∣t=0

(−kφj (t)−k−1φ′

j (t))

= c k(k + 1)

N1∑

j=1

φj (0)−k−2(φ′j (0))2 − c k

N1∑

j=1

φj (0)−k−1φ′′j (0)

= c k φ(0)−k−2

((k + 1)

N1∑

j=1

(φ′j (0))2 − φ(0)

N1∑

j=1

φ′′j (0)

).

We now fix j = 1, . . . , N1 and t ∈ R. We look for simple expressions for

v(x ◦ Exp (tXj )), z(x ◦ Exp (tXj )).

We recall that, since G is a step two nilpotent group, the following Campbell–Hausdorff formula holds:

Exp (A) ◦ Exp (B) = Exp

(A + B + 1

2[A,B]

)∀ A, B ∈ g.

Thus we have

Exp(v(x ◦ Exp (tXj )) + z(x ◦ Exp (tXj ))

)(by (18.18))

= x ◦ Exp (tXj )

= Exp (Log (x)) ◦ Exp (tXj ) (by Campbell–Hausdorff)

= Exp

(Log (x) + tXj + t

2[Log (x),Xj ]

)(by (18.18))

= Exp

(v(x) + z(x) + tXj + t

2[v(x) + z(x),Xj ]

)(z(x) ∈ z, ∀ x ∈ G)

= Exp

(v(x) + tXj + z(x) + t

2[v(x),Xj ]

).

Comparing the first term of this equality to the last one and observing that

v(x) + tXj ∈ b, z(x) + t

2[v(x),Xj ] ∈ z,

from (18.18) we get


v(x◦Exp (tXj )) = v(x)+tXj , z(x◦Exp (tXj )) = z(x)+ t

2[v(x),Xj ]. (18.20a)

Hence we have

φj (t) = (|v(x) + tXj |2 + ε2)2 + 16|z(x) + t

2[v(x),Xj ]|2

= (|v(x)|2 + t2 + 2 t 〈v(x),Xj 〉 + ε2)2

+ 16

(|z(x)|2 + t2

4|[v(x),Xj ]|2 + t 〈z(x), [v(x),Xj ]〉

).

This gives

φ′j (t) = 2

(|v(x)|2 + t2 + 2 t 〈v(x),Xj 〉 + ε2) · (2t + 2 〈v(x),Xj 〉)+ 16

(t

2|[v(x),Xj ]|2 + 〈z(x), [v(x),Xj ]〉

);

φ′′j (0) = 8〈v(x),Xj 〉2 + 4(|v(x)|2 + ε2) + 8|[v(x),Xj ]|2;

φ′j (0) = 4(|v(x)|2 + ε2) · 〈v(x),Xj 〉 + 16 〈z(x), [v(x),Xj ]〉

= 4(|v(x)|2 + ε2) · 〈v(x),Xj 〉 + 16 〈Jz(x)(v(x)),Xj 〉= 4〈(|v(x)|2 + ε2) · v(x) + 4 Jz(x)(v(x)),Xj 〉.

In particular, we obtain

N1∑

j=1

(φ′j (0))2 = 16

N1∑

j=1

〈(|v(x)|2 + ε2) · v(x) + 4 Jz(x)(v(x)),Xj 〉2

(by (18.17a)) = 16 |(|v(x)|2 + ε2) · v(x) + 4 Jz(x)(v(x))|2(by (18.4a)) = 16 (|v(x)|2 + ε2)2 · |v(x)|2 + 162 |Jz(x)(v(x))|2(by (18.4c)) = 16 |v(x)|2 {(|v(x)|2 + ε2)2 + 16 |z(x)|2}

= 16 |v(x)|2 φ(0);N1∑

j=1

φ′′j (0) = 8

N1∑

j=1

〈v(x),Xj 〉2 + 4 N1 (|v(x)|2 + ε2) + 8N1∑

j=1

|[v(x),Xj ]|2

(by (18.17a)) = 8 |v(x)|2 + 4 N1 (|v(x)|2 + ε2) + 8N1∑

j=1

|[v(x),Xj ]|2.

Now [v(x),Xj ] ∈ [b, b] = z and thus, from (18.17b) we have

N1∑

j=1

∣∣[v(x),Xj ]∣∣2 =

N1∑

j=1

N2∑

i=1

〈Zi, [v(x),Xj ]〉2 =N1∑

j=1

N2∑

i=1

〈JZi(v(x)),Xj 〉2

=N2∑

i=1

N1∑

j=1

〈JZi(v(x)),Xj 〉2 =

N2∑

i=1

|Zi |2 |v(x)|2 = N2 |v(x)|2.


(In the fourth equality, we used (18.17a) and (18.4c).) As a consequence,

N1∑

j=1

φ′′j (0) = 8 |v(x)|2 + 4 N1 (|v(x)|2 + ε2) + 8 N2 |v(x)|2

= 16(k + 1) |v(x)|2 + 4 N1 ε2,

and then

L(Φε)(x) = c k φ(0)−k−2(16(k + 1) |v(x)|2 φ(0)

− 16(k + 1) |v(x)|2 φ(0) − 4 N1 ε2 φ(0))

= −4 N1 c k ε2 φ(0)−k−1

= −4 N1 c k ε2 ((|v(x)|2 + ε2)2 + 16|z(x)|2)(−2−Q)/4. (18.20b)

For ε = 0, we clearly have

LΦ(x) = 0 ∀ x �= 0. (18.20c)

Moreover, from Φ = c d2−Q we immediately get

Φ ∈ L1loc(G). (18.20d)

In order to prove Theorem 18.3.1, we are only left to show that LΦ = −Dirac0(Dirac0 denotes the Dirac mass at the origin) in the sense of distributions. Let us firstobserve that, since v and z are δλ-homogeneous of degree one and two, respectively,we immediately get

L(Φε)(δε(x)) = ε−QL(Φ1)(x). (18.20e)

We now observe that

L(Φ1)(x) = −c (Q − 2)N1((|v(x)|2 + 1)2 + 16|z(x)|2)(−2−Q)/4

is a C∞ function on G whose norm is bounded at infinity by a function δλ-homogeneous of degree −2 − Q. Thus L(Φ1)(x) has finite integral on G, and wecan choose c > 0 such that

∫

G

L(Φ1)(x) dx = −1. (18.20f)

Let us fix f ∈ C∞0 (G). We want to prove that

∫

G

Φ(x)Lf (x) dx = −f (0). (18.20g)

By dominated convergence, since |Φε| ≤ Φ ∈ L1loc, we immediately get

limε→0+

∫

G

Φε(x)Lf (x) dx =∫

G

Φ(x)Lf (x) dx.


On the other hand, since Φε ∈ C∞(G), f has compact support and L is self-adjoint,we have

∫

G

Φε(x)Lf (x) dx =∫

G

f (x)LΦε(x) dx

(by (18.20e) with the change of variable x = δε(z))

=∫

G

f (δεz)LΦ1(z) dz

ε→0+−→∫

G

f (0)LΦ1(z) dz = −f (0) (by (18.20f)).

This proves (18.20g) and completes the proof of Theorem 18.3.1. �Remark 18.3.5. In the above proof (see (18.20b)), we have incidentally proved that,for every ε > 0, the function

x �→ ((|v(x)|2 + ε2)2 + 16|z(x)|2)(2−Q)/4

is, up to a multiplicative constant, a solution of the semilinear differential equation

−Lu = u(Q+2)/(Q−2).

18.4 H-type Groups of Iwasawa-type

We begin with the following definition.

Definition 18.4.1 (H-type Iwasawa group). Let H be an H-type group, and let,as usual, h = b ⊕ z be the decomposition of its algebra, as in Definition 18.1.1(page 681). For any Z ∈ z, let JZ be the endomorphism of b defined in (18.1). ThenH is called an H-type Iwasawa group (in the sequel, an Iwasawa group) if, for everyB ∈ b and for every Z, Z′ ∈ z with Z⊥Z′, there exists Z′′ ∈ z (depending onB,Z,Z′) such that

JZ(JZ′(B)) = JZ′′(B). (18.21)

In literature, condition (18.21) is referred to as the “J 2-condition” (see, e.g.[CDKR91,CK84]).

Remark 18.4.2 (Hn is an Iwasawa group). Any Heisenberg–Weyl group Hk is an

H-type group of Iwasawa-type. Indeed, we know that (see Remark 3.6.5, page 172)the classical Heisenberg–Weyl group H

k on R2k+1 is canonically isomorphic to the

(prototype) H-type group H corresponding to the case m = 2k, n = 1 and

B(1) =(

0 −Ik

Ik 0

).

In turn, H is obviously an Iwasawa group since its center z is one-dimensional, andif Z,Z′ ∈ z are orthogonal, then at least one of them is the null vector, so that (forevery B ∈ b) we have

18.4 H-type Groups of Iwasawa-type 703

JZ(JZ′(B)) = 0

and (18.21) is satisfied by choosing Z′′ = 0. This proves that any H-type group withone-dimensional center is an Iwasawa group. �

We now explicitly write the J 2-condition (when H is a prototype H-type group;see Theorem 18.2.1 and Remark 18.2.2 for all the details) in terms of the usual ma-trices U(r)’s defining the composition law in H. First, we fix orthonormal bases{B1, . . . , Bm}, {Z1, . . . , Zn} of b and z, respectively, and we recall that, if Z =∑n

r=1 zrZr ∈ z, then we have

JZ : b → b,

JZ

(m∑

j=1

vjBj

)=

m∑

j=1

(n∑

r=1

m∑

i=1

viU(r)j,i zr

)Bj . (18.22)

More explicitly, JZ is represented (w.r.t. the basis {B1, . . . , Bm} of b) by the m × m

matrixn∑

r=1

zr U(r).

Now, if we write

B =m∑

j=1

vjBj , Z =n∑

r=1

zrZr, Z′ =n∑

r=1

z′rZr , Z′′ =

n∑

r=1

z′′r Zr ,

then it is easy4 to see that (18.21) writes as{

∀ v ∈ Rm, ∀ z, z′ ∈ R

n :n∑

r=1

zrz′r = 0

}

�⇒{

∃ z′′ ∈ Rn :

( ∑

1≤s<r≤n

(zsz′r − zrz

′s)U

(s)U(r)

)v =

(n∑

r=1

z′′r U

(r)

)v

}.

(18.23)

Example 18.4.3 (An H-type non-Iwasawa group). Let us consider the prototype H-type group H on R

6 defined by taking m = 4, n = 2 and

U(1) =⎛

⎜⎝

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

⎞

⎟⎠ , U(2) =⎛

⎜⎝

0 0 −1 00 0 0 11 0 0 00 −1 0 0

⎞

⎟⎠ .

4 The reader is invited to verify (18.23) by making use of the well-known properties of thematrices U(r)’s

(U(r))2 = −Im, U(r)U(s) = −U(s)U(r) (r �= s)

and (since Z⊥Z′) ∑nr=1 zrz

′r = 0.


More explicitly, H has the composition law

⎛

⎜⎜⎜⎜⎜⎝

x1x2x3x4t1t2

⎞

⎟⎟⎟⎟⎟⎠◦

⎛

⎜⎜⎜⎜⎜⎝

ξ1ξ2ξ3ξ4τ1τ2

⎞

⎟⎟⎟⎟⎟⎠=

⎛

⎜⎜⎜⎜⎜⎝

x1 + ξ1x2 + ξ2x3 + ξ3x4 + ξ4

t1 + τ1 + 12 (x2ξ1 − x1ξ2 + x4ξ3 − x3ξ4)

t2 + τ2 + 12 (−x3ξ1 + x4ξ2 + x1ξ3 − x2ξ4)

⎞

⎟⎟⎟⎟⎟⎠.

Then it is easy to see that (18.23) has no solution z′′ ∈ R2 if we choose

v = (0, 0, 1, 0) ∈ R4, Z = (1, 0) ∈ R

2, Z′ = (0, 1).

Hence H is an H-type group which is not an Iwasawa group. �Note 18.4.4. If g is a simple Lie algebra of rank one and n is the Lie algebra in theso-called Iwasawa decomposition

k ⊕ a ⊕ n

of g (also referred to as the KAN decomposition), then (equipped with a suitableinner product) n is an H-type algebra satisfying the “J 2-condition” (18.21), and viceversa. More precisely, by [CDKR91, Theorem 1.1], the Iwasawa n-components ofsimple Lie groups of real rank one are exactly the H-type algebras which satisfy the“J 2-condition”. This motivates the name of Iwasawa-type.

The importance of the Iwasawa-type groups appears naturally in dealingwith Kelvin-type transforms, which we shall briefly treat in the next section(see [CDKR91]).

18.5 The Inversion and the Kelvin Transform on H-type Groups

In the classical theory of harmonic functions, the map

RN \ {0} x �→ σ(x) := −x/|x|2 ∈ R

N \ {0}is the so-called inversion map (with respect to the unit sphere centered at the ori-gin). This map is used to define the Kelvin-transform of a function u (defined, forsimplicity, on R

N with N ≥ 3), namely

u∗ : RN \ {0} → R, u∗(x) := |x|2−N u(σ(x)).

As it is well known (a simple but tedious computation is enough to the purpose), if u

is a solution to the classical Laplace equation on RN , then the same is true of u∗ on

RN \ {0}.

Unfortunately (at present), a well-behaved analogue of the Kelvin transform foran arbitrary Carnot group is not known, except for the case of H-type groups of

18.5 The H-inversion and the H-Kelvin Transform 705

Iwasawa-type. Indeed, in this section we explicitly write a suitable inversion mapfor a general H-type group: it turns out that the relevant Kelvin transform is well-behaved (namely, preserves the L-harmonicity) only for H-type groups of Iwasawa-type (hence, in particular, for Heisenberg–Weyl groups).

We fix the notation: throughout the remaining of this section, H = (Rm+n, ◦, δλ)

is a fixed prototype H-type group (any general H-type group is naturally isomorphicto a prototype one, as we proved in Theorem 18.2.1) with usual coordinates (x, t),x = (x1, . . . , xm) ∈ R

m, t = (t1, . . . , tn) ∈ Rn, and the composition map

(x, t) ◦ (ξ, τ ) =(

x + ξ, t1 + τ1 + 1

2〈U(1)x, ξ 〉, . . . , tn + τn + 1

2〈U(n)x, ξ 〉

)

where U(1), . . . , U(n) are fixed m × m matrices with the following properties:

(H1) U(r) is an m × m skew-symmetric and orthogonal matrix for every r ≤ n;(H2) U(r) U(s) = −U(s) U(r) for every r, s ∈ {1, . . . , n} with r �= s.

Moreover, the dilation group is given by

δλ(x, t) = (λ x, λ2t).

As usual, ΔH denotes the canonical sub-Laplacian for H. We recall that (see The-orem 18.3.1 and Remark 18.3.3) the fundamental solution for ΔH has the formΓ = c d2−m−2n for a suitable positive constant c and the following gauge on H

d(x, t) = (|x|4 + 16 |t |2)1/4.

It is known that d defines an actual distance on H, i.e. d satisfies the pseudo-triangleinequality (see Proposition 5.1.7, page 231) with the constant c = 1 (see, e.g.[Cyg81]). We are now ready to explicitly write the inversion map and the Kelvintransform.

Definition 18.5.1 (H-inversion and H-Kelvin transform). Let H be an (prototype)H-type group. Following all the above notation, we set

σ : H \ {0} → H \ {0},(18.24)

σ(x, t) :=(

− |x|2x − 4∑n

k=1 tkU(k)x

|x|4 + 16|t |2 ,− t

|x|4 + 16|t |2)

.

If σ is as above, for any function u : H → R we let

u∗ : H \ {0} → R,(18.25)

u∗(x, t) := d(x, t)2−m−2n u(σ (x, t)).

We call σ the H-inversion map on H and u∗ the H-Kelvin transform of u.


The remarkable rôle played by the H-Kelvin transform on Iwasawa-type groupsis comparable to that of the classical Kelvin transform.

The definitions of the inversion map and the Kelvin transform in the abstractsetting of an (general) H-type group (not necessarily written in the exponential co-ordinates) are slightly more complicated. Our (non-restrictive) choice of prototypecoordinates makes it much simpler to give the very explicit definitions in (18.24) and(18.25).

We collect some easy properties of the H-inversion map in the following propo-sition.

Proposition 18.5.2. Let σ be the H-inversion map defined in (18.24). We also writeσ(x, t) = (σ (1)(x, t), σ (2)(x, t)) with the usual stratified splitting of the coordinates.Then, for every (x, t) ∈ H \ {0}, we have:

1) if | · | denotes the Euclidean norm (in Rm or R

n, accordingly), it holds

|σ (1)(x, t)| = |x|d2(x, t)

, |σ (2)(x, t)| = |t |d4(x, t)

; (18.26)

2) for every λ > 0, we have

σ(δλ(x, t)) = δ1/λ

(σ(x, t)

);3) the inversion is involutive, i.e. σ(σ (x, t)) = (x, t); in particular, σ is a bijection

of H \ {0} onto itself;4) it holds

d(σ (x, t)) = 1

d(x, t), (18.27)

whence σ maps the punctured d-ball

Bd(0, 1) \ {0} = {(x, t) ∈ H : 0 < d(x, t) < 1}onto H \ Bd(0, 1), and vice versa, and maps the d-sphere ∂Bd(0, 1) onto itself.

Proof. Throughout the proof, (x, t) is a fixed point of H \ {0} and (for the sake ofbrevity) we write d instead of d(x, t).

1) From the very definition of σ we have

|σ (1)(x, t)|=

∣∣∣∣|x|2x − 4

∑nk=1 tkU

(k)x

|x|4 + 16|t |2∣∣∣∣

= d−4

∣∣∣∣∣|x|2x − 4n∑

k=1

tkU(k)x

∣∣∣∣∣

= d−4

⟨(|x|2 Im − 4

n∑

k=1

tkU(k)

)x,

(|x|2 Im − 4

n∑

k=1

tkU(k)

)x

⟩1/2

18.5 The H-inversion and the H-Kelvin Transform 707

= d−4

⟨(|x|2 Im − 4

n∑

k=1

tkU(k)

)T

·(

|x|2 Im − 4n∑

k=1

tkU(k)

)x, x

⟩1/2

= d−4

⟨(|x|2 Im + 4

n∑

k=1

tkU(k)

)·(

|x|2 Im − 4n∑

k=1

tkU(k)

)x, x

⟩1/2

= d−4

⟨(|x|4 Im + 4

n∑

k=1

tkU(k) − 4

n∑

k=1

tkU(k)

− 16n∑

j,k=1

tj tk U(k)U(j)

)x, x

⟩1/2

= d−4〈(|x|4 Im + 16 |t |2 Im)x, x〉1/2 = d−4 d2 |x| = |x|d−2,

thus proving the first identity in (18.26). In the fifth equality, we used the skew-symmetry of the U(r)’s; in the seventh equality, we used the following fact

n∑

j,k=1

tj tk U(k)U(j)

=( ∑

1≤j=k≤n

+∑

1≤j<k≤n

+∑

1≤k<j≤n

)tj tk U(k)U(j)

=n∑

j=1

t2j (U(j))2 +

∑

1≤r<s≤n

tr ts U(s)U(r) +∑

1≤r<s≤n

ts tr U(r)U(s)

= −n∑

j=1

t2j Im +

∑

1≤r<s≤n

tr ts (U(s)U(r) + U(r)U(s)) = −|t |2 Im, (18.28)

by the assumptions (H1)–(H2) on the matrices U(r)’s defining an H-type group. Asfor the second identity in (18.26), we immediately have

|σ (2)(x, t)| =∣∣∣∣

t

|x|4 + 16|t |2∣∣∣∣ = |t | d−4.

This proves 1).2) A simple computation.3) From the results in (18.26), we infer

σ (1)(σ (x, t)) = −|σ (1)(x, t)|2 σ (1)(x, t) − 4∑n

k=1 σ (2)(x, t)k U(k)σ (1)(x, t)

|σ (1)(x, t)|4 + 16|σ (2)(x, t)|2

= −|x|2 d−4 σ (1)(x, t) + 4∑n

k=1 tk d−4 U(k)σ (1)(x, t)

|x|4 d−8 + 16|t |2 d−8

= −(

|x|2σ (1)(x, t) + 4n∑

k=1

tk U(k)σ (1)(x, t)

)


= d−4

(|x|2

(|x|2x − 4

n∑

j=1

tjU(j)x

)

+ 4n∑

k=1

tk U(k)

(|x|2x − 4

n∑

j=1

tjU(j)x

))

= d−4

(|x|4x − 16

{n∑

j,k=1

tj tk U(k)U(j)

}x

)

= d−4 (|x|4x + 16 |t |2x) = d−4 d4 x = x.

To derive the sixth equality, we used the computations in (18.28). Moreover, it holds

σ (2)(σ (x, t)) = − σ (2)(x, t)

|σ (1)(x, t)|4 + 16|σ (2)(x, t)|2

= − −t d−4

|x|4 d−8 + 16|t |2 d−8= t.

This proves 3).4) From (18.26) we have

d(σ (x, t)) = (|σ (1)(x, t)|4 + 16 |σ (2)(x, t)|2)1/4

= (|x|4 d−8(x, t) + 16 |t |2 d−8(x, t))1/4

= d−2(x, t) (|x|4 + 16 |t |2)1/4 = d−2(x, t) d(x, t) = d−1(x, t),

which proves (18.27). The second part of what asserted in 4) immediately followsfrom (18.27) and the results in 3). This ends the proof. �

We end the section by stating the following result on the H-Kelvin transform inthe setting of Iwasawa-type groups. We denote by ΔH the canonical sub-Laplacianof the H-type group H fixed at the beginning of the section.

Theorem 18.5.3 (Cowling et al., [CDKR91], Theorem 4.2). The H-Kelvin trans-form of every ΔH-harmonic function is always ΔH-harmonic if and only if H is ofIwasawa-type. In this case, the following formula holds

ΔH(u∗(x, t)) = d−2−m−2n(x, t) (ΔHu)(σ (x, t)) (18.29)

for every u ∈ C2(H \ {0}) and every (x, t) ∈ H \ {0}.Remark 18.5.4. We explicitly remark that formula (18.29) and the “if” part of The-orem 18.5.3 could also be proved by a direct computation, since both the canonicalsub-Laplacian ΔH and the H-Kelvin transform are explicitly written for our proto-type groups (see (18.14), page 692, for the expression of ΔH).


Bibliographical Notes. H-type groups have been introduced by A. Kaplan in 1980(see [Kap80]). The definition we have given in this chapter is not exactly the originalone given by Kaplan, but it is the one that is usually adopted in the most recentliterature. It is not difficult to prove the equivalence of the two definitions.

The inversion map and the Kelvin transform were first introduced on Heisen-berg–Weyl groups by A. Korányi [Kor82] and then generalized to the H-type groups,see [CK84,CDKR91]. It is far from our scopes here to provide complete results andan exhaustive list of references from the existing literature on H-type groups and therelated topics, which is nowadays still growing fast.

The result asserting that the H-Kelvin transform of a ΔH-harmonic function isΔH-harmonic if and only if H is of Iwasawa-type (which is not proved in this chap-ter) is demonstrated in the paper by M. Cowling, A. H. Dooley, A. Korányi andF. Ricci [CDKR91].

Some of the topics presented in this chapter also appear in [BU04a].


Ex. 1) Prove that (18.21) writes as (18.23) when H is an (prototype) H-type group.Ex. 2) Prove in details the assertions made in Example 18.4.3, page 703.Ex. 3) Verify that the group introduced in Example 18.4.3 is indeed an H-type

group and that an orthonormal basis for its algebra is the Jacobian basis

X1 = ∂x1 + 1

2(x2∂t1 − x3∂t2),

X2 = ∂x2 + 1

2(−x1∂t1 + x4∂t2),

X3 = ∂x3 + 1

2(x4∂t1 + x1∂t2),

X4 = ∂x4 + 1

2(−x3∂t1 − x2∂t2),

T1 = ∂t1 , T2 = ∂t2 .

Ex. 4) Verify that the H-inversion map σ for the group introduced in Exam-ple 18.4.3 is given by

σ(x, t) = −(|x|4 + 16 |t |2)−1

⎛

⎜⎜⎜⎜⎜⎝

|x|2 x1 + 4(−t1 x2 + t2 x3)

|x|2 x2 + 4(t1 x1 − t2 x4)

|x|2 x3 + 4(−t1 x4 − t2 x1)

|x|2 x4 + 4(t1 x3 + t2 x2)

t1t2

⎞

⎟⎟⎟⎟⎟⎠.


Ex. 5) Write the explicit expression of the H-inversion map σ for the Heisenberg–Weyl group H

1.Ex. 6) Prove formula (18.29) for the Heisenberg–Weyl group H

1 (using the calcu-lus formulas given at the end of Ex. 6, Chapter 1, page 77).

Ex. 7) Let X := {X1, . . . , Xm} be a given set of smooth vector fields on RN . If

Ω ⊆ G is an open set with smooth boundary, we recall that the character-istic set (related to X) of Ω is the set

X-Char(Ω) := {x ∈ ∂Ω |Xi(x) ∈ Tx(∂Ω), i = 1, . . . , m},Tx(∂Ω) being the tangent space to ∂Ω at the point x.Provide the details for the following result.Let G be an H-type group. Choose coordinates on G as in Theorem 18.2.1.Let ΔG = ∑m

j=1 X2j be the canonical sub-Laplacian of G, as in Proposi-

tion 18.2.4. Finally, let X = {X1, . . . , Xm}. Consider the general half-spaceΠ of G

Π = {(x, t) ∈ G | 〈a, x〉 + 〈b, t〉 > c},where a ∈ R

m, b ∈ Rn and c ∈ R are fixed. Then Π possesses X-

characteristic points if and only if b �= 0. Moreover, any half-space withcharacteristic points can be left-translated into a half-space of the type

{(x, t) ∈ G |〈b, t〉 > 0}.Indeed, a point (x, t) ∈ ∂Π is characteristic if and only if (why? usealso (18.13))

aj + 1

2

n∑

s=1

(m∑

i=1

xi U(s)i,j

)bs = 0 for every j ≤ m,

or equivalently,

(Ch)

⎧⎪⎨

⎪⎩

〈a, x〉 + 〈b, t〉 = c,(n∑

s=1

bsU(s)

)T

x = −2a.

If b = 0, then a �= 0, and the second equation in (Ch) has clearly no solu-tion (why?). If b �= 0, then (why?) the matrix (

∑ns=1 bs/|b|U(s))T is non-

singular. As a consequence, the second equation in (Ch) admits a solutionx ∈ R

m. The characteristic set for Π is then given by

{(x, t) ∈ G | x = x, 〈b, t〉 = c − 〈a, x〉}.Finally, let b �= 0. If (ξ, τ ) ∈ G is fixed, we set Π = (ξ, τ ) ◦ Π , i.e. Π isthe set of all points (x, t) ∈ G such that (why?)

n∑

j=1

bj (tj + τj ) +m∑

j=1

aj ξj +m∑

j=1

xj

(aj + 1

2

n∑

s=1

bs

m∑

i=1

ξiU(s)i,j

)> c.


There exists (why?) ξ ∈ Rm such that

aj + 1

2

n∑

s=1

bs

m∑

i=1

ξ iU(s)i,j = 0

for every j ≤ m. Then, there exists (why?) τ ∈ Rn such that

n∑

j=1

bj τj = c −m∑

j=1

aj ξj .

If Π = (ξ , τ )−1 ◦ Π , we have Π = {(x, t) ∈ G |〈b, t〉 > 0}.Ex. 8) Let (G, ◦) be a H-type group. Fix z0 ∈ G. We denote by Zz0 the following

vector field on G

Zz0I (z) = d

dh

∣∣∣∣h=0

((hz0) ◦ z).

Prove that, if z0 = (x0, t0), then the above vector field Zz0I (x, t) is givenby (we follow our usual notation on ◦)

Zz0I (x, t) =(

x0j , j = 1, . . . , m

t0j + 1

2 〈x0, U(j)x〉, j = 1, . . . , n

).

Ex. 9) Denote by ξ = (z, t) = (x + iy, t) ≡ (x, y, t) the points of the Heisenberggroup H

n = Cn × R ≡ R

2n+1. The group law on Hn is given by

ξ ◦ ξ ′ = (z + z′, t + t ′ + 2 Im(zz′)).

Define

F : Hn → S

2n+1 \ {(0, . . . , 0,−1)},F (ξ) = F(z, t) =

(2iz

t + i(1 + |z|2) ,−t + i(1 − |z|2)t + i(1 + |z|2)

).

Here S2n+1 denotes the unit sphere of C

n+1. The map F is a CR equivalencebetween the Heisenberg group and the sphere minus the south pole, whichis the analogue in CR geometry of the stereographic projection. Define now

K : Hn \ {0} → H

n \ {0}to be the map conjugate to the inversion in C

n+1 of S2n+1 through the equiv-

alence F , i.e.K := F ◦ (−IdCn+1) ◦ F−1.

Prove that K has the following expression

K(z, t) =( −iz

t + i|z|2 ,− t

d4

), where d := (|z|4 + t2)1/4.


Compare with the H-inversion map defined in (18.24). Finally compute theJacobian determinant of K and recognize that

det(JK(z, t)) = 1

d2Q,

where Q = 2n + 2 is the homogeneous dimension of Hn.

Ex. 10) Let us consider the Heisenberg group Hn with the usual notation. An appli-

cation ρ : Hn → H

n is called a rotation around the t-axis if there exists acomplex unitary transformation F : C

n → Cn such that

ρ(z, t) = (F (z), t).

Let Ω be a half-space of Hn, and let us define

Ω1 = {(x, y, t) ∈ R2n+1 | x1 > 0}

andΩt = {(x, y, t) ∈ R

2n+1 | t > 0}.Prove that one of the two following cases always occurs:(i) there exists ξ0 ∈ H

n such that either Ω = τξ0(Ωt ) or Ω = τξ0(−Ωt),(ii) there exist ξ0 ∈ H

n and a rotation ρ around the t-axis such that Ω =ρ(τξ0(Ω1)).

Prove, in particular, that the first case occurs when ∂Ω and the t-axis in-tersect, the second one when they are parallel. Finally prove that ΔHn isinvariant with respect to rotations around the t-axis, i.e.

ΔHn(u ◦ ρ) = (ΔHnu) ◦ ρ

for every rotation ρ around the t-axis.Ex. 11) Let us consider the Heisenberg group H

n with the usual notation, and letΩ = {ξ = (x, y, t) ∈ R

2n+1 | x1 > 0}. For every ξ ∈ Ω , set

r(ξ) = x1

2, Bξ = Bd(ξ, r(ξ)),

K : Ω × Ω → R, K(ξ, ξ ′) = md

ψ(ξ, ξ ′)r(ξ)Q

χBξ

(ξ ′),

where χBξ

is the characteristic function of Bξ and

ψ(ξ, ξ ′) = |z − z′|2d(ξ, ξ ′)2

.

For u ∈ L1loc(Ω), define

T u : Ω → R, (T u)(ξ) = (Mr(ξ)u)(ξ) =∫

Ω

K(ξ, ξ ′) u(ξ ′) dξ ′.

(Compare to (5.50f) on page 259 where md was also introduced.)


a) Observe that T u = u for any ΔHn -harmonic function u and deduce that∫

Ω

K(ξ, ξ ′) dξ ′ = 1 ∀ ξ ∈ Ω.

b) Prove that the integration of K(ξ, ξ ′) with respect to the variable ξ alsoyields a constant value, i.e. there exists a constant β such that

∫

Ω

K(ξ, ξ0) dξ = β

for every ξ0 ∈ Ω .c) Recognize now that T is a bounded linear operator in Lp(Ω) for every

p ∈ [1,+∞]. More precisely,

‖T u‖∞ ≤ ‖u‖∞ ∀ u ∈ L∞(Ω),

‖T u‖pp ≤ β ‖u‖p

p ∀ u ∈ Lp(Ω), ∀p ∈ [1,+∞[.Moreover, ∫

Ω

T u = β

∫

Ω

u ∀ u ∈ L1(Ω).

d) Prove that β > 1.e) Prove the following Liouville type theorem. If u is a non-negative ΔHn -

superharmonic function such that

u ∈ L1(Ω),

then u ≡ 0 in Ω .f) Using Ex. 10, prove that the Liouville-type theorem stated in (e) holds

for any half-space Ω of Hn whose boundary does not intersect the cen-

ter of Hn.

(Hint: In order to prove (b), use the invariance of Ω (and of r) under theleft translations “parallel” to the boundary. To prove (d), test the solid meanvalue formula (5.51), page 259, with a suitable ΔHn -subharmonic functionv (for instance, the function d−Q, suitably left translated) in order to obtainthat, chosen a compact subset K of Ω with |K| > 0, there exists ε > 0 suchthat T v ≥ v + ε in K . Use then (c) and deduce (d). To prove (e), observethat u ≥ T u gives ∫

Ω

u ≥∫

Ω

T u = β

∫

Ω

u,

and then use (d). All the details of this exercise can be found in [Ugu99].)Ex. 12) a) Prove that the Liouville-type theorem stated in Ex. 11 no longer holds

on every Hn if we replace the hypothesis u ∈ L1(Ω) with u ∈ Lp(Ω).

b) Prove that the function

−∂t (d(ξ)2−Q) = n t (|z|4 + t2)−1− n2


is ΔHn -harmonic away from the origin, and in the half-space {t > 1} itis positive and belongs to Lp for all p > 1 and also to the weak L1.

(Hint: To prove (a), take p ∈ ]1,+∞], n > 1p−1 and consider the funda-

mental solution of −ΔHn with pole outside Ω .)

19

The Carathéodory–Chow–Rashevsky Theorem

The aim of this chapter is to furnish the proof of the so-called Carathéodory–Chow–Rashevsky theorem, i.e. the result asserting the connectedness of R

N with respectto a family X of Hörmander vector fields. We shall restrict to the particular but sig-nificant case when the given vector fields are the generators of the first layer of ahomogeneous Carnot group G.

To this aim, we shall make use of two crucial tools. First, it will intervenethe Carnot–Carathéodory distance related to X (already introduced in Section 5.2,page 232). Second, we shall exploit the deep properties of the Campbell–Hausdorffoperation � recalled in Section 14.2.1 (page 584). In particular, we shall make use ofa result asserting that, roughly speaking, we can approximate any nested commutator

[Xq, · · · [X2, X1] · · ·]of the vector fields Xj ’s in the algebra of G by means of a �-composition of thevector fields ±X1, . . . ,±Xq (up to commutators of length ≥ q + 1).

As an application of the connectivity theorem, we shall prove that any Carnotgroup G admits the decomposition

G = Exp (g1) ∗ · · · ∗ Exp (g1)

into a precise number of factors which are the images under the exponential mapof the first layer g1 of the stratification of g (see Theorem 19.2.1). We shall make acrucial use of this result again in Chapter 20, where a suitable version of Lagrangemean value theorem in the stratified setting will be provided.

19.1 The Carathéodory–Chow–Rashevsky Theorem for StratifiedVector Fields

In this section, we give a proof of the Carathéodory–Chow–Rashevsky connectivitytheorem in the case of stratified vector fields. First of all, we recall the definitions ofX-subunit path and of X-connectedness, given in Section 5.2, page 232.

716 19 The Carathéodory–Chow–Rashevsky Theorem

Definition 19.1.1 (X-subunit path). Let X = {X1, . . . , Xm} be any family of vectorfields in R

N . A piecewise regular path γ : [0, T ] → RN is said to be X-subunit if

〈γ (t), ξ 〉2 ≤m∑

j=1

〈XjI (γ (t)), ξ 〉2 ∀ ξ ∈ RN,

for almost every t ∈ [0, T ].We denote by S(X) the set of all X-subunit paths, and we put

l(γ ) = T

if [0, T ] is the domain of γ ∈ S(X).

We explicitly remark that every integral curve of ±Xj , j ∈ {1, . . . , m} isX-subunit.

Definition 19.1.2 (X-connectedness. Carnot–Carathéodory distance). We say thatR

N is X-connected if and only if, for every x, y ∈ RN , there exists

γ ∈ S(X), γ : [0, T ] → RN such that γ (0) = x and γ (T ) = y.

If RN is X-connected, the following definition makes sense

dX(x, y) := inf{T > 0 | ∃ γ : [0, T ] → R

N, γ ∈ S(X), γ (0) = x, γ (T ) = y}

for every x, y ∈ RN . Moreover, the function (x, y) �→ dX(x, y) is a metric1 on R

N ,called the X-control distance or the Carnot–Carathéodory distance related to X.

In what follows, we shall simply write d instead of dX.We now specialize to the case of stratified vector fields. From now on, throughout

this section, we always denote by G = (RN, ◦, δλ) a fixed homogeneous Carnotgroup with the Lie algebra g. The dilations {δλ}λ>0 of G are given by


Ni , 1 ≤ i ≤ r.

We shall denote by m := N1 the number of generators of G and by

Q = N1 + 2 N2 + · · · + r Nr

the homogeneous dimension of G.The following theorem then holds.

Theorem 19.1.3 (Carathéodory–Chow–Rashevsky theorem). Let the precedinghypotheses and notation hold. Let us suppose that Z1, . . . , Zm are vector fields ing such that

g = Lie{Z1, . . . , Zm}1 See Proposition 5.2.3, page 232.

19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields 717

(for example, we can choose Z1, . . . , Zm to be the first m vector fields of the Jacobianbasis for g, i.e. the left invariant vector fields on G such that Zj (0) = (∂/∂xj )|0,j = 1, . . . , m). Then, setting Z = {Z1, . . . , Zm}, we have that

G is Z-connected.

Moreover, the Z-connectedness is given by paths that are piecewise integral curves ofthe vector fields Z1, . . . , Zm. In particular, the Z-control distance d is well-defined.Finally, setting

d0 := d(·, 0),

d0 turns out to be a homogeneous norm2 on G.

(Note. We explicitly remark that X-connectivity also holds when X is a system ofvector fields in R

N satisfying Hörmander’s rank condition. See the BibliographicalNotes at the end of this chapter for references.)

A main step in the proof of Theorem 19.1.3 is Lemma 19.1.4 below which,roughly speaking, states that we can approximate any nested commutator

[Xq, · · · [X2, X1] · · ·]of vector fields Xj ’s in g by means of a composition of the vector fields ±X1, . . . ,

±Xq (up to commutators of length ≥ q + 1). The composition is meant in the senseof iterated application of the Campbell–Hausdorff operation � (see Definition 2.2.11on page 128; see also Section 14.2.1, page 584). More precisely, we have

[Xq, · · · [X2, X1] · · ·] = Xj1 � · · · � Xjc(q)+ R(X1, . . . , Xq),

where c(q) is an integer depending only on q (namely, c(q) = 3 · 2q−1 − 2) and notdepending on the Xj ’s, and

Xj1, . . . , Xjc(q)∈ {±X1, . . . ,±Xq}

can be chosen in a “universal” way, again not depending on the Xj ’s. Finally, theremainder R(X1, . . . , Xq) is a linear combination of brackets of height ≥ q+1 of thevector fields X1, . . . , Xq , and the form of R can, once more, be chosen independentlyof the Xj ’s.

In order to be more precise, we need to introduce some more notation and defi-nitions from algebra. We leave the details to the interested reader. (See also Defini-tion 15.2.1, page 600, and Section 1.1, page 3.)

Let q ∈ N, let x1, . . . , xq be non-commuting letters and let us denote by a theset of the formal words in the letters x1, . . . , xq . For example, an element of a isx3

1x2x3x41x3. Let also a be equipped with its natural structure of associative algebra3

(a, ·) over R and then of a Lie algebra (a, [·, ·]) by setting [x, y] := xy − yx forevery x, y ∈ a.

2 See Section 5.1, page 229, for the definition of homogeneous norm on G.3 For example, x1x2

3x4 · x34x1x7 = x1x2

3x44x1x7 and x1x2

3 · x47 = x1x2

3x47 , etc.


For every k ∈ N and any given multi-index J = (j1, . . . , jk) ∈ {1, . . . , q}k , weset

xJ := [xj1, · · · [xjk−1 , xjk] · · ·].

We say that xJ is a commutator of length (or height) k of x1, . . . , xq . If J = j1, wealso say that xJ := xj1 is a commutator of length 1 of x1, . . . , xq .

For any q, k ∈ N, we shall use the notation

Pk[x1, . . . , xq ] := span{xJ = [xj1, . . . [xjk−1 , xjk

] . . . ] : J ∈ {1, . . . , q}k}

to denote the vector space generated by the symbols xJ , J ∈ {1, . . . , q}k , i.e. theset of linear combinations with real coefficients of the formal objects xJ . We shallrefer to Pk[x1, . . . , xq ] as the set of formal Lie-polynomials homogeneous of lengthk on the (non-commuting) indeterminates x1, . . . , xq , and xJ will be called a formal(nested) Lie-monomial (homogeneous of length k on x1, . . . , xq ). For a fixed q ∈ N,any sum of elements in Pk[x1, . . . , xq ] with a finite number of k’s will be referred toas a formal Lie-polynomial on the indeterminates x1, . . . , xq .

Let us now denote by h any (abstract) nilpotent Lie algebra. For example, h willbe soon equal to g, the Lie algebra of a homogeneous Carnot group G. For anyq, k ∈ N and every P ∈ Pk[x1, . . . , xq ], we wish to give a precise meaning toP(X1, . . . , Xq), whenever X1, . . . , Xq ∈ h. Let P ∈ Pk[x1, . . . , xq ]. There exist asubset I of {1, . . . , q}k and real numbers aJ ’s (for every J ∈ I) such that

P =∑

J∈IaJ xJ .

We setP(X1, . . . , Xq) :=

∑

J∈IaJ XJ

if X1, . . . , Xq ∈ g, where

(�) XJ := [Xj1, · · · [Xjk−1, Xjk] · · ·]

if xJ = [xj1 , · · · [xjk−1 , xjk] · · ·]. Note that the signs [·, ·] in (�) denote the Lie brack-

ets in h.For example, if h = g, then, given

P = [x1, [x2, x3]] − [x3, [x4, x1]] + 2[x2, [x3, x3]] ∈ P3[x1, . . . , x4]and given vector fields X1, . . . , X4 ∈ g, we have

P(X1, . . . , X4) = [X1, [X2, X3]] − [X3, [X4, X1]] ∈ g.

If r is the step of nilpotency of h, and s is any fixed integer in {1, . . . , r}, we denoteby

r⊕

k=s

Pk[x1, . . . , xq ]


the set of formal sums∑r

k=s P (k), where P (k) ∈ Pk[x1, . . . , xq ]. If h = g,X1, . . . , Xq ∈ g and Rs ∈ ⊕r

k=s Pk[x1, . . . , xq ], then

Rs(X1, . . . , Xq)

will denote a sort of “remainder”, a linear combination of brackets of height ≥ s ofthe vector fields X1, . . . , Xq . Vice versa, recalling Proposition 1.1.7, page 12, anylinear combination of brackets of height ≥ s of the vector fields X1, . . . , Xq ∈ g canbe written as Rs(X1, . . . , Xq) for some Rs ∈ ⊕r

k=s Pk[x1, . . . , xq ].With the above notation at hand, the Campbell–Hausdorff operation � in every

nilpotent Lie algebra h can be written as

X � Y = X + Y + H2(X, Y )

= X + Y + 1

2[X, Y ] + H3(X, Y ) ∀ X, Y ∈ h, (19.1)

where

H2 ∈r⊕

k=2

Pk[x1, x2], H3 ∈r⊕

k=3

Pk[x1, x2]

depend4 only on the step of nilpotency r of h.Let us now henceforth fix h = g, the Lie algebra of the homogeneous Carnot

group G, nilpotent of step r . We want to show some notable algebraic properties ofthe � operation on g in the simple cases q = 2 and q = 3.

• For q = 2, we claim that there exists

R3 ∈r⊕

k=3

Pk[x1, x2]

such that

[Y,X] = Y � X � (−Y) � (−X) + R3(X, Y ) ∀ X, Y ∈ g (19.2)

(we recall that � is associative!). Indeed, (19.1) gives

(Y � X) � ((−Y) � (−X))

= Y � X + (−Y) � (−X) + 1

2[Y � X, (−Y) � (−X)]

+ H3(Y � X, (−Y) � (−X))

=(

Y + X + 1

2[Y,X] + H3(Y,X)

)+

(−Y − X + 1

2[Y,X] + H3(−Y,−X)

)

+ 1

2[Y + X + H2(Y,X),−(Y + X) + H2(−Y,−X)] + P3(X, Y )

4 The “universality” of H2, H3 follows from the “universality” of the Campbell–Hausdorffoperation (14.2), page 585.


= [Y,X] + H3(Y,X) + H3(−Y,−X)

+ 1

2([Y + X,−(Y + X)] + P ′

3(X, Y )) + P3(X, Y )

= [Y,X] + P ′′3 (X, Y )

for some P3, P′3, P

′′3 ∈ ⊕r

k=3 Pk[x1, x2] depending only on the step r of G.• For q = 3, we claim that there exists

R4 ∈r⊕

k=4

Pk[x1, x2, x3]

such that

[Z, [Y,X]] = Z � Y � X � (−Y) � (−X) � (−Z) � X � Y � (−X) � (−Y)

+ R4(X, Y,Z) for every X, Y,Z ∈ g. (19.3)

In order to prove (19.3), we start by recalling that −W is the �-inverse of W (for anyW ∈ g), i.e. −W is the unique element of g such that

(−W) � W = 0 = W � (−W).

As a consequence, we get

X � Y � (−X) � (−Y) = −(Y � X � (−Y) � (−X)), (19.4)

since

(X � Y � (−X) � (−Y)) � (Y � X � (−Y) � (−X))

= (X � Y � (−X)) � ((−Y) � Y) � (X � (−Y) � (−X))

= (X � Y) � ((−X) � X) � ((−Y) � (−X))

= · · · = 0,

and analogously

(Y � X � (−Y) � (−X)) � (X � Y � (−X) � (−Y)) = 0.

We now introduce the notation

α(X, Y ) := Y � X � (−Y) � (−X) ∀ X, Y ∈ g, (19.5)

so that (19.2) reads

[Y,X] = α(X, Y ) + R3(X, Y ) ∀ X, Y ∈ g. (19.6)

From (19.4) and (19.6) it follows that


Z � (Y � X � (−Y) � (−X)) � (−Z) � (X � Y � (−X) � (−Y))

= Z � α(X, Y ) � (−Z) � (−α(X, Y ))

= α(α(X, Y ), Z)

= [Z, α(X, Y )] − R3(α(X, Y ), Z)

= [Z, [Y,X] − R3(X, Y )] − R3([Y,X] − R3(X, Y ), Z)

= [Z, [Y,X]] + P4(X, Y,Z),

for some P4 ∈ ⊕rk=4 Pk[x1, x2, x3] depending only on the step r of nilpotency of g.

This proves (19.3). ��Let us now consider the case of general q. Further (in particular, in (19.7) below)

we use the following notation: if X1, . . . , Xq is a family of vector fields, we agree tolet

X−i := −Xi ∀ i ∈ {1, . . . , q}.Lemma 19.1.4. Let the notation in the preceding paragraphs be fixed. Let q ∈ N,q ≥ 2, and set c(q) = 3 · 2q−1 − 2. Recall also that r denotes the step of the (fixed)Carnot group G. Then there exists an “indexing map”

jq : {1, . . . , c(q)} → {−q, . . . ,−1, 1, . . . , q}and there exists

Rq+1 ∈r⊕

k=q+1

Pk[x1, . . . , xq ],

Rq+1 =r∑

k=q+1

R(q)k , R

(q)k ∈ Pk[x1, . . . , xq ]

(we agree to let Rq+1 = 0 if q ≥ r) such that

[Xq, · · · [X2, X1] · · ·] = Xjq(1) � · · · � Xjq(c(q)) + Rq+1(X1, . . . , Xq) (19.7)

for every X1, . . . , Xq ∈ g.

For example:• when q = 2, we have (see (19.2)) c(2) = 4 and

j2 : {1, 2, 3, 4} → {−2,−1, 1, 2},j2(1) = 2, j2(2) = 1, j2(3) = −2, j2(4) = −1,

so that

Xj2(1) � Xj2(2) � Xj2(3) � Xj2(4)

= X2 � X1 � X−2 � X−1

= X2 � X1 � (−X2) � (−X1);


• when q = 3, we have c(3) = 10 and (see (19.3))

j3 : {1, . . . , 10} → {−3,−2,−1, 1, 2, 3},j3(1) = 3, j3(2) = 2, j3(3) = 1, j3(4) = −2, j3(5) = −1,

j3(6) = −3, j3(7) = 1, j3(8) = 2, j3(9) = −1, j3(10) = −2.

Proof (of Lemma 19.1.4). (We explicitly remark that the proof makes a crucial useof the “universality” of the Campbell–Hausdorff operation �.)

We argue by induction on q ≥ 2. The case q = 2 follows from (19.2). Let usnow prove the statement for q + 1 assuming it to be true for q. First of all, let usobserve that, arguing as in (19.4), we get

(−Xjq(c(q))) � · · · � (−Xjq(1)) = −(Xjq(1) � . . . � Xjq(c(q))).

Hence, recalling (19.5) and using (19.6), we have

Xq+1 � (Xjq(1) � · · · � Xjq(c(q))) � (−Xq+1) � ((−Xjq(c(q))) � · · · � (−Xjq(1)))

= α(Xjq(1) � · · · � Xjq(c(q)), Xq+1)

= [Xq+1, Xjq(1) � · · · � Xjq(c(q))] − R3(Xjq(1) � · · · � Xjq(c(q)), Xq+1)

= [Xq+1, [Xq, · · · [X2, X1] · · ·] − Rq+1(X1, . . . , Xq)

] + Pq+2(X1, . . . , Xq+1)

= [Xq+1, [Xq, · · · [X2, X1] · · ·]] + P ′q+2(X1, . . . , Xq+1),

for some Pq+2, P′q+2 ∈ ⊕r

k=q+2 Pk[x1, . . . , xq+1] depending only on the step r

of G. Since

2c(q) + 2 = 2(3 · 2q−1 − 2) + 2 = 3 · 2q − 2 = c(q + 1),

the proof is complete. ��We are now ready to prove Theorem 19.1.3.

Proof (of Theorem 19.1.3). Using g = Lie{Z1, . . . , Zm} and recalling again Propo-sition 1.1.7 on page 12, we conclude that there exist multi-indices

J1 ∈ {1, . . . , m}q1 , . . . , JN ∈ {1, . . . , m}qN , 1 ≤ qi ≤ r,

such that

g = span{ZJ1, . . . , ZJN}. (19.8a)

For every i ∈ {1, . . . , N}, we also consider the multi-index

J−i := (Ji(1), . . . , Ji(qi − 2), Ji(qi), Ji(qi − 1))

obtained by interchanging the last two entries of Ji , so that

ZJ−i= −ZJi

. (19.8b)


By means of Lemma 19.1.4, for every i ∈ {−N, . . . ,−1, 1, . . . , N} there exists anindexing map

hi : {1, . . . , c(qi)} → {−m, . . . ,−1, 1, . . . , m}(q−i := qi) and there exists

Wik ∈ Pk[x1, . . . , xm] for every k ∈ {qi + 1, . . . , r}

such that, setting

Wi(t) :=r∑

k=qi+1

R(qi)k (t1/qi ZJi(qi ), . . . , t

1/qi ZJi(1))

=r∑

k=qi+1

tk/qi W ik(Z1, . . . , Zm) ∀ t ≥ 0 (19.8c)

(we agree to let Wi ≡ 0 if qi ≥ r), we have

t ZJi= [t1/qi ZJi(1), . . . [t1/qi ZJi(qi−1), t

1/qi ZJi(qi )] . . .]= (t1/qi Zhi(1)) � · · · � (t1/qi Zhi(c(qi ))

)

+ Rqi+1(t1/qi ZJi(qi ), . . . , t

1/qi ZJi(1))

= (t1/qi Zhi(1)) � · · · � (t1/qi Zhi(c(qi )))

+r∑

k=qi+1

R(qi)k (t1/qi ZJi(qi ), . . . , t

1/qi ZJi(1))


+r∑

k=qi+1

tk/qi R(qi )k (ZJi(qi ), . . . , ZJi(1))


+r∑

k=qi+1

tk/qi W ik(Z1, . . . , Zm)

= (t1/qi Zhi(1)

) � · · · � (t1/qi Zhi(c(qi ))) + Wi(t) (19.8d)

for every t ≥ 0. As usual, we agree to let Z−j := −Zj for every j ∈ {1, . . . , m}.For every i ∈ {−N, . . . ,−1, 1, . . . , N}, we now define

Ei : G × [0,+∞[→ G,

Ei(x, t) := (exp(t1/qi Zhi(c(qi ))) ◦ · · · ◦ exp(t1/qi Zhi(1)))(x). (19.8e)

(Here ◦ denotes the usual composition of maps.) Let us recall that, by means ofTheorem 15.1.1, we have


exp(Y )(

exp(X)(x)) = exp(X � Y)(x) ∀ X, Y ∈ g, ∀ x ∈ G. (19.8f)

As a consequence, we can write

Ei(x, t) = exp((

t1/qi Zhi(1)

) � · · · � (t1/qi Zhi(c(qi ))

))(x)

= exp(tZJi

− Wi(t))(x).

We have used here (19.8d). We now recall5 formula (1.9), page 8,

exp(X)(x) =r∑

k=0

1

k! XkI (x) ∀ X ∈ g, ∀ x ∈ G. (19.8g)

Taking X = tZJi− Wi(t), we obtain

Ei(x, t) = x + (tZJi

− Wi(t))I (x) +

r∑

k=2

1

k!(tZJi

− Wi(t))k

I (x)

= x + tZJiI (x) + ωi(x, t), (19.8h)

for some remainder function ωi : G × [0,+∞[ → G. From the very defini-tion (19.8c) of Wi it follows that such remainder function has the following regu-larity properties:

ωi ∈ C1(G × [0,+∞[, G

),

ωi(x, t) = t1+1/qi ωi(x, t), ωi ∈ C(G × [0,+∞[, G

), (19.8i)

(∂tωi)(x, 0) = 0.

Roughly speaking, formula (19.8h) above states that a translation along a commuta-tor ZJ of arbitrary length can be approximated by a composition of a finite numberof elementary translations along ±Z1, . . . ,±Zm, at least for small positive times.We now want to take into account negative times too (see formula (19.8l) below).For every i ∈ {1, . . . , N}, we set

ri : G × R → G, ri(x, t) :={

ωi(x, t) if t ≥ 0,

ω−i (x,−t) if t < 0.

From (19.8i) it follows that ri has the following regularity properties:

ri ∈ C1(G × R, G),

ri(x, t) = |t |1+1/(2qi ) ri (x, t), ri ∈ C(G × R, G), (19.8j)

(∂t ri)(x, 0) = 0.

5 We explicitly remark that (19.8g) is a particular case of (1.9), page 8, because Xj I ≡ 0 forevery j > r , since (for those j ’s) Xj is a sum of differential operators δλ-homogeneous ofdegree > r , whereas the component functions of the identity map I are δλ-homogeneousof degree at most r .


For every i ∈ {1, . . . , N}, we now define

Bi : G × R → G, Bi(x, t) :={

Ei(x, t) if t ≥ 0,

E−i (x,−t) if t < 0.(19.8k)

Recalling (19.8h) and (19.8b), we can write

Bi(x, t) = x + tZJiI (x) + ri(x, t) ∀ (x, t) ∈ G × R. (19.8l)

As a consequence, it turns out that

Bi ∈ C1(G × R, G).

Moreover, we have

Bi(x, 0) = x,(19.8m)

JBi(x, 0) = (IN,ZJi

I (x)).

We now fix x0 ∈ G, and we define

F : RN → R

N ≡ G,(19.8n)

F(t) := BN

(. . . B2(B1(x0, t1), t2) . . . , tN

).

We haveF ∈ C1(RN, R

N),

JF (t) = JBN(BN−1(. . . B2(B1(x0, t1), t2) . . . , tN−1))

· diag[. . . diag[JB2(B1(x0, t1), t2) diag[∂t1B1(x0, t1), 1], 1] . . . , 1],where, for any p ×q matrix A, we have denoted by diag[A, 1] the (p + 1)× (q + 1)

matrix

diag[A, 1] :=(

A 00 1

).

Using (19.8m), we obtain

JF (0) = JBN(x0, 0) diag[. . . diag[JB2(x0, 0) diag[ZJ1I (x0), 1], 1] . . . , 1]

= (IN,ZJNI (x0))

· diag[. . . diag[(IN,ZJ2I (x0)) diag[ZJ1I (x0), 1], 1] . . . , 1]= (ZJ1I (x0), . . . , ZJN

I (x0)).

As a consequence, recalling the choice of J1, . . . , JN (see (19.8a)), we get

detJF (0) �= 0,

and F is a diffeomorphism from a neighborhood V of the origin to a neighborhoodW of x0 = F(0).


In particular, for every y ∈ W , there exists t ∈ V such that y = F(t). Wenow observe that (by the definition of F , see (19.8n), (19.8k), (19.8e)) F(t) is thefinal point of a (Z1, . . . , Zm)-subunit path (recall Definition 19.1.1 of subunit path)starting from x0. Furthermore, such a path is piecewise an integral curve of the vectorfields Z1, . . . , Zm. As a consequence, there exists a neighborhood of x0 whose pointsare (Z1, . . . , Zm)-connected to x0 along paths that are piecewise integral curves ofthe vector fields Z1, . . . , Zm.

A standard argument now allows to pass from local to global connectivity. Weset Z = (Z1, . . . , Zm) and, once x0 ∈ R

N is fixed, we consider the set E of pointsZ-connected to x0 along paths that are piecewise integral curves of the vector fieldsof Z. Such a set E is non-empty, since x0 ∈ E. Moreover, E is open. Indeed, ify ∈ E, from the above arguments it follows that there exists an open neighborhoodof y whose points are Z-connected to y along paths that are piecewise integral curvesof the vector fields of Z.

Therefore, x0 is Z-connected to any point of such a neighborhood of y. Finally,E is closed. In fact, let yn be a sequence in E such that yn → y. By the aboveargument, there exists an open neighborhood W of y whose points are Z-connectedto y along paths that are piecewise integral curves of the vector fields of Z.

Moreover, there exists n ∈ N such that yn ∈ W . As a consequence, x0 is Z-connected to y (passing through yn) along paths that are piecewise integral curves ofthe vector fields of Z, and then y ∈ E. This completes the proof of the (Z1, . . . , Zm)-connectivity of G.

We now want to prove the second statement of Theorem 19.1.3, i.e. that d0 =d(·, 0) is a homogeneous norm on G. By means of Proposition 5.2.6, page 234, weonly need to prove that d0 is continuous. For every i ∈ {−N, . . . ,−1, 1, . . . , N}, wehave

d(x,Ei(x, t)) ≤ c(qi) t1/qi . (19.8o)

Indeed, for every fixed t ≥ 0, the point Ei(x, t) is connected to x by a (Z1, . . . , Zm)-subunit path, which is piecewise an integral curve of one of the vector fieldsZ1, . . . , Zm. Each piece of this path connects two points which have d-distance≤ t1/qi . Hence, from the triangle inequality for d we get

d(x,Ei(x, t)) ≤c(qi )∑

j=1

t1/qi = c(qi) t1/qi .

Recalling the definition of F (see (19.8n), (19.8k)) and setting

x1 := B1(x0, t1), x2 := B2(x1, t2), . . . , xN := BN(xN−1, tN ) = F(t),

from the triangle inequality we also get (for |t | ≤ 1)

d(x0, F (t)) ≤ d(x0, x1) + d(x1, x2) + · · · + d(xN−1, xN)

≤ c(q1) |t1|1/q1 + c(q2) |t2|1/q2 + · · · + c(qN) |tN |1/qN

≤ c |t |1/r .

19.2 An Application of Carathéodory–Chow–Rashevsky Theorem 727

Therefore,

|d0(x) − d0(x0)| = |d(x, 0) − d(x0, 0)|≤ d(x0, x) = d(x0, F (F−1(x))) ≤ c|F−1(x)|1/r

−→ c|F−1(x0)|1/r = 0 as x → x0.


19.2 An Application of Carathéodory–Chow–RashevskyTheorem

As an application of Theorem 19.1.3, we have the following result, which has aninterest in its own.

Theorem 19.2.1. Let (G, ∗) be a homogeneous Carnot group, let g1 be the first layerof a stratification of the algebra g of G and let Z = {Z1, . . . , Zm} be a basis for g1.Finally, fix a homogeneous norm d on G.

Then there exists an absolute constant M ∈ N (only depending on G) such that

G = Exp (g1) ∗ · · · ∗ Exp (g1)︸︷︷︸M times

.

More precisely, for every x ∈ G, there exist x1, . . . , xM ∈ Exp (g1) with the followingproperties:

x = x1 ∗ · · · ∗ xM, (19.9a)

d(xj ) ≤ c0 d(x) for all j = 1, . . . , M, (19.9b)

for every j = 1, . . . , M, there exist tj ∈ R and ij ∈ {1, . . . , m}such that xj = Exp (tj Zij ). (19.9c)

Here c0 > 0 is a constant depending on G, d and Z , but not depending on x and ofthe xj ’s.

Before proving Theorem 19.2.1, we need a simple preliminary remark (used alsoin the proof of Theorem 20.3.1, page 746).

Remark 19.2.2. Let us recall Definition 2.2.6, page 125. Let g = g1 ⊕ · · · ⊕ gr

be a stratification of the algebra g of a homogeneous Carnot group G. For everyk = 1, . . . , r , let us fix a basis for gk , say X

(k)1 , . . . , X

(k)Nk

(where Nk = dim(gk)). Asusual, we say that

{X1, . . . , XN } := {X(1)1 , · · · , X

(1)N1

, · · · , X(r)1 , · · · , X

(r)Nr

}is a basis of g adapted to the stratification. For every ξ = (ξ1, . . . , ξN ) =(ξ (1), . . . , ξ (r)), we set


ξ · X =N∑

j=1

ξj Xj =r∑

k=1

Nk∑

j=1

ξ(k)j X

(k)j . (19.10a)

If the dilations on G are indifferently denoted by

δλ(x) = (λσ1x1, . . . , λσN xN) = (λx(1), . . . , λrx(r)),

we can define a dilation on g (see Definition 1.3.24, page 46) still denoted by δλ,extending by linearity the map such that δλ(Xj ) := λσj Xj for every j = 1, . . . , N .In other words, we have

δλ

(N∑

j=1

ξj Xj

)=

N∑

j=1

ξj λσj Xj =r∑

k=1

Nk∑

j=1

ξ(k)j λk X

(k)j .

Following (19.10a), this can be rewritten as

δλ(ξ · X) = δλ(ξ) · X. (19.10b)

Now, in (4.12) (page 196), we have proved that, for every x ∈ G, ξ ∈ RN and λ > 0,

we haveδλ

(exp(ξ · X)(x)

) = exp((δλξ) · X)(δλ(x)).

Following (19.10b), this also reads as

δλ

(exp(ξ · X)(x)

) = exp(δλ(ξ · X)

)(δλ(x)). (19.10c)

In particular, if we take as ξ the i-th coordinate vector of the standard basis for RN ,

this gives

δλ

(exp(Xi)(x)

) = exp(λσiXi

)(δλ(x)). (19.10d)

Here λ > 0, x ∈ G and (due to the arbitrariness of the adapted basis) Xi can be anyleft-invariant (non identically vanishing) vector field in g which is δλ-homogeneousof degree σi . As a consequence, if λ = t1/σi , we have

δt1/σi

(exp(Xi)(x)

) = exp(t Xi

)(δt1/σi (x)) for all t > 0 and x ∈ G,

so that, if d is any homogeneous norm on G, we have

d(

exp(t Xi)(δt1/σi (x))) = t1/σi d

((exp(Xi)(x)

)). (19.10e)

Proof (of Theorem 19.2.1). We claim that the proof of this theorem is containedin the proof of the Carathéodory–Chow–Rashevsky Theorem 19.1.3. We closely an-alyze that proof. We know that the following map (take x0 = 0 in (19.8n))

F : RN → R

N ≡ G,

F (t) := BN

(. . . B2(B1(0, t1), t2) . . . , tN

)

19.2 An Application of Carathéodory–Chow–Rashevsky Theorem 729

is a diffeomorphism from a neighborhood V of the origin to a neighborhood W ofthe origin. It is not restrictive to replace V by

V1 := V ∩ {t ∈ G : |t1|1/q1 + · · · + t

1/qN

N < 1}

and take W1 := F(V1). (We notice that the qi’s and the map F itself depend only onthe structure of G.)

We have (see (19.8k) and (19.8e)): for every 1 ≤ i ≤ N , Bi(x, ti) is the compo-sition of c(qi) = 3 · 2qi−1 − 2 exponential maps (starting from x) along one of thefollowing vector fields

±t1/qi

i Z1, . . . ,±t1/qi

i Zm.

But since 1 ≤ qi ≤ r for every 1 ≤ i ≤ N , we can infer that F(t) is the compositionof at most

M := N(3 · 2r−1 − 2) ≥N∑

i=1

(3 · 2qi−1 − 2) =N∑

i=1

c(qi)

vector fields of the above type.Thus (setting for brevity τi := 0 if needed) any ζ ∈ W1 can be written as

ζ = exp(τMZiM)(· · · exp(τ2Zi2)(exp(τ1Zi1)(0)) · · ·), (19.11a)

where i1, . . . , iM ∈ {1, . . . , m} and |τi | < 1 are suitably chosen. We set

ζ1 := exp(τ1Zi1)(0), ζk := exp(τkZik )(ζk−1), k = 2, . . . , M. (19.11b)

By the well-known formula exp(Z)(x) = x ∗ Exp (Z), from (19.11b) we have ζk :=ζk−1 ∗ Exp (τkZik ) (here ζ0 := 0), so that from (19.11b) we also derive

ζ = ζM = ζM−1 ∗ Exp (τMZiM)

= ζM−2 ∗ Exp (τM−1ZiM−1) ∗ Exp (τMZiM) = [· · ·]= Exp (τ1Zi1) ∗ · · · ∗ Exp (τM−1ZiM−1) ∗ Exp (τMZiM).

As a consequence, we have the decomposition

ζ = ξ1 ∗ · · · ∗ ξM, where ξj = Exp (τjZij ). (19.11c)

We explicitly remark that, since the Zij ’s all belong to the first layer of g,from (19.10e) we get

d(ξj ) = d(Exp (τjZij )

) = |τj | d(Exp (±Zij )

).

Recalling that |τj | < 1 and setting

αd := max{d(Exp (±Z1)

), . . . , d

(Exp (±Zm)

)},

we have proved the estimate


d(ξj ) ≤ αd for every j = 1, . . . , M. (19.11d)

Thus, (19.11c) and (19.11d) demonstrate (19.9a)–(19.9c) for points in W1.Finally, take any x ∈ G. We shall use a simple δλ-homogeneity argument. Since

W1 is a neighborhood of the origin, there exists r > 0 such that Bd(0, r) ⊂ W1.Consequently, we have ζ := δr/d(x)(x) ∈ W1. Thus, from what we proved above,one gets (setting λ := d(x)/r)

x = δλ(ζ ) = δλ(ξ1 ∗ · · · ∗ ξM) = δλ(ξ1) ∗ · · · ∗ δλ(ξM) =: x1 ∗ · · · ∗ xM.

We notice that (by using (19.10d))

xj = δλ(ξj ) = δλ

(Exp (τjZij )

) = Exp (λτj Zij ),

whence (19.9c) follows by setting tj = λτj . Moreover, it holds

d(xj ) = d(δλ(ξj )) = λ d(ξj ) = d(x)

rd(ξj ) ≤ αd

rd(x)

(using (19.11d)), so that (19.9b) follows as well by taking c0 := αd/r . This ends theproof of the theorem. ��Bibliographical Notes. The connectivity Theorem 19.1.3 (indeed, a more generalversion of it) is due to W.L. Chow [Cho39] and P.K. Rashevsky [Ras38] (the versionof this theorem in the R

3 case and for two Hörmander vector fields is also known asCarathéodory’s theorem, see [Car09]). For modern proofs, see also [Bel96,Gro96,Her68,NSW85,VSC92]. For applications, see e.g., P. Hajłasz and P. Koskela [HK00].We explicitly remark that X-connectivity also holds when X is a system of vectorfields in R

N satisfying Hörmander’s rank condition (see the references above).For a proof of Theorem 19.2.1, see also G.B. Folland and E.M. Stein [FS82,

Lemma 1.40] and P. Pansu [Pan89, Corollary]. Theorem 19.2.1 is called “generatingproperty” by V. Magnani [Mag06, Proposition 3.12] and is applied in [Mag06] toderive properties of convex functions in Carnot groups.


Ex. 1) Let Ω ⊆ RN be an open and connected set, and let X = {X1, . . . , Xm} be any

family of locally Lipschitz-continuous vector fields defined in Ω . Obviously,the Definition 5.2.1 of X-subunit path γ ∈ S(X) makes sense (see page 232).Given x, y ∈ Ω , we let, by definition, dX(x, y) = ∞ iff there does notexist any γ ∈ S(X), γ : [0, T ] → Ω , such that γ (0) = x and γ (T ) = y.Otherwise, dX(x, y) is defined as in Definition 5.2.2, page 232.

Prove that Ω splits into a (possibly infinite and not denumerable) family ofsets Ω = ⋃

i Ωi such that:


a) x, y ∈ Ωi iff there exists γ ∈ S(X) connecting x and y;b) for every i, (Ωi, dX|Ωi×Ωi

) is a metric space;c) if x ∈ Ωi and y ∈ Ωj with i �= j , then dX(x, y) = ∞.

Are the Ωi’s open sets?With the above notation, let Ω = R

2, X = {∂x1}. Prove that dX(x, y) < ∞iff x and y both lie on the same line parallel to the x1-axis. Find the relevantΩi’s.

Ex. 2) Write down and prove a simplified version of Lemma 19.1.4 in the case ofstep two Carnot groups.

Ex. 3) Write down explicitly the indexing map jq of Lemma 19.1.4 in the casesq = 4 and q = 5.

20

Taylor Formula on Homogeneous Carnot Groups

The aim of this chapter is to prove the Lagrange mean value theorem and, conse-quently, several versions of the Taylor formula for smooth functions on homogeneousCarnot groups.

We begin with a brief sketch of polynomial functions and derivatives of a homo-geneous Carnot group. Then, we introduce and investigate the Taylor polynomialsand we prove suitable versions of the Lagrange mean value theorem and of the Tay-lor formula.

The latter will be derived from the former by an adaptation of standard argu-ments, whereas the proof of the Lagrange mean value theorem will deeply rely onthe Carathéodory–Chow–Rashevsky connectivity theorem in Chapter 19.

Due to the importance of the Lagrange mean value theorem on stratified groups,we provide two proofs of it both relying on the cited Theorem 19.1.3. The first onemakes also use of the Carnot–Carathéodory distance dZ , the second one does notinvolve dZ , but it makes use of some precise estimates in the connectivity theorem.

Finally, Taylor formulas with remainder of Lagrange-type, of Peano-type and inintegral form will be given.

Throughout the chapter, we fix the following notation. G = (RN, δλ, ∗) is a ho-mogeneous Carnot group nilpotent of step r , whose points are denoted indifferentlyby

x = (x1, . . . , xN) = (x(1), . . . , x(r)), (20.1)

where x(i) ∈ RNi . Following our usual notation for the dilations δλ : R

N → RN ,

δλ(x) = (λσ1x1, . . . , λσN xN) = (λ x(1), . . . , λrx(r)). (20.2)

α will always denote a multi-index with N entries, i.e. α = (α1, . . . , αN) withα1, . . . , αN ∈ N ∪ {0}. As usual, if α is a multi-index, we set

xα = xα11 · · · xαN

N , (Dx)α = ∂α1

x1· · · ∂αN

xN, and α! = α1! · · · αN !.

We shall also use the following notation

|α| = α1 + · · · + αN, |α|G = α1 σ1 + · · · + αN σN (20.3)

734 20 Taylor Formula on Carnot Groups

to denote, respectively, the Euclidean length (also said isotropic length) of α and thehomogeneous length (also said δλ-length or G-length) of α.

Furthermore, we denote by {Z1, . . . , ZN } the Jacobian basis for the algebra g

of G, i.e. for every j ∈ {1, . . . , N}, Zj is the left invariant vector field on G such that

Zj |0 = ∂xj|0.

When ξ = (ξ1, . . . , ξN ) ∈ RN , we use the useful notation ξ ·Z to denote

∑Nj=1 ξj Zj ,

the vector field in g whose Jacobian coordinates are given by (ξ1, . . . , ξN ). Finally,we denote by

Exp : g → G, Log : G → g,

respectively, the usual exponential and logarithmic maps related to (G, ∗).

20.1 Polynomials and Derivatives on Homogeneous CarnotGroups

20.1.1 Polynomial Functions on GGG

We begin with a very natural definition; although, despite this naturalness, we shallabandon it very shortly (after some due remarks).

Definition 20.1.1 (GGG-polynomial). A function P : G → R is called a G-polynomialif p := P ◦Exp is a polynomial function on the vector space g, i.e. p is a polynomialfunction when expressed in coordinates w.r.t. any (or equivalently, w.r.t. at least one)basis for g.

Since {Z1, . . . , ZN } is a basis for g, the set of G-polynomials is given by{P = p ◦ Log | p(ξ) := p(ξ · Z) is a polynomial in ξ

}. (20.4)

We now recall formulas (1.75a) and (1.75b) (page 50), where we proved that Expand Log have the following coordinate form

Exp (ξ · Z) =

⎛

⎜⎜⎝

ξ1ξ2 + B2(ξ1)

...

ξN + BN(ξ1, . . . , ξN−1)

⎞

⎟⎟⎠ ∀ ξ ∈ RN,

Log (x) =

⎛

⎜⎜⎝

x1x2 + C2(x1)

...

xN + CN(x1, . . . , xN−1)

⎞

⎟⎟⎠ · Z ∀ x ∈ G, (20.5)

where the Bi’s and Ci’s are polynomial functions on RN (δλ-homogeneous of degree

σi) completely determined by the composition law on G. It is then immediately seenthat

20.1 Polynomials and Derivatives on Homogeneous Carnot Groups 735

the set of G-polynomials coincides with the set of polynomial functionswith respect to the fixed coordinate system (x1, . . . , xN) on G.

Indeed, if Q is any polynomial function on G, then the function q := Q ◦ Exp is apolynomial on g, for (thanks to (20.5))

q(ξ · Z) = Q(Exp (ξ · Z)) = Q(ξ1, ξ2 + B2(ξ1), . . . , ξN + BN(ξ1, . . . , ξN−1))

is evidently a polynomial function in ξ . Vice versa, let P be a G-polynomial. Then(setting p = P ◦ Exp and following the notation in (20.4))

P(x) = (P ◦ Exp ◦ Log )(x) = p((x1, . . . , xN + CN(x1, . . . , xN−1)) · Z)

= p(x1, x2 + C2(x1), . . . , xN + CN(x1, . . . , xN−1))

is a polynomial function in x, since p and the Ci’s are polynomials.If p = p(x) is any polynomial function in the fixed coordinates (x1, . . . , xN)

on G, we set (also following notation (20.3))

deg(p) := max

{|α| : p(x) =

∑

α

cα xα with cα �= 0 for every α

},

degG (p) := max

{|α|G : p(x) =

∑

α

cα xα with cα �= 0 for every α

}

to denote, respectively, the Euclidean degree (also said isotropic degree) of p and thehomogeneous degree (also said δλ-degree or G-degree) of p. For any n ∈ N ∪ {0},we give the following definition

Pn := {p polynomial on G : degG (p) ≤ n}. (20.6)

Obviously, Pn is a vector space over the field R. We denote its dimension by

μn := dimR(Pn). (20.7)

For every n ∈ N ∪ {0}, we introduce the set of multi-indices

In := {α multi-index in (N ∪ {0})N : |α|G ≤ n}. (20.8)

With the above definitions, we have the following proposition, whose simple proofis left as an exercise.

Proposition 20.1.2 (On the vector space Pn). For every n ∈ N ∪ {0}, it holds Pn =span{xα : α ∈ In}, so that μn equals the cardinality of In. Moreover, if In has beenordered in some fixed way, then the map

Pn → Rμn, p → ((Dx)

αp(0))α∈In

is a vector space isomorphism. More explicitly, if β ∈ In, the above map sendsxβ into (0, . . . , β!, . . . , 0), where the only non-vanishing entry appears at the sameplace where β appears in the fixed ordering of In.


Example 20.1.3. If G = H1 is the Heisenberg–Weyl group on R

3 (whose points aredenoted by (x, y, t)) with dilations δλ(x, y, t) = (λx, λy, λ2t), we have

I0 = {(0, 0, 0)},I1 = {(0, 0, 0), (1, 0, 0), (0, 1, 0)},I2 = I1 ∪ {(0, 0, 1), (2, 0, 0), (0, 2, 0), (1, 1, 0)},I3 = I2 ∪ {(1, 0, 1), (0, 1, 1), (3, 0, 0), (0, 3, 0), (2, 1, 0), (1, 2, 0)},P0 = span{1}, μ0 = 1,

P1 = span{1, x, y}, μ1 = 3,

P2 = span{1, x, y; t, x2, y2, xy}, μ1 = 7,

P3 = span{1, x, y; t, x2, y2, xy; xt, yt, x3, y3, x2y, xy2}, μ1 = 13.

20.1.2 Derivatives on GGG

We now relate the usual Euclidean derivatives on RN ≡ G to the derivatives along

the Jacobian basis related to G. We recall formulas (1.37) and (1.38) (page 22) wherewe showed that, for any smooth function u on G and every x ∈ G, it holds

(Z1u(x), . . . , ZNu(x)

) = (∂x1u(x), . . . , ∂xNu(x)

) · Jτx (0),(∂x1u(x), . . . , ∂xN

u(x)) = (Z1u(x), . . . , ZNu(x)

) · Jτx−1 (0), (20.9)

where

Jτx (0) =

⎛

⎜⎜⎜⎜⎝

1 0 · · · 0

a2,1(x) 1. . .

......

. . .. . . 0

aN,1(x) · · · aN,N−1(x) 1

⎞

⎟⎟⎟⎟⎠

is the Jacobian matrix of the left translation by x on G (i.e. τx(y) = x ∗ y) and theai,j ’s are polynomial functions on G. More explicitly, the usual partial derivativesw.r.t. the fixed coordinates xi’s on G and the vector fields of the Jacobian basis arerelated by the following “triangle-shaped” formulas:

Z1 = ∂x1 + a2,1(x) ∂x2 + a3,1(x) ∂x3 + · · · + aN,1(x) ∂xN,

Z2 = ∂x2 + a3,2(x) ∂x3 + · · · + aN,2(x) ∂xN,

......

ZN = ∂xN(20.10)

and, setting bi,j (x) := ai,j (x−1),

∂x1 = Z1 + b2,1(x) Z2 + b3,1(x) Z3 + · · · + bN,1(x) ZN,

∂x2 = Z2 + b3,2(x) Z3 + · · · + bN,2(x) ZN,

......

∂xN= ZN. (20.11)


Taking into account the δλ-degrees of the Zh’s and of the ∂xh’s we have the formulas

Zh = ∂xh+

∑

j : σj >σh

aj,h(x) ∂xj,

∂xh= Zh +

∑

j : σj >σh

bj,h(x) Zj . (20.12)

For example, if G is the Heisenberg–Weyl group on R3 (see Example 20.1.3),

Z1 = ∂x + 2y∂t , Z2 = ∂y − 2x∂t , Z3 = ∂t

is the Jacobian basis. It is immediately seen that

Z1 = ∂x + 2y ∂t ,

Z2 = ∂y − 2x ∂t ,

Z3 = ∂t ,

∂x = Z1 − 2y Z3,

∂y = Z2 + 2x Z3,

∂t = Z3.

For any multi-index α, we introduce the higher order Z-derivative setting

Zα := Zα11 · · ·ZαN

N . (20.13)

Obviously, Zα is a left invariant differential operator (not necessarily a vector field)and, as a differential operator, it is δλ-homogeneous of degree

α1σ1 + · · · + αNσN = |α|G(for Zj is δλ-homogeneous of degree σj ).

We explicitly observe that, a priori, Z2Z1 is not necessarily contained in the setof the Zα’s. Nonetheless, we have the following proposition.

Proposition 20.1.4. For every k ∈ N, every i1, . . . , ik ∈ {1, . . . , N}, and everyβ1, . . . , βk ∈ N ∪ {0}, we have

(Zi1)β1 · · · (Zik )

βk ∈ span{(Z1 · · ·ZN)α : α multi-index}.More precisely, the multi-indices in the right-hand side run over the set of the α’ssuch that1 |α|G = σi1β1 + · · · + σikβk .

Proof. First, we write

(Zi1)β1 · · · (Zik )

βk = Zi1 · · ·Zi1︸︷︷︸β1 times

· · ·Zik · · ·Zik︸︷︷︸βk times

= Zi1 · · · (ZiZj ) · · · Zik ,

1 Here, the σij ’s are the exponents in (20.2).


where by i and j we have indicated an arbitrary couple of contiguous indices. Now,since ZiZj = [Zi, Zj ] + ZjZi , we have

Zi1 · · · (ZiZj ) · · · Zik = Zi1 · · · [Zi, Zj ] · · · Zik + Zi1 · · · (ZjZi) · · · Zik . (20.14)

Since [Zi, Zj ] is a left invariant vector field, it can be written as a linear combinationof the Zk’s, say [Zi, Zj ] = ∑

r cr Zr , so that the first summand in the right-handside of (20.14) equals

∑r cr Zi1 · · ·Zr · · ·Zik . Moreover, the Zr ’s can be chosen so

that σr = σi + σj .As a consequence, we have written the left-hand side of (20.14) as the sum of a

term with Zi and Zj interchanged plus a linear combination of summands each withβ1 +· · ·+βk −1 factors of the type Zh’s. Also, these two terms are δλ-homogeneousoperators of the same degree as the left-hand side (namely, σi1β1+· · ·+σikβk). Now,this argument can be iterated and, arguing by induction, the assertion can be easilyproved. �

For example, if G is the Heisenberg–Weyl group on R3 (see Example 20.1.3) and

Z1 = ∂x + 2y∂t , Z2 = ∂y − 2x∂t , Z3 = ∂t is the Jacobian basis, then Z2Z1 does notbelong to the set of the Zα’s in (20.13). Indeed, it holds

Z21 = ∂x,x + 4y ∂x,t + 4y2 ∂t,t ,

Z22 = ∂y,y − 4x ∂y,t + 4x2 ∂t,t ,

Z1Z2 = ∂x,y − 2 ∂t − 2x ∂x,t + 2y ∂y,t − 4xy ∂t,t ,

Z2Z1 = ∂x,y + 2 ∂t + 2y ∂y,t − 2x ∂x,t − 4xy ∂t,t .

Though, as asserted in the above proposition, it holds

Z2Z1 = Z1Z2 + 4 Z3 = (Z1Z2Z3)(1,1,0) + 4 (Z1Z2Z3)

(0,0,1).

We now provide a higher-order version of formula (20.12).

Proposition 20.1.5 (Zα’s and (Dx)α’s). For every multi-index α, we have

Zα = (Dx)α +

∑

β: β �=α, |β|≤|α|, |β|G≥|α|Gaβ,α(x) (Dx)

β, (20.15)

where the aβ,α’s are polynomials, δλ-homogeneous of degree |β|G − |α|G.

Example 20.1.6. If G is the Heisenberg–Weyl group on R3 (see Example 20.1.3) and

Z1 = ∂x + 2y∂t , Z2 = ∂y − 2x∂t , Z3 = ∂t is the Jacobian basis, we have

(Z1Z2Z3)(1,1,0) = ∂x,y − 2∂t − 2x∂x,t + 2y∂y,t − 4xy∂t,t

= (Dx,y,t )(1,1,0) − 2(Dx,y,t )

(0,0,1) − 2x(Dx,y,t )(1,0,1)

+ 2y(Dx,y,t )(0,1,1) − 4xy(Dx,y,t )

(0,0,2).

We compare this to (20.15). We notice that the far right-hand side contains (Dx)α

plus 4 summands having (Dx)β ’s terms with β �= α: the first term satisfies |β| < |α|

and |β|G = |α|G, the second, third and fourth terms satisfy |β| = |α| and |β|G >

|α|G.


Proof (of Proposition 20.1.5). We argue by induction on the (isotropic) length of α.If |α| = 1, then α = eh (the h-th vector of the standard basis of R

N ) and Zα =Zh. In this case, (20.15) immediately reduces to (20.12) (see also Proposition 1.3.5,page 34).

Suppose now that (20.15) has been proved for every multi-index α with |α| ≤ k.Let us prove it for a multi-index α with |α| = k + 1. We have

α = (0, . . . , 0, αh, . . . , αN), with αh �= 0.

In particular, it holds Zα = Zh Zα , where α = α − eh, so that |α| = k. Moreover,we have |α|G = |α|G − σh. As a consequence, by the inductive hypothesis, we have

Zα = Zh

( ∑

β: |β|≤|α|, |β|G≥|α|Gaβ,α(x) (Dx)

β

)

=( ∑

σj ≥σh

aj ∂xj

)( ∑

|β|≤k, |β|G≥|α|Gaβ (Dx)

β

)

=∑

σj ≥σh, |β|≤k, |β|G≥|α|Gaj (∂xj

aβ) (Dx)β

+∑

σj ≥σh, |β|≤k, |β|G≥|α|Gaj aβ (Dx)

β+ej .

Now we treat each of the two sums appearing in the above far right-hand side. First,we notice that aβ is δλ-homogeneous of degree |β|G − |α|G = |β|G − |α|G + σh

and that aj is δλ-homogeneous of degree σj −σh. Then the G-degree of aj (∂xjaβ) is

σj −σh +|β|G −|α|G +σh −σj = |β|G −|α|G. Moreover, ∂xjaβ ≡ 0 whenever the

G-degree of aβ is < σj , i.e. whenever |β|G − |α|G < σj − σh. So we can supposethat |β|G − |α|G ≥ σj − σh ≥ 0.

Finally, as for the second sum, the G-degree of aj aβ is σj − σh + |β|G − |α|G +σh = σj + |β|G − |α|G = |β + ej |G − |α|G. Moreover, |β + ej |G = |β|G + σj ≥|α|G + σh = |α|G.

We now isolate the coefficient of (Dx)α . In the first sum, this term does not

appear, since the sum runs over the β’s with |β| ≤ k, whereas |α| = k + 1. In thesecond sum, this term appears only when j and β (satisfying also σj ≥ σh, |β| ≤ k

and |β|G ≥ |α|G − σh) are such that β + ej = α. In particular, β = α − ej , sothat |β|G = |α|G − σj , which is ≥ |α|G − σh only if σj ≤ σh. But since the sumruns over the j ’s such that σj ≥ σh, it must be σj = σh. Now, from the first equalityin (20.12), we see that the only summand with σj = σh is the one with j = h, andin this case ah = 1. This gives β = α − eh = α, so that

aj aβ = aj aβ,α = aj aα,α = 1,

also using the inductive hypothesis aα,α = 1. This completes the proof. � A completely analogous proof (this time making use of the second formula

in (20.12)) demonstrates the following result.


Proposition 20.1.7 ((Dx)α’s and Zα’s). For every multi-index α, we have

(Dx)α = Zα +

∑

β: β �=α, |β|≤|α|, |β|G≥|α|Gbβ,α(x) Zβ, (20.16)

where the bβ,α’s are polynomials, δλ-homogeneous of degree |β|G − |α|G.

From Propositions 20.1.5 and 20.1.7 we derive the following corollary.

Corollary 20.1.8. Let u be a smooth real-valued function on G. Then u is a polyno-mial if and only if there exists k ∈ N such that Zαu ≡ 0 for every multi-index α with|α|G ≥ k (or with |α| ≥ k).

The same assertion holds if there exists k ∈ N such that Zαu ≡ 0 for everymulti-index α with |α|G = k.

Proof. First, we remark that (see (20.3))

σ1 |α| ≤ |α|G ≤ σN |α|. (20.17)

As a consequence, the “only if” part follows from (20.15). Vice versa, if Zβu ≡ 0 forevery multi-index β with |β|G ≥ k, then (20.16) proves that (Dx)

αu ≡ 0 whenever|α|G ≥ k. Thanks to (20.17), if |α| ≥ k/σ1, then |α|G ≥ k, so that (Dx)

αu ≡ 0.Hence u is a polynomial.

As for the last statement of the corollary, we only need to prove that if there existsk ∈ N such that Zαu ≡ 0 whenever |α|G = k, then there exists k∗ ∈ N such thatZαu ≡ 0 whenever |α|G ≥ k∗. Now, it suffices to take k∗ large enough dependingon k and σ1, . . . , σN . Indeed2, we argue by induction noticing that

Zα = Zαi0i0

Zα1+i01+i0

· · ·ZαN

N = Zαi0 −1i0

(Z

αi0 −1i0

Zα1+i01+i0

· · · ZαN

N

)

(here i0 is the first index such that αi0 �= 0) and∣∣(0, . . . , 0, αi0 − 1, α1+i0 , . . . , αN)

∣∣G

= |α|G − σi0 .

This ends the proof. � We now recall that, since G is a Carnot group, then the elements of the Jaco-

bian basis laying in the first layer of the stratification of g Lie-generate the whole g.Setting m := N1 in (20.1), we denote these vectors by Z1, . . . , Zm. We thus have

Lie{Z1, . . . , Zm} = span{Z1, . . . , ZN } = g. (20.18)


2 Notice that the choice k∗ = k will not do. For instance, in the Heisenberg group on R3 of

Example 20.1.6, if we know that Zαu ≡ 0 whenever |α|G ≥ 5, then we cannot immediatelydeduce by induction that Zαu ≡ 0 whenever |α|G = 6. For example, Z3

3 = Z(0,0,3) and

|(0, 0, 3)|G = 6, but Z33 = Z3 Z2

3 = Z3(Z(0,0,2)) and |(0, 0, 2)|G = 4.

20.2 Taylor Polynomials on Homogeneous Carnot Groups 741

Proposition 20.1.9. It holds

span{Zi1 · · ·Zik : i1, . . . , ik ∈ {1, . . . , m}, k ∈ N}= span{(Z1 · · · ZN)α : α multi-index}.

Proof. The inclusion ⊆ follows from Proposition 20.1.4. In order to prove the oppo-site inclusion, it suffices to prove that Z1, . . . , ZN can all be written as linear com-binations of terms of the type Zi1 · · · Zik . This fact now easily follows from (20.18),since

[· · · [[Zi1 , Zi2], Zi3 ] · · · Zik ] ∈ span{Zi1 · · ·Zik : ij ∈ {1, . . . , m}, k ∈ N}.This ends the proof. �

With the aid of Lemma 20.1.11 below (whose proof is left as an exercise), it canimmediately be proved the following result (compare it to Corollary 1.5.5, page 68).

Corollary 20.1.10. Let u be a smooth real-valued function on G. Then u is a poly-nomial if and only if there exists k ∈ N such that Zi1 · · · Ziku ≡ 0 for everyi1, . . . , ik ∈ {1, . . . , m}.Proof. It follows by collecting together Corollary 20.1.8, Proposition 20.1.9 andLemma 20.1.11 below. � Lemma 20.1.11. If H1, . . . , Hn are (not identically vanishing) differential operatorson R

N ≡ G which are δλ-homogeneous of degrees d1, . . . , dn, respectively, withdi �= dj for every i �= j , then they are linearly independent.

Example 20.1.12. Let G be the Heisenberg–Weyl group on R3 (see Example 20.1.6).

Then, for example, we know that u is a polynomial provided that

Z21u, Z1Z2u, Z2

2u, Z3u

all vanish identically: this is stated in the last part of Corollary 20.1.8. Also, we knowthat u is a polynomial function provided that

Z21u, Z1Z2u, Z2Z1u, Z2

2u

all vanish identically: this is stated in Corollary 20.1.10. These conditions are equiv-alent, though different in appearance.

20.2 Taylor Polynomials on Homogeneous Carnot Groups

We begin with another result on the derivatives on G, helpful for our Taylor formula.We follow the notation in the previous sections (see in particular (20.6)–(20.8) and(20.13)).


Theorem 20.2.1. Let n ∈ N ∪ {0} and consider the space Pn of the polynomialsδλ-homogeneous of degree at most n. Moreover, let In be ordered in some fixed way.Then the map

Pn → Rμn, p → ((Zαp)(0))α∈In

is a vector space isomorphism.

Proof. We consider (20.15) of Proposition 20.1.5. Since aβ,α(x) is δλ-homoge-neous of degree |β|G − |α|G, we have aβ,α(0) = 0 whenever |β|G > |α|G andaβ,α(x) =constant (say, aβ,α) whenever |β|G = |α|G. Setting aα,α = 1, from (20.15)we thus derive

Zα|x=0 =∑

β: |β|≤|α|, |β|G=|α|Gaβ,α (Dx)

β |x=0. (20.19)

Now, we recall the dual formula

(Dx)α|x=0 =

∑

β: |β|≤|α|, |β|G=|α|Gbβ,α Zβ |x=0 (20.20)

implied by (20.16) of Proposition 20.1.7. These facts prove the following result: Ifwe indicate by Φ : Pn → R

μn the map in Proposition 20.1.2 and by Ψ : Pn →R

μn the map in Theorem 20.2.1, then (20.19) shows that there exists a linear mapφ : R

μn → Rμn such that

Ψ (p) = φ(Φ(p)) for every p ∈ Pn;analogously, (20.20) shows that there exists a linear map ψ : R

μn → Rμn such that

Φ(p) = ψ(Ψ (p)) for every p ∈ Pn.

These facts together demonstrate that ψ and φ are inverse to each other, so thatΨ = φ ◦ Φ is a linear isomorphism since Φ is (see Proposition 20.1.2). �

Theorem 20.2.1 has the following important corollary.

Corollary 20.2.2. Let n ∈ N ∪ {0}. Let, as usual, Pn be the space of polynomials,δλ-homogeneous of degree at most n, and let μn be the dimension of Pn. Then, forany μn-tuple of real numbers z, there exists one and only one polynomial p in Pn

such that((Zαp)(0))α∈In

= z. (20.21)

Here, as usual, we have fixed an ordering on In, the set of multi-indices α such that|α|G ≤ n.

Corollary 20.2.2 proves that the following definition is well-posed.

Definition 20.2.3 (Z-Mac Laurin polynomial). Let f ∈ C∞(G, R). Let, as usual,Z1, . . . , ZN denote the Jacobian basis related to G. Then, for every n ∈ N ∪ {0},there exists one and only one polynomial p, δλ-homogeneous of degree at most n,such that


(Z1 · · ·ZN)α(p)(0) = (Z1 · · ·ZN)α(f )(0) for every α ∈ In. (20.22)

We say that p(x) = Pn(f, 0)(x) is the Z-Mac Laurin polynomial of δλ-degree n,related to f .

Indeed, it suffices to apply Corollary 20.2.2 with

z = ((Zαf )(0))α∈In.

Remark 20.2.4. In (20.56) of Section 20.3.2, we shall provide an explicit formula forPn(f, 0).

An example is in order.

Example 20.2.5. Let us consider G = H1, the Heisenberg–Weyl group on R

3 (seealso Example 20.1.3), with its Jacobian basis Z1 = ∂x + 2y∂t , Z2 = ∂y − 2x∂t ,Z3 = ∂t . Let f ∈ C∞(H1, R). Let us find p = P1(f, 0). Since p ∈ P1, there exista, b, c ∈ R such that p(x, y, t) = a + b x + c y. Then (20.22) gives

a = p(0) = f (0),

b = Z1p(0) = Z1f (0),

c = Z2p(0) = Z2f (0),

so thatP1(f, 0)(x, y, t) = f (0) + Z1f (0) x + Z2f (0) y.

Let us now find q = P2(f, 0). Since q ∈ P2, there exist a, b, c, d, e, f, g ∈ R suchthat q(x, y, t) = a + b x + c y + d t + e x2 + f y2 + g xy. Then (20.22) gives

a = q(0) = f (0),

b = Z1q(0) = Z1f (0),

c = Z2q(0) = Z2f (0),

d = Z3q(0) = Z3f (0),

2e = (Z1)2q(0) = (Z1)

2f (0),

2f = (Z2)2q(0) = (Z2)

2f (0),

g − 2d = Z1Z2q(0) = Z1Z2f (0),

so that

P2(f, 0)(x, y, t) = f (0) + Z1f (0) x + Z2f (0) y + Z3f (0) t

+ 1

2(Z1)

2f (0) x2 + 1

2(Z2)

2f (0) y2 + (2Z3f (0)

+ Z1Z2f (0)) xy.

Since in this case Z3 = − 14 (Z1Z2 − Z2Z1), this formula can be rewritten as


P2(f, 0)(x, y, t)

= f (0) + Z1f (0) x + Z2f (0) y − 1

4(Z1Z2 − Z2Z1)f (0) t

+ 1

2(Z1)

2f (0) x2 + 1

2(Z2)

2f (0) y2 + 1

2(Z1Z2 + Z2Z1)f (0)xy. (20.23)

Notice that, though unique, the Z-Mac Laurin polynomial can be written in manydifferent symbolic ways.

For instance, observe that (20.23) can be rewritten in the suggesting form

P2(f, 0)(x, y, t) = f (0) + ∇Gf (0) ·(

xy

)

+ Z3f (0) t + 1

2(x, y) · Hesssymf (0) ·

(xy

), (20.24)

where ∇Gf (0) = (Z1f (0), Z2f (0)) is the horizontal gradient of f at 0 and

Hesssymf (0) :=(

Z21f (0) 1

2 (Z1Z2f (0) + Z2Z1f (0))12 (Z1Z2f (0) + Z2Z1f (0)) Z2

2f (0)

)

is the so-called symmetrized horizontal Hessian of f at 0. See Ex. 4 for the computa-tion of P3(f, 0) in the present case. See Ex. 6 for a suitable generalization of (20.24)to other Carnot groups and see Section 20.3.2 for a general formula for Pn(f, 0) forevery n and every Carnot group. �

We leave as an exercise (see Ex. 7 at the end of the chapter) to prove the followingassertion.

Remark 20.2.6. Let Pn be the vector space of the polynomials with G-degree atmost n. Let also P be the set of polynomial functions on G. Since the set of allmulti-indices I = (N ∪ {0})N can be decomposed as I = ⋃

n≥0 In, the followingmap is well-defined

πn : P −→ Pn, p → pn,

where

p(x) =∑

α

cα xα =∑

α∈In

cα xα +∑

α/∈In

cα xα =: pn(x) + p∗n(x).

Then, if Qn(f, 0)(x) denotes the usual Euclidean Mac Laurin polynomial of f , it iseasy to see that

Pn(f, 0)(x) = πn

(Qn(f, 0)(x)

).

In other words, the Z-Mac Laurin polynomial of f coincides with the sum ofthe terms, in the usual Euclidean Mac Laurin polynomial of f , which are δλ-homogeneous of degree at most n.

The following result is not so evident, a priori.


Proposition 20.2.7 (Nested property of the Z-Mac Laurin polynomials). Letn ∈ N and f ∈ C∞(G, R). Then Pn(f, 0) − Pn−1(f, 0) is a δλ-homogeneous poly-nomial of degree n. In other words,

Pn(f, 0) = Pn−1(f, 0) + {a polynomial of G-degree = n}.Proof. Let us write Pn(f, 0) = p + q, where p has G-degree ≤ n − 1, so that q hasG-degree = n (unless it vanishes identically). We have to prove that p = Pn−1(f, 0).By Definition 20.2.3, this follows if we show that

Zα(p)(0) = Zα(f )(0) for every α ∈ In−1. (20.25)

If α ∈ In−1, then α ∈ In, so that by the very definition of Pn(f, 0), we have

Zα(f )(0) = Zα(Pn(f, 0))(0) = Zα(p + q)(0) = Zαp(0) + Zαq(0).

Then, (20.25) will follow if we show that Zαq(0) = 0. This fact is a consequenceof the fact that Zαq is a polynomial, δλ-homogeneous of positive degree, preciselyn − |α|G (recall that q has G-degree = n and |α|G ≤ n − 1, since α ∈ In−1). Thisends the proof. �

The following definition is definitely not unexpected.

Definition 20.2.8 (Z-Taylor polynomial). Let f ∈ C∞(G, R), x0 ∈ G and n ∈N ∪ {0} be fixed. Let us consider the Z-Mac Laurin polynomial of the function

x → f (x0 ∗ x),

i.e. the polynomial Pn(f ◦ τx0, 0). We say that the polynomial

Pn(f, x0)(x) := Pn(f ◦ τx0 , 0)(x−10 ∗ x) (20.26)

is the Z-Taylor polynomial of G-degree n related to f and centered at x0.

Remark 20.2.9. In (20.56) of Section 20.3.2, we shall provide an explicit formula forPn(f, x0).

The following characterization of Pn(f, x0) holds.

Proposition 20.2.10 (Characterization). With the notation of the above definition,Pn(f, x0) is characterized by being the only polynomial p, δλ-homogeneous of de-gree at most n, such that

(Z1 · · ·ZN)α(p)(x0) = (Z1 · · ·ZN)α(f )(x0) for every α ∈ In. (20.27)

Proof. An immediate consequence of Definitions 20.2.3, 20.2.8 and the left invari-ance of Z1, . . . , ZN . �


With reference to Example 20.2.5, setting z0 = (x0, y0, t0), we have

P2(f ◦ τz0, 0) = f (z0) + Z1f (z0) x + Z2f (z0) y + Z3f (z0) t

+ 1

2(Z1)

2f (z0) x2 + 1

2(Z2)

2f (z0) y2

+ 1

2(Z1Z2 + Z2Z1)f (z0) xy,

so that

P2(f, z0)(x, y, t) = P2(f ◦ τz0, 0)(z−10 ∗ (x, y, t))

= f (z0) + Z1f (z0) (x − x0) + Z2f (z0) (y − y0)

+ Z3f (z0) (t − t0 − 2y0x + 2x0y) + 1

2(Z1)

2f (z0) (x − x0)2

+ 1

2(Z2)

2f (z0) (y − y0)2

+ 1

2(Z1Z2 + Z2Z1)f (z0) (x − x0)(y − y0).

20.3 Taylor Formula on Homogeneous Carnot Groups

The aim of this section is to provide our G-version of Taylor formula. In order todo this, we use a suitable version of the mean value theorem, using paths along vec-tor fields of the first layer of the stratification of G. For this reason, we shall call itthe stratified Lagrange mean value theorem. To derive this theorem, we need someimportant prerequisites. Indeed, we shall make a crucial use of the results in Chap-ter 19 (and in particular, of Lemma 19.1.4, a subtle consequence of the Campbell–Hausdorff formula).

Throughout the section, (G, ∗) is a fixed homogeneous Carnot group with m

generators. We take Z1, . . . , Zm (the first m vectors of the Jacobian basis) as gener-ators for the algebra of G. Moreover, we shall denote by Exp : g → G the usualexponential map related to (G, ∗) and by d any homogeneous norm on G.

Theorem 20.3.1 (The stratified Lagrange mean value theorem). There exist ab-solute constants c1, b > 0, depending only on the Carnot group G and on the homo-geneous norm d , such that

|f (x ∗ h) − f (x)| ≤ c1 d(h) supz: d(z)≤b d(h)

|(Z1f (x ∗ z), . . . , Zmf (x ∗ z))|,(20.28)

for all f ∈ C1(G, R) and every x, h ∈ G.

Proof. Due to the importance of this result, we provide two proofs of it, both relyingon the Carathéodory–Chow–Rashevsky connectivity Theorem 19.1.3 in Chapter 19.

20.3 Taylor Formula on Homogeneous Carnot Groups 747

The first one is very simple, but it makes also use of the Carnot–Carathéodory dis-tance dZ related to the set of vector fields Z = {Z1, . . . , Zm} (see Section 5.2, Chap-ter 5, page 232). The second one does not involve dZ but it makes use of the preciseestimate in Theorem 19.2.1, page 727.

First proof. Since all homogeneous norms are equivalent (see Proposition 5.1.4,page 230) we can replace d by the Carnot–Carathéodory distance dZ related toZ = {Z1, . . . , Zm}. By the very definition of dZ (see (5.6), page 232), there ex-ists a sequence Tn ↓ dZ(x ∗ h, x) = dZ(h, 0) and, for every n ∈ N, there exists anabsolutely continuous curve γn : [0, Tn] → G connecting x to x ∗ h (i.e. γn(0) = x

and γn(Tn) = x ∗ h) which is {Z1, . . . , Zm}-subunit, i.e.

〈γn(t), ξ 〉2 ≤m∑

j=1

〈XjI (γn(t)), ξ 〉2 ∀ ξ ∈ RN

almost everywhere in [0, T ].Then we have (also setting Zf (y) = (Z1f (y), . . . , Zmf (y)))

|f (x ∗ h) − f (x)| =∣∣∣∣∫ Tn

0

d

ds(f (γn(s))) ds

∣∣∣∣ =∣∣∣∣∫ Tn

0〈(∇f )(γn(s)), γn(s)〉 ds

∣∣∣∣

≤∫ Tn

0

(m∑

j=1

〈ZjI (γn(s)), (∇f )(γn(s))〉2

)1/2

ds

=∫ Tn

0

∣∣(Zf )(γn(s))∣∣ ds ≤ Tn sup

s∈[0,Tn]|Zf (γn(s))|. (20.29)

Now, by the very definition of dZ , for every s ∈ [0, Tn] we obviously have

dZ(x, γn(s)) ≤ dZ(x, γn(Tn)) = dZ(x, x ∗ h) = dZ(h, 0),

so (20.29) yields

|f (x ∗ h) − f (x)| ≤ Tn supy: dZ(x,y)≤dZ(h,0)

|Zf (y)|. (20.30)

Let now n → ∞ in (20.30), and recall that Tn ↓ dZ(h, 0). This gives3

|f (x ∗ h) − f (x)| ≤ dZ(h, 0) supy: dZ(x,y)≤dZ(h,0)

|Zf (y)|. (20.31)

Finally, (20.31) implies (20.28) by setting y = x∗z and by taking b = c2 and c1 = c,where c is the constant such that c−1 d ≤ dZ ≤ c d on G × G.

Second proof. First, suppose that h ∈ Exp (span{Zj }) for some j ∈ {1, . . . , m}.We can suppose that h = Exp (tZj ) for some t > 0 (the case t < 0 followsby replacing Zj by −Zj ). Then (19.10e) gives d(h) = t d(Exp (Zj )) (recall that{Z1, . . . , Zm} span the first layer of the stratification, i.e. σj = 1 for 1 ≤ j ≤ m).Let us set

3 (20.31) shows that the stratified mean-value theorem in (20.28) holds with constants c1 = 1and b = 1 when d is the Carnot–Carathéodory distance.


c := max{d−1(Exp (Z1)), . . . , d

−1(Exp (Zm))}.

With this position, we have

t = d(h) d−1(Exp (Zj )) ≤ c d(h).

Thus, for every x ∈ G, it holds

|f (x ∗ h) − f (x)|= ∣∣f (x ∗ Exp (tZj )

)− f (x)∣∣ = ∣∣f ( exp(tZj )(x)

)− f (x)∣∣

(set γ (t) := exp(tZj )(x))

=∣∣∣∣∫ t

0

d

ds(f (γ (s))) ds

∣∣∣∣ =∣∣∣∣∫ t

0(Zjf )(γ (s)) ds

∣∣∣∣≤ t sup

s∈[0,t]|Zjf (x ∗ Exp (sZj ))| ≤ c d(h) sup

z: d(z)≤d(h)

|Zf (x ∗ z)|. (20.32)

Here we have set Zf (y) = (Z1f (y), . . . , Zmf (y)), and we have used again the factthat, for every s ∈ [0, t], one has

d(Exp (sZj )) = s d(Exp (Zj )) ≤ t d(Exp (Zj )) = d(Exp (tZj )) = d(h).

In the case of an arbitrary h ∈ G, we first remark that by Theorem 19.2.1(page 727), we have h = h1 ∗ · · · ∗ hM, where any hj ’s belong to Exp (span{Zi}) forsome i ∈ {1, . . . , m} and d(hj ) ≤ c0 d(h) for all j = 1, . . . , M. Thus, using (20.32)repeatedly, one gets (also setting h0 := 0)

|f (x ∗ h) − f (x)| = ∣∣f(x ∗ h1 ∗ · · · ∗ hM

)− f (x)∣∣

≤M∑

j=1

∣∣f(x ∗ h1 ∗ · · · ∗ hj

)− f (x ∗ h1 ∗ · · · ∗ hj−1)∣∣

(by (20.32)) ≤M∑

j=1

c d(hj ) supz: d(z)≤d(hj )

|Zf (x ∗ h1 ∗ · · · ∗ hj−1 ∗ z)|. (20.33)

By the pseudo-triangle inequality for d , d(x ∗ y) ≤ c (d(x) + d(y)), we have

d(h1 ∗ · · · ∗ hj−1 ∗ z) ≤ cj−1 (d(h1) + · · · + d(hj−1) + d(z))

(since d(z) ≤ d(hj )) ≤ cj−1 (d(h1) + · · · + d(hj ))

(since d(hj ) ≤ c0 d(h)) ≤ cj−1 j c0 d(h) ≤ cj−1 M c0 d(h) =: b d(h).

As a consequence, (20.33) proves that

|f (x ∗ h) − f (x)| ≤ M c c0 d(h) supζ : d(ζ )≤b d(h)

|Zf (x ∗ ζ )|,

so that the position c1 := M c c0 finally gives (20.28). �


With Theorem 20.3.1 at hand, we can prove the following theorem. Note that hereand in the proof below, α is a multi-index with m entries (not N , as in the previousparagraphs). Note that the Euclidean length |α| coincides with

|(α, 0, . . . , 0)|G,

the 0’s occurring N − m times.

Theorem 20.3.2 (Stratified Taylor inequality). Let (G, ∗) be a homogeneousCarnot group and d a homogeneous norm on G. For every n ∈ N ∪ {0}, there existsa constant cn > 0 (depending only on n, G and d) such that

|f (x ∗ h) − Pn(f, x)(x ∗ h)| ≤ cn dn(h)

× supd(z)≤bn d(h)

{∣∣(Z1, . . . , Zm)αf (x ∗ z) − (Z1, . . . , Zm)αf (x)∣∣ : |α| = n

}

(20.34)

for all f ∈ Cn(G, R) and every x, h ∈ G. Here b is as in Theorem 20.3.1.

Proof. We fix x ∈ G and n ∈ N ∪ {0}, and we let

Φ(h) := f (x ∗ h) − Pn(f, x)(x ∗ h).

By Proposition 20.2.10, it holds Zβ(f − Pn(f, x))(x) = 0 whenever β is a N -dimensional multi-index with |β|G ≤ n, i.e. (by using the left-invariance of the Zβ ’s)

(Z1, . . . , ZN)βΦ(0) = 0 ∀ β ∈ (N ∪ {0})N, |β|G ≤ n.

By taking a multi-index β with 0 entries from the (m + 1)-th one, we get

(Z1, . . . , Zm)αΦ(0) = 0 ∀ α : |α| ≤ n. (20.35)

Let us prove by induction on j = 0, 1, . . . , n that if |α| = n − j then∣∣(Z1, . . . , Zm)αΦ(h)

∣∣ ≤ cj dj (h)

× supd(z)≤bj d(h)

{∣∣(Z1, . . . , Zm)αf (x ∗ z) − (Z1, . . . , Zm)αf (x)∣∣ : |α| = n

}.

(20.36)

When j = n (whence |α| = 0, so that (Z1, . . . , Zm)αΦ(h) = Φ(h)) this willobviously prove (20.34).

Step zero: For the sake of brevity, we set Z = (Z1, . . . , Zm) (to distinguish itfrom the former Z = (Z1, . . . , ZN)). If j = 0, i.e. |α| = n, then ZαPn(f, x)(·) hasG-degree = 0, i.e. it is constant, so that

(ZαPn(f, x)

)(x ∗ h) = (ZαPn(f, x)

)(x) = (Zαf )(x),

whence the far left-hand side of (20.36) equals


∣∣Zαf (x ∗ h) − Zαf (x)∣∣.

In this case (20.36) is trivially true (for the left-hand side is one of the elements inthe curly brackets when z = h).

Step of induction: Suppose that (20.36) holds true for indices 0 ≤ · · · ≤ j −1 andlet us prove it for j . Let α be a multi-index with |α| = n − j . We apply the stratifiedmean-value Theorem 20.3.1 to the function f = ZαΦ and the points x = 0, h ∈ G

(so that f (x) = (ZαΦ)(0) = 0 thanks to (20.35)). Then from (20.28) we infer

|ZαΦ(h)| ≤ c1 d(h) supz: d(z)≤b d(h)

|(Z1ZαΦ(z), . . . , ZmZαΦ(z))| =: (�). (20.37)

But Z1Zα, . . . , ZmZα are linear combinations of the Zβ ’s with |β| = |α| + 1 =

n − (j − 1) (see the proof of Proposition 20.1.4 and apply, if needed, a simplehomogeneity argument). As a consequence, we can apply the inductive hypothesisto estimate the right-hand side of (20.37). We get (here c(Z) denotes a structuralconstant depending only on the algebraic structure properties of the set of vectorfields {Z1, . . . , ZN }; see the proof of Proposition 20.1.4)

(�) ≤ c1 d(h) supz: d(z)≤b d(h)

c(Z) cj−1 dj−1(z)

× supζ : d(ζ )≤bj−1 d(z)

|α|=n

{|(Z1, . . . , Zm)αf (x ∗ ζ ) − (Z1, . . . , Zm)αf (x)|}

≤ c1 c(Z) cj−1 bj−1 dj (h)

× supζ : d(ζ )≤bj d(h)

|α|=n

{|(Z1, . . . , Zm)αf (x ∗ ζ ) − (Z1, . . . , Zm)αf (x)|}.

This is precisely (20.36). If we set

cj := c1 c(Z) cj−1 bj−1,

the proof is complete. � Starting from Theorems 20.3.1 and 20.3.2 it is a standard matter to prove the

following stratified Taylor formula with remainder. Here, α is a multi-index with m

entries (not N , as in the paragraphs preceding Theorem 20.3.2).

Theorem 20.3.3 (Stratified Taylor formula). Let (G, ∗) be a homogeneous Carnotgroup. For every n ∈ N ∪ {0}, there exists a constant cn > 0 (depending only on n

and G) such that

|f (x ∗ h) − Pn(f, x)(x ∗ h)| ≤ cn dn+1(h)

× supd(z)≤bn+1 d(h)

{|(Z1, . . . , Zm)αf (x ∗ z)| : |α| = n + 1} (20.38)

for all f ∈ Cn+1(G, R) and every x, h ∈ G. Here b is as in Theorem 20.3.1.


Proof. First, apply the stratified Taylor inequality in Theorem 20.3.2, and then es-timate the right-hand side of (20.34) by means of the stratified mean value Theo-rem 20.3.1. This ends the proof. � Example 20.3.4. Consider the Heisenberg–Weyl group H

1 on R3. The relevant Jaco-

bian basis is

Z1 = ∂x1 + 2x2 ∂x3 , Z2 = ∂x2 − 2x1 ∂x3 , Z3 = ∂x3 ,

and a homogeneous norm on H1 is, for example,

d(x) = |x1| + |x2| + |x3|1/2.

Hence formula (20.38) in Theorem 20.3.3 gives the estimate∣∣u(x) − (u(0) + x1 Z1u(0) + x2 Z2u(0)

)∣∣

≤ c1(|x1|2 + |x2|2 + |x3|

)× supd(z)≤b2 d(x)

{|Z21u|(z), |Z2

2u|(z), |Z1Z2u|(z)}.

As an application, if u ∈ C2(G, R) is such that

Z21u, Z1Z2u, Z2

2u ≡ 0 in H1,

thenu(x) = u(0) + x1 Z1u(0) + x2 Z2u(0).

In particular, Z3u ≡ 0 whence also Z2Z1u = Z1Z2u + 4Z3u ≡ 0. This is quitesurprising, since we have derived an information on {Z3, Z2Z1} starting from aninformation only on {Z2

1, Z1Z2, Z22}, but

Z3, Z2Z1 /∈ span{Z21, Z1Z2, Z

22}.

See Ex. 7 for a generalization of the above comments.

20.3.1 Stratified Taylor Formula with Peano Remainder

From Theorem 20.3.3 we immediately derive the following corollary.

Corollary 20.3.5 (Stratified Taylor formula–Peano remainder). Let (G, ∗) be ahomogeneous Carnot group. For every f ∈ Cn+1(G, R), x0 ∈ G and n ∈ N ∪ {0},we have

f (x) = Pn(f, x0)(x) + Ox→x0

(dn+1(x−1

0 ∗ x)). (20.39)

There is another way to prove the stratified Taylor formula with Peano remainderin (20.39), by starting from the usual Euclidean Taylor formula.

This alternative proof, which we now describe, is much simpler than the proofgiven in Section 20.3. Nonetheless, it does not provide any estimate of the remainder(in terms of the sole Zα’s, |α|G = n + 1) as in Theorem 20.3.3. We begin with asimple lemma.


Lemma 20.3.6. Let f be a smooth function on RN such that

f (x) = Ox→0(|x|n+1),

where | · | is the Euclidean norm on RN . Then Zαf (0) = 0 for every multi-index α

with |α|G ≤ n.

Proof. It is known4 that the assertion holds in the Euclidean setting, i.e. when Zα =(Dx)

α . In the present setting, note that, by (20.15),

Zα|0 = (Dx)α|0 +

∑

β: β �=α, |β|≤|α|, |β|G=|α|Gaβ,α(0) (Dx)

β |0.

Hence, Zαf (0) is a linear combination of terms (Dx)βf (0) with |β| ≤ |α|. Note

that if |α|G ≤ n, then |α| ≤ n, for (in general) it holds |α| ≤ |α|G. Consequently,Zαf (0) is a linear combination of terms (Dx)

βf (0) with |β| ≤ n, and the proof iscomplete, thanks to the cited Euclidean version of the lemma. �

By the classical Mac Laurin formula, for a given f ∈ C∞(RN, R), we have

f (x) = Qn(f, 0)(x) + R(x), where R(x) = Ox→0(|x|n+1). (20.40)

Here Qn(f, 0)(x) =: q(x) is the (only) polynomial in x of ordinary degree n suchthat

(Dx)αq(0) = (Dx)

αf (0) for all |α| ≤ n.

Now, we decompose (in a unique way) q as q1+q2, where q1 contains only monomi-als which are δλ-homogeneous of degree ≤ n. Note that q2 contains only monomialsof δλ-degree ≥ n + 1. Consequently,

f = q1 + (q2 + R). (20.41)

Next, we take any multi-index α with |α|G ≤ n. Then

Zαf = Zαq1 + Zαq2 + ZαR,

so thatZαf (0) = Zαq1(0) + Zαq2(0) + ZαR(0). (20.42)

Now, Zαq2 is a polynomial of δλ-degree ≥ n + 1 − |α|G ≥ 1, so that Zαq2(0) = 0.Moreover, since R(x) = Ox→0(|x|n+1), by means of Lemma 20.3.6, it follows thatZαR(0) = 0, whence (20.42) yields

Zαf (0) = Zαq1(0) whenever |α|G ≤ n

i.e. by definition, q1 is the Z-Mac Laurin polynomial of f .

4 One of the simplest way to see this is to apply the Euclidean Taylor formula. Note thatwhen N = 1, a Cauchy-type theorem is needed.


Consider now the following δλ-homogeneous norm on G

d(x) = d(x(1), x(2), . . . , x(r)) = |x(1)| + |x(2)|1/2 + · · · + |x(r)|1/r

(here | · | denotes the Euclidean norm on any RNi , i = 1, . . . , r). We have

q2(x), R(x) = Ox→0(d(x)n+1). (20.43)

Indeed, the estimate of q2 follows by recalling that q2 contains only monomials ofδλ-degree ≥ n + 1. Moreover, noticing that (for every i = 1, . . . , r) it holds |x(i)| ≤|x(i)|1/i near the origin, we can use (20.40) to estimate R.

By collecting together (20.41) and (20.43), we have proved the formula

f (x) = q1(x) + Ox→0(dn+1), (20.44)

where q1 is the Z-Mac Laurin polynomial of f or, equivalently, the “projection”on Pn of the usual Mac Laurin polynomial of f , Pn being the vector space of thepolynomials of δλ-degree ≤ n.

Finally, if d is any homogeneous norm on G, then d is equivalent to d , so that(20.44) gives back (20.39). (Incidentally, we have also demonstrated the assertion inRemark 20.2.6.)

The following remark shows that there is only one polynomial of δλ-degree atmost n which plays the role of Pn(f, x0) in (20.39).

Remark 20.3.7. Suppose f ∈ Cn+1(G, R), x0 ∈ G and n ∈ N ∪ {0}. If p(x) is apolynomial of δλ-degree at most n such that

(�) f (x) = p(x) + Ox→x0

(dn+1(x−1

0 ∗ x)),

then p coincides with Pn(f, x0). Indeed, (�) and (20.39) give

q(x) := Pn(f, x0)(x) − p(x) = Ox→x0

(dn+1(x−1

0 ∗ x)).

We remark that q is a polynomial of δλ-degree at most n. Setting q(y) := q(x0 ∗ y),the above is equivalent to

q(y) = Oy→0(dn+1(y)

). (20.45)

Now, since q is a polynomial of δλ-degree at most n and the i-th component func-tion of x0 ∗ y is a polynomial function in y of δλ-degree5 at most σi (see Theo-rem 1.3.15-(3), page 39), then q(y) = q(x0 ∗ y) is a polynomial function in y ofδλ-degree at most n. Hence, (20.45) can hold if and only if q ≡ 0, i.e. q(x0 ∗ y) = 0for every y ∈ G. Recalling the definition of q, this is in turn equivalent to

p(x0 ∗ y) = Pn(f, x0)(x0 ∗ y) ∀ y ∈ G.

Obviously, this means p = Pn(f, x0).

5 Here, we are writing, as usual, the dilation δλ on G as

δλ(x) = (λσ1x1, . . . , λσN xN ).


20.3.2 Stratified Taylor Formula with Integral Remainder

In this section, we derive a stratified Taylor formula with integral remainder whichwill furnish yet another proof of the Taylor formulas in the previous sections. Theresulting Taylor polynomials will be apparently different from the Pn(f, x0)’s intro-duced in Section 20.2, but they will prove to be equivalent, thanks to the uniquenessresult in Remark 20.3.7. Thus, we shall obtain another way to represent the Z-Taylorpolynomials.

As usual, G = (RN, ∗, δλ) is a homogeneous Carnot group of step r and m

generators and Z1, . . . , ZN is the relevant Jacobian basis of g, the Lie algebra of G.We denote by Exp and Log the exponential and logarithmic maps related to G.Furthermore, we denote, as usual, the dilation δλ on G by

δλ(x) = (λσ1x1, . . . , λσN xN).

We begin by recalling a known result. Let X ∈ g be given, and suppose that γ (t)

is an arbitrary integral curve of X, i.e. γ (t) = XI (γ (t)) for every t ∈ R. Then, forevery u ∈ Cn+1(G, R), we have (see Ex. 8 in Chapter 1)

u(γ (t)) =n∑

k=0

tk

k! (Xku)(γ (0))

+ 1

n!∫ t

0(t − s)n

(Xn+1u

)(γ (s)) ds. (20.46)

Indeed, it suffices to apply the ordinary Taylor formula with integral remainder tot → u(γ (t)), by noticing that

dk

dtk

(u(γ (t))

) = (Xku)(γ (t)).

We thus immediately get the following assertion.

Lemma 20.3.8. Let x, h ∈ G. Let also u ∈ Cn+1(G, R). Then we have

u(x ∗ h) =n∑

k=0

1

k! ((Log h)ku)(x)

+ 1

n!∫ 1

0(1 − s)n((Log h)n+1u)(x ∗ Exp (s Log h)) ds. (20.47)

Proof. We know that (see Corollary 1.2.24, page 24) the integral curve γ of X :=Log h ∈ g starting at x is given by

s → γ (s) = exp(s Log h

)(x) = x ∗ Exp

(s Log h

).

Note that γ (0) = x, γ (1) = x ∗ Exp(Log h

) = x ∗ h, so that, by applying (20.46)with t = 1, we get (20.47). �


A restatement of the lemma above gives the following corollary.

Corollary 20.3.9. Let x, h ∈ G and u ∈ Cn+1(G, R). Let (X1, . . . , XN) be anybasis of g. Then we have

u(x ∗ h) = u(x) +n∑

k=1

N∑

i1,...,ik=1

ζi1(h) · · · ζik (h)

k! Xi1 · · · Xiku(x)

+N∑

i1,...,in+1=1

ζi1(h) · · · ζin+1(h)

n!

×∫ 1

0

(Xi1 · · ·Xin+1u

)(

x ∗ Exp

(N∑

i=1

s ζi(h)Xi

))(1 − s)n ds. (20.48)

Here, we have used the following notation

Log h = ζ1(h)X1 + · · · + ζN(h)XN ∀ h ∈ G, (20.49)

i.e. ζ1(h), . . . , ζN (h) are the components of Log h w.r.t. (X1, . . . , XN).

Proof. Obviously, there exist (polynomial) functions

G � h → ζi(h) ∈ R, i = 1, . . . , N,

such that (20.49) holds. For every k ∈ N, we thus obtain

(Log h)k =(

N∑

i=1

ζi(h)Xi

)k

=N∑

i1,...,ik=1

ζi1(h) · · · ζik (h)Xi1 · · ·Xik .

Now, (20.48) follows directly from (20.47). � Next, we recall that the Lie algebra of G is equipped with the stratification

g = V (1) ⊕ · · · ⊕ V (r) with [V (1), V (i)] ={

V (i+1) if i = 1, . . . , r − 1,

{0} if i = r .

We say that a basis of g is adapted to the stratification if it has the form

X = (X(1)1 , . . . , X

(1)N1

; . . . ; X(r)1 , . . . , X

(r)Nr

), (20.50)

where, for every i = 1, . . . , r ,

X(i)1 , . . . , X

(i)Ni

is a basis of V (i).

If X is a basis for g adapted to the stratification as in (20.50), then we can define ong a group of dilations, still denoted by {δλ}λ>0, such that


δλ(X(i)j ) = λi X

(i)j for every i = 1, . . . , r and every j = 1, . . . , Ni .

We then know that (argue as in Theorem 1.3.28 on page 49)

δλ

(Log (x)

) = Log(δλ(x)

), δλ

(Exp (X)

) = Exp(δλ(X)

) ∀ x ∈ G, X ∈ g.

(20.51)In the proof of Proposition 20.3.11 below, we shall need the following lemma havingan interest in its own.

Lemma 20.3.10. Let d be any homogeneous norm on G. Then there exists a constantc = c(d, G) such that, for every x ∈ G and every s ∈ [0, 1], we have d(γ (s)) ≤c d(x), where γ is the integral curve of Log (x) starting at the origin. More explicitly,

d(Exp (sLog (x))

) ≤ c d(x) for every x ∈ G and every s ∈ [0, 1]. (20.52)

Proof. We know that the integral curve γ of Log (x) starting at the origin is s →Exp (sLog (x)), hence we have to prove (20.52). It is not restrictive to suppose x �= 0,otherwise (20.52) holds for any c.

Let X be any basis for g adapted to the stratification (for example, the Jaco-bian basis), and let δλ denote both the dilation on g defined above and the usualdilation on G. Then, thanks to (20.51), the linearity of δλ and the fact that d isδλ-homogeneous of degree 1, we have

d(Exp (sLog (x)))

d(x)= d(δ1/d(x)Exp (sLog (x))) = d(Exp (δ1/d(x)(sLog (x))))

= d(Exp (sδ1/d(x)(Log (x)))) = d(Exp (sLog (δ1/d(x)(x)))).

As a consequence, being d(δ1/d(x)(x)) = 1, we have

d(Exp (sLog (x)))

d(x)≤ max

ξ∈G: d(ξ)=1

s∈[0,1]

d(Exp (sLog (ξ))) =: c.

Note that c is finite, for {ξ : d(ξ) = 1} is compact and d , Exp , Log are continuousfunctions. �

We are in a position to prove the following result.

Proposition 20.3.11 (Stratified Mac Laurin formula—integral remainder). Letn ∈ N ∪ {0}, and let u ∈ Cn+1(G, R). Suppose X = (X1, . . . , XN) is any basis of g

adapted to the stratification of g. Then, following the notation in (20.49), we have

u(x) = u(0) +n∑

h=1

∑

k=1,...,n

i1,...,ik≤N

σi1 +···+σik=h

ζi1(x) · · · ζik (x)

k! Xi1 · · ·Xiku(0) + Rn(x), (20.53)


where

Rn(x) ={

n∑

k=1

∑

i1,...,ik≤N

σi1 +···+σik≥n+1

ζi1(x) · · · ζik (x)

k! Xi1 · · · Xiku(0)

+N∑

i1,...,in+1=1

ζi1(x) · · · ζin+1(x)

n!

×∫ 1

0

(Xi1 · · ·Xin+1u

)(

Exp

(N∑

i=1

s ζi(x)Xi

))(1 − s)n ds

}. (20.54)

Moreover, for every fixed homogeneous norm d on G and every n ∈ N ∪ {0}, thereexists cn > 0 (depending on n, G, d and the basis X ) such that, for x near 0,

|Rn(x)| ≤ cn dn+1(x) × supd(z)≤c d(x)

{|Xi1 · · · XiM u(z)| :n + 1 ≤ M ≤ r(n + 1), i1, . . . , iM ∈ {1, . . . , m}}. (20.55)

Finally, when X is the Jacobian basis related to G, the polynomial function

qn(x) = u(0) +n∑

h=1

∑

k=1,...,n

i1,...,ik≤N

σi1+···+σik=h

ζi1(x) · · · ζik (x)

k! Xi1 · · ·Xiku(0) (20.56)

coincides with the Z-Mac Laurin polynomial of δλ-degree n related to u.

Proof. Let us begin by taking x = 0 and h = x in (20.48). We get

u(x) = u(0) +n∑

k=1

N∑

i1,...,ik=1

ζi1(x) · · · ζik (x)

k! Xi1 · · ·Xiku(0)

+N∑

i1,...,in+1=1

ζi1(x) · · · ζin+1(x)

n!

×∫ 1

0(Xi1 · · ·Xin+1u)

(Exp

(N∑

i=1

s ζi(x)Xi

))(1 − s)n ds. (20.57)

If Z = (Z1, . . . , ZN) is the Jacobian basis of G, then it is easily seen that (with theobvious notation), for every i = 1, . . . , r ,

Z(i)1 , . . . , Z

(i)Ni

is a basis of V (i).

Hence, there exist r non-singular square matrices M(1), . . . , M(r) such that, for i =1, . . . , r , M(i) has order Ni and such that the matrix


⎛

⎝M(1) · · · 0

.... . .

...

0 · · · M(r)

⎞

⎠

represents the transformation of the coordinates between the bases X and Z. Hence,by the corresponding results for the Jacobian basis in Theorem 1.3.28 (page 49), wecan infer that, following the notation in (20.49), ζi(x) is a polynomial function in x,δλ-homogeneous of degree σi . In particular, if d is any homogeneous norm on G,there exists c = c(d, Log ) > 0 such that

c−1 dσi (x) ≤ |ζi(x)| ≤ c dσi (x) ∀ x ∈ G, i ≤ N. (20.58)

As a consequence, for every k ∈ N∪{0}, ζi1(x) · · · ζik (x) is a polynomial functionin x, δλ-homogeneous of degree

σi1 + · · · + σik .

Observe that this is also the δλ-degree of Xi1 · · ·Xik as a differential operator. Inparticular, ζi1(x) · · · ζin+1(x) is δλ-homogeneous of degree ≥ n+1 and, in the integralsummands, there appear only higher order derivatives of u with δλ-height ≥ n + 1.

We rewrite (20.57) pointing out, in the right-hand side, the polynomial of δλ-degree ≤ n

u(x) = u(0) +n∑

h=1

∑

k=1,...,n

i1,...,ik≤N

σi1+···+σik=h

ζi1(x) · · · ζik (x)

k! Xi1 · · · Xiku(0)

+{

n∑

k=1

∑

i1,...,ik≤N

σi1+···+σik≥n+1

ζi1(x) · · · ζik (x)

k! Xi1 · · ·Xiku(0)

+N∑

i1,...,in+1=1

ζi1(x) · · · ζin+1(x)

n!

×∫ 1

0(Xi1 · · ·Xin+1u)

(Exp

(N∑

i=1

s ζi(x)Xi

))(1 − s)n ds

}

=: qn(x) + {Rn(x)}. (20.59)

As we remarked above, the function

qn(x) =n∑

h=0

q(h)n (x),


where

q(h)n (x) =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

u(0) if h = 0,∑

k=1,...,n

i1,...,ik≤N

σi1 +···+σik=h

ζi1(x) · · · ζik (x)

k! Xi1 · · ·Xiku(0) if h = 1, . . . , n,(20.60)

is a polynomial function of δλ-degree ≤ n and each q(h)n (x) is a homogeneous poly-

nomial of δλ-degree = h.We now turn our attention to Rn(x). The first sum in braces (ranging over k =

1, . . . , n) is a polynomial function of δλ-degree ≥ n + 1. The remaining summandcan be estimated as follows (for x in a neighborhood of 0)

∣∣∣∣∣

N∑

i1,...,in+1=1

ζi1(x) · · · ζin+1(x)

n!

×∫ 1

0(Xi1 · · ·Xin+1u)

(Exp

(N∑

i=1

s ζi(x)Xi

))(1 − s)n ds

∣∣∣∣∣

≤N∑

i1,...,in+1=1

|ζi1(x)| · · · |ζin+1(x)|(n + 1)!

× supd(z)≤c d(x)

{|Xi1 · · ·Xin+1u(z)| : i1, . . . , in+1 ∈ {1, . . . , N}}.

Indeed, we recall that, by definition, it holds

N∑

i=1

s ζi(x)Xi = s Log (x).

Then, we can apply Lemma 20.3.10 to derive that

Exp

(N∑

i=1

s ζi(x)Xi

)∈ Bd(0, c d(x)),

so that∣∣∣∣∣(Xi1 · · · Xin+1u)

(Exp

(N∑

i=1

s ζi(x)Xi

))∣∣∣∣∣

≤ supd(z)≤c d(x)

{|Xi1 · · ·Xin+1u(z)| : i1, . . . , in+1 ∈ {1, . . . , N}}.

As a consequence, Rn(x) can be estimated as follows (for x near 0).


|Rn(x)| ≤n∑

k=1

∑

i1,...,ik≤N

σi1 +···+σik≥n+1

|ζi1(x)| · · · |ζik (x)|k!

∣∣Xi1 · · · Xiku(0)∣∣

+N∑

i1,...,in+1=1

|ζi1(x)| · · · |ζin+1(x)|(n + 1)!

× supd(z)≤c d(x)

{|Xi1 · · ·Xin+1u(z)| : i1, . . . , in+1 ∈ {1, . . . , N}}

≤ cn dn+1(x) × supd(z)≤c d(x)

{|Xi1 · · ·Xiku(z)| :k ≤ n + 1, i1, . . . , ik ≤ N, σi1 + · · · + σik ≥ n + 1}

≤ cn dn+1(x) × supd(z)≤c d(x)

{|Xi1 · · ·XiM u(z)| :n + 1 ≤ M ≤ r(n + 1), i1, . . . , iM ∈ {1, . . . , m}}.

Here, we have used the fact that (thanks to the stratification of g) any Xj (for j ∈{1, . . . , N}) is a precise linear combination (with scalars fixed together with g andthe basis (X1, . . . , XN)) of terms like

Xi1 · · ·Xih

with 1 ≤ h ≤ r (r being the step of nilpotency of g) and i1, . . . , ih ∈ {1, . . . , m}.Since the above estimate of Rn gives

Rn(x) = O(dn+1(x)) as x → 0,

the decomposition u(x) = qn(x) + Rn(x) in (20.59) rewrites as

u(x) = qn(x) + O(dn+1(x)) as x → 0.

Thus (being qn a polynomial of δλ-degree ≤ n) via an application of what has beenproved in Remark 20.3.7, we infer that qn(x) is necessarily the Z-Mac Laurin poly-nomial of δλ-degree n related to u. This ends the proof. �

For an improvement of (20.55) for large x’s, see Exercise 9 at the end of thechapter. For the future references, we state the explicit formula for Pn(u, 0) provedabove in the following corollary.

Corollary 20.3.12 (Explicit form of the Z-Mac Laurin polynomial). Let n ∈ N ∪{0}, and let u ∈ C∞(G, R). Let Z = (Z1, . . . , ZN) be the Jacobian basis relatedto G. We set

Log (x) = ζ1(x) Z1 + · · · + ζN(x) ZN ∀ x ∈ G,

where Log is the logarithmic map, i.e. ζ1(x), . . . , ζN (x) are the components of Log h

w.r.t. Z or, equivalently,6

6 See Remark 1.2.20 on page 21.


ζj (x) = Log (x)Ij (0) ∀ j ∈ {1, . . . , N}.Then the Z-Mac Laurin polynomial of δλ-degree n related to u, i.e. the only polyno-mial p of δλ-degree ≤ n such that

(Z1 · · ·ZN)α(p)(0) = (Z1 · · · ZN)α(u)(0),

for every multi-index α with |α|G ≤ n, is given by the following formula

Pn(u, 0)(x) = u(0) +n∑

h=1

∑

k=1,...,n

i1,...,ik≤N

σi1+···+σik=h

ζi1(x) · · · ζik (x)

k! Zi1 · · ·Ziku(0).

For example, if n = 2, the above formula for P2(u, 0) rewrites as

P2(u, 0)(x) = u(0) +∑

i: σi=1

ζi(x) (Ziu)(0) +∑

i: σi=2

ζi(x) (Ziu)(0)

+∑

i,j : σi=σj =1

ζi(x) ζj (x)

2(ZiZju)(0). (20.61)

Example 20.3.13. Consider the Heisenberg–Weyl group H1 on R

3. In order to obtainthe expansion of a smooth u on H

1 by Proposition 20.3.11, we must take n = 1.Hence, if we consider the Jacobian basis

Z1 = ∂x1 + 2x2 ∂x3 , Z2 = ∂x2 − 2x1 ∂x3 , Z3 = ∂x3 ,

then we have

Exp (ξ1Z1 + ξ2Z2 + ξ3Z3) = (ξ1, ξ2, ξ3), Log (x1, x2, x3) = x1Z1 +x2Z2 +x3Z3),

so that formula (20.53) gives

u(x) = u(0) + x1 Z1u(0) + x2 Z2u(0)

+{

x3 Z3u(0) +3∑

i,j=1

xixj

∫ 1

0(XiXju)(sx) (1 − s) ds

}. (20.62)

This shows that formula (20.53), despite its very simple proof, is not as “optimal”as formula (20.38) in Theorem 20.3.3. Indeed, note that the remainder in braces in(20.62) contains also derivatives of u of order 4 (w.r.t. Z1, Z2) such as

Z3Z3u = 1

16

(Z2Z1Z2Z1 − Z2Z

21Z2 − Z1Z

22Z1 + Z1Z2Z1Z2

)u.

Instead, formula (20.38) gives the estimate∣∣u(x) − (u(0) + x1 Z1u(0) + x2 Z2u(0)

)∣∣

≤ c1(|x1|2 + |x2|2 + |x3|

)× supd(z)≤b2 d(x)

{|Z21u|(z), |Z2

2u|(z), |Z1Z2u|(z)}.


If x0 ∈ G is any given point, by replacing u and x in the above results respectivelywith u ◦ τx0 and x−1

0 ∗ x, we obtain the corresponding results for the stratified Taylorformula. We state them without further comment, recalling only that

Pn(u, x0) = Pn(u ◦ τx0, 0) ◦ τx−1

0.

Proposition 20.3.14 (Stratified Taylor formula-integral remainder). Let n ∈ N ∪{0}, u ∈ Cn+1(G, R), x0 ∈ G. Suppose X = (X1, . . . , XN) is any basis of g adaptedto the stratification of g. Then, following the notation in (20.49), we have

u(x) = u(x0) +n∑

h=1

∑

k=1,...,n

i1,...,ik≤N

σi1 +···+σik=h

ζi1(x−10 ∗ x) · · · ζik (x

−10 ∗ x)

k!

× (Xi1 · · · Xiku)(x0) + Rn(x, x0), (20.63)

where

Rn(x, x0)

={

n∑

k=1

∑

i1,...,ik≤Nσi1 +···+σik

≥n+1

ζi1(x−10 ∗ x) · · · ζik (x

−10 ∗ x)

k!

× (Xi1 · · ·Xiku)(x0) +

N∑

i1,...,in+1=1

ζi1(x−10 ∗ x) · · · ζin+1(x

−10 ∗ x)

n!

×∫ 1

0

(Xi1 · · ·Xin+1u

)(

x0 ∗ Exp

(N∑

i=1

s ζi(x−10 ∗ x)Xi

))(1 − s)n ds

}.

(20.64)

Moreover, for every fixed homogeneous norm d on G and every n ∈ N ∪ {0}, thereexists cn > 0 (depending on n, G, d and the basis X ) such that (for x near x0)

|Rn(x, x0)| ≤ cn dn+1(x−10 ∗ x) × sup

d(z)≤c d(x−10 ∗x)

{|(Xi1 · · ·XiM u)(x0 ∗ z)| :n + 1 ≤ M ≤ r(n + 1), i1, . . . , iM ∈ {1, . . . , m}}. (20.65)

Finally, when X is the Jacobian basis related to G, the polynomial function

qn(x, x0) = u(x0) +n∑

h=1

∑

k=1,...,n

i1,...,ik≤N

σi1 +···+σik=h

ζi1(x−10 ∗ x) · · · ζik (x

−10 ∗ x)

k!

× (Xi1 · · ·Xiku)(x0) (20.66)

coincides with the Z-Taylor polynomial of δλ-degree n related to u and x0.

For an improvement of (20.65) for large x’s, see Exercise 9 at the end of thechapter.


Corollary 20.3.15 (Explicit form of the Z-Taylor polynomial). Let n ∈ N ∪ {0},u ∈ C∞(G, R) and x0 ∈ G. Let Z = (Z1, . . . , ZN) be the Jacobian basis relatedto G. If the ζi’s are as in Corollary 20.3.12, then the Z-Taylor polynomial of δλ-degree n related to u and x0, i.e. the only polynomial p of δλ-degree ≤ n such that

(Z1 · · ·ZN)α(p)(x0) = (Z1 · · · ZN)α(u)(x0),

for every multi-index α with |α|G ≤ n, is given by the following formula

Pn(u, x0)(x) = u(x0) +n∑

h=1

∑

k=1,...,n

i1,...,ik≤N

σi1+···+σik=h

ζi1(x0 ∗ x) · · · ζik (x0 ∗ x)

k!

× (Zi1 · · ·Ziku)(x0).

By suitably choosing the “increment” h in Corollary 20.3.9 to belong to the “firstlayer” of G, we obtain the following “horizontal” Taylor formula.

Corollary 20.3.16 (Horizontal Taylor formula). Let x ∈ G, and let u ∈Cn+1(G, R). Let (X1, . . . , Xm) be any basis of the first layer of the stratificationof g. Then, for every ξ1, . . . , ξm ∈ R, we have

u

(x ∗ Exp

(m∑

j=1

ξjXj

))

= u(x) +n∑

k=1

m∑

i1,...,ik=1

ξi1 · · · ξik

k! Xi1 · · ·Xiku(x)

+m∑

i1,...,in+1=1

ξi1 · · · ξin+1

n!

×∫ 1

0(Xi1 · · · Xin+1u)

(x ∗ δs

(Exp

(m∑

j=1

ξj Xj

)))(1 − s)n ds. (20.67)

Proof. It suffices to apply (20.67) with

h = Exp

(m∑

j=1

ξjXj

)

and to observe that, in this case, Log (h) = ∑mj=1 ξjXj , so that (with the notation

in (20.67))

ζj (h) ={

ξj if j ∈ {1, . . . , m},0 if j ∈ {m + 1, . . . , N}.

Finally, notice that (see, e.g. (19.10d) on page 728)


Exp

(m∑

j=1

s ξj Xj

)= Exp

(δs

(m∑

j=1

ξj Xj

))= δs

(Exp

(m∑

j=1

ξj Xj

)).

This ends the proof. � For example, if we take n = 1 in the above corollary, we get the horizontal Taylor

formula with the integral remainder of order two

u

(x ∗ Exp

(m∑

j=1

ξjXj

))

= u(x) +m∑

j=1

ξj Xju(x)

+∫ 1

0

m∑

h,k=1

ξhξk(XhXku)

(x ∗ δs

(Exp

(m∑

j=1

ξj Xj

)))(1 − s) ds (20.68)

for every ξ1, . . . , ξm ∈ R. Since, for a given arbitrary m × m matrix A = (ah,k)h,k ,it holds

m∑

h,k=1

ξhξk ah,k =m∑

h,k=1

ξhξk

ah,k + ak,h

2=⟨A + AT

2ξ, ξ

⟩,

(20.68) also rewrites as

u(x ∗ Exp (ξ · X)) = u(x) + (ξ · X)u(x)

+∫ 1

0〈Hesssymu(x ∗ δs(Exp (ξ · X))) ξ, ξ 〉 (1 − s) ds,

(20.69)

where we have set

ξ · X =m∑

j=1

ξjXj

and

Hesssymu =(

XhXku + XkXhu

2

)

h,k=1,...,m

,

the so-called symmetrized horizontal Hessian of u.

As an application of Taylor’s formula, we give the following result.

Proposition 20.3.17 (L-harmonicity of Taylor polynomials). Let (G, ∗) be a ho-mogeneous Carnot group, and let Z = (Z1, . . . , ZN) be the Jacobian basis relatedto G. Let also L be a sub-Laplacian on G. Suppose that u ∈ C∞(G, R) is an L-harmonic function.

Then, for every n ∈ N ∪ {0} and every x0 ∈ G, the function x → Pn(u, x0)(x) isL-harmonic in G.


Proof. Suppose we have proved the assertion for x0 = 0. Then, by applying theassertion for u ◦ τx0 (which is L-harmonic on G since u is) we infer that y →Pn(u ◦ τx0, 0)(y) is L-harmonic on G. As a consequence, since (20.26) gives

Pn(u, x0) = Pn(u ◦ τx0 , 0) ◦ τx−1

0,

we infer that x → Pn(u, x0)(x) is L-harmonic in G.The above argument shows that it suffices to prove the assertion when x0 = 0.

By Lemma 20.3.18 below, we know that, for every n ≥ 2,

L(Pn(u, 0)) = Pn−2(Lu, 0) = Pn−2(0, 0) = 0.

Since L(Pn(u, 0)) = 0 also for n = 0, 1 (since P0 and P1 are polynomials of G-degree ≤ 1), this completes the proof. � Lemma 20.3.18. Let u ∈ C∞(G, R). Then we have

L(Pn(u, 0)) = Pn−2(Lu, 0) for every n ≥ 2. (20.70)

Proof. By Proposition 20.3.11, we have u(x) = Pn(u, 0)(x) + Rn(x), so that

(�) Lu(x) = L(Pn(u, 0))(x) + LRn(x).

If we show that, for every n ≥ 2, one has

(��) LRn(x) = Ox→0(dn−1),

then, by Remark 20.3.7, (�) will give (20.70). We are then left with the proof of (��).Denote the expression of Rn in (20.54) as follows

Rn(x) =n∑

k=1

Qk(x) +∑

I=(i1,...,in+1)

AI (x) × BI (x),

where BI is the integral in the far right-hand side of (20.54).Clearly, each Qk is a δλ-homogeneous polynomial function of degree ≥ n + 1,

so that LQk is a δλ-homogeneous polynomial function of degree ≥ n − 1, whenceLQk(x) = Ox→0(d

n−1). Dropping the subscript I in AI and BI , we have (see Ex. 6,Chapter 1)

L(AB) = B LA + ALB + 2 〈∇LA,∇LB〉.Since A(x) = ζi1(x) · · · ζin+1(x)/n!, it clearly holds

LA = Ox→0(dn−1), A = Ox→0(d

n+1(x)), |∇LA| = Ox→0(dn).

This proves that L(AB) = Ox→0(dn−1), since B, LB and ∇LB are bounded. This

completes the proof. �


Bibliographical Notes. For the topics presented in Section 20.2, we are much in-debted to the presentation of the same subject in [FS82] by G.B. Folland andE.M. Stein. For results concerning Taylor’s formula on Carnot groups of step two,see G. Arena, A. Caruso, A. Causa [ACC06]. For other results of calculus on Carnotgroups, see [CM06,Hei95a].


Ex. 1) Prove Proposition 20.1.2, page 735. (Hint: Show that the map introducedtherein is linear and sends a basis into a basis.)

Ex. 2) Retracing the proof of Proposition 20.1.5, prove in details Proposition 20.1.7.Ex. 3) Prove Lemma 20.1.11, page 741. (Hint: If

∑j cjHj ≡ 0, then 0 = (

∑j cjHj )×

(u(δλ(x))) =∑j cjλdj (Hju)(δλ(x)). Now take as u any function of the type

xα and complete the argument.)Ex. 4) With reference to Example 20.2.5, let us now find r = P3(f, 0). Since r ∈

P3, there exist real numbers, a, b, . . . , m such that r(x, y, t) = a + b x +c y + d t + e x2 + f y2 + g xy + h xt + i yt + j x3 + k y3 + l x2y + m xy2.Then (20.22) gives

a = r(0) = f (0),

b = Z1r(0) = Z1f (0),

c = Z2r(0) = Z2f (0),

d = Z3r(0) = Z3f (0),

2e = (Z1)2r(0) = (Z1)

2f (0),

2f = (Z2)2r(0) = (Z2)

2f (0),

g − 2d = Z1Z2r(0) = Z1Z2f (0),

h = Z1Z3r(0) = Z1Z3f (0),

i = Z2Z3r(0) = Z2Z3f (0),

3!j = (Z1)3r(0) = (Z1)

3f (0),

3!k = (Z2)3r(0) = (Z2)

3f (0),

2l − 4h = (Z1)2Z2r(0) = (Z1)

2Z2f (0),

2m − 4i = Z1(Z2)2r(0) = Z1(Z2)

2f (0),

so that

P3(f, 0)(x, y, t) = f (0) + Z1f (0) x + Z2f (0) y + Z3f (0) t

+ 1

2(Z1)

2f (0) x2 + 1

2(Z2)

2f (0) y2 + (2Z3f (0)


+ Z1Z2f (0)) xy + Z1Z3f (0) xt + Z2Z3f (0) yt

+ 1

3! (Z1)3f (0) x3 + 1

3! (Z2)3 y3 + (2Z1Z3f (0)

+ 1

2Z2

1Z2f (0)) x2y + (2Z2Z3f (0) + 1

2Z1Z

22f (0)) xy2.

Ex. 5) Consider the homogeneous Carnot group G on R3 with the composition law

x ◦ y = (x1 + y1, x2 + y2, x3 + y3 + x1 y2).

Prove that the related Jacobian basis is given by

Z1 = ∂1, Z2 = ∂2 + x1 ∂3, Z3 = [Z1, Z2] = ∂3.

Show that

P2(f, 0)(x1, x2, x3) = f (0) + Z1f (0) x1 + Z2f (0) x2 + Z3f (0) x3

+ 1

2(Z1)

2f (0) x21 + 1

2(Z2)

2f (0) x22

+ (Z2Z1)f (0)x1 x2 (20.71)

is the Z-Mac Laurin polynomial of G-degree 2.Deduce that formula (20.24) does not apply in this case.(Indeed, this would hold true iff 1

2 (Z1Z2 +Z2Z1) = Z2Z1, i.e. [Z1, Z2] = 0which is false.) This shows that (20.24) strongly depends on the propertiesof the fixed coordinate system on G or, equivalently, on the properties of thecomposition law.Notice that the inverse on G differs from −x (and the coordinate system onG is not the logarithmic one). See also Ex. 6 below.

Finally, compare to (20.61). Notice that here we have

Exp (ξ1Z1 + ξ2Z2 + ξ3Z3) = (ξ1, ξ2, ξ3 + ξ1 ξ2/2),

so that

Log (x1, x2, x3) = x1Z1 + x2Z2 + (x3 − x1x2/2)Z3.

With the notation of Corollary 20.3.12, this means that

ζ1(x) = x1, ζ2(x) = x2, ζ3(x) = x3 − x1x2/2,

whence (20.61) gives

P2(f, 0)(x1, x2, x3)

= f (0) + Z1f (0) x1 + Z2f (0) x2

+ Z3f (0) (x3 − x1x2/2) + 1

2(Z1)

2f (0) x21 + 1

2(Z2)

2f (0) x22


+ 1

2(Z1Z2)f (0)x1 x2 + 1

2(Z2Z1)f (0)x2 x1

= f (0) + Z1f (0) x1 + Z2f (0) x2 + Z3f (0) x3

+ 1

2(Z1)

2f (0) x21 + 1

2(Z2)

2f (0) x22

+ 1

2(Z1Z2 + Z2Z1 − Z3)f (0)x1 x2,

which is (20.71), for

Z1Z2 + Z2Z1 − Z3 = Z1Z2 + Z2Z1 − [Z1, Z2] = 2Z2Z1.

Ex. 6) (Symmetrized Z-Mac Laurin formula of δλ-degree two). Suppose that G

is a homogeneous Carnot group (of step r ≥ 2) for which the inversion equals−x. Then7 the Z-Mac Laurin polynomial of δλ-degree two is given by

P2(f, 0)(x(1), x(2), x(3), . . . , x(r)) = f (0) + ∇(1)f (0) · x(1)

+ ∇(2)f (0) · x(2) + 1

2

⟨x(1), Hesssymf (0) · x(1)

⟩. (20.72)

Here

Hesssymf (0) :=(

1

2{Z(1)

h Z(1)k f (0) + Z

(1)k Z

(1)h f (0)}

)

h,k=1,...,N1

is the so-called symmetrized horizontal Hessian of f at 0.For example, a sufficient condition for x−1 = −x to hold (and hence, for(20.72) to hold) is that G is equipped with logarithmic coordinates. Note that(20.72) holds for the Heisenberg–Weyl group H

n since the inversion on Hn

is −x (even if Hn is not generally equipped with logarithmic coordinates).

Compare also to the previous Ex. to show that (20.72) does not hold in gen-eral unless suitable hypotheses on the coordinates are made. Hint: In orderto prove (20.72), by Definition 20.2.3, it suffices to show that the polynomial(say q) in the right-hand side of (20.72) has δλ-degree two (obvious) and thefollowing facts hold:

(i) q(0) = f (0),

(ii) Z(1)h q(0) = Z

(1)h f (0) for h = 1, . . . , N1,

(iii) Z(2)h q(0) = Z

(2)i f (0) for h = 1, . . . , N2,

(iv) (Z(1)h Z

(1)k )q(0) = (Z

(1)h Z

(1)k )f (0) for 1 ≤ h ≤ k ≤ N1.

7 Here we used the following notation. The usual notation (1.79a) (page 56) for the “strati-fied” coordinates on a homogeneous group; moreover, for i = 1, 2,

∇(i)f (0) =(Z

(i)1 f (0), . . . , Z

(i)Ni

f (0)),

i.e. Z(i)j

is the left-invariant vector field coinciding with ∂x

(i)j

at 0 or, equivalently,

Z(i)1 , . . . , Z

(i)Ni

are the vector fields of the Jacobian basis which are δλ-homogeneous of

degree i. Notice that ∇(1)f (0) is the horizontal gradient of f at 0.


Now, (i) is trivial, whereas (ii) and (iii) are simple consequences of the factthat Z

(i)h |0 = (∂/∂ x

(i)h )|0. In order to show (iv), recall (20.10), so that

Z(1)h = ∂

x(1)h

+∑

j : σj >1

aj,h(x) ∂xj, h = 1, . . . , N1,

where

(aj,h

)j,h=1,...,N

=

⎛

⎜⎜⎜⎜⎝

1 0 · · · 0

a2,1(x) 1. . .

......

. . .. . . 0

aN,1(x) · · · aN,N−1(x) 1

⎞

⎟⎟⎟⎟⎠= Jτx (0)

is the Jacobian matrix of the left translation by x on G (i.e. τx(y) = x ∗ y)and the aj,h’s are polynomial functions, δλ-homogeneous of degree σj − σh.

As a consequence, Z(1)h Z

(1)k coincides with

∂2

∂ x(1)h x

(1)k

+∑

j : σj =2

∂ aj,k

∂ x(1)h

(0) ∂xj= ∂2

∂ x(1)h x

(1)k

+N2∑

j=1

∂ a(2)j,k

∂ x(1)h

(0) ∂x

(2)j

at 0. Here, we used a “stratified” notation for Jτx (0), namely

Jτx (0) =

⎛

⎜⎜⎜⎜⎝

IN1 0 · · · 0

A(2)1 (x) IN2

. . ....

.... . .

. . . 0A

(r)1 (x) · · · A

(r)r−1(x) INr

⎞

⎟⎟⎟⎟⎠.

Notice that(a

(2)j,k(x))1≤j≤N2,1≤k≤N1 = A

(2)1 (x).

This shows that (iv) is equivalent to

1

2{Z(1)

h Z(1)k f (0) + Z

(1)k Z

(1)h f (0)} +

N2∑

j=1

∂ a(2)j,k

∂ x(1)h

(0) Z(2)j f (0)

?= ∂2f (0)

∂ x(1)h x

(1)k

+N2∑

j=1

∂ a(2)j,k

∂ x(1)h

(0) ∂x

(2)j

f (0).

Arguing as above, this is in turn equivalent to

∂2f (0)

∂ x(1)h x

(1)k

+N2∑

j=1

∂ a(2)j,k

∂ x(1)h

(0) ∂x

(2)j

f (0)


+ 1

2

N2∑

j=1

{∂ a

(2)j,k

∂ x(1)h

(0) + ∂ a(2)j,h

∂ x(1)k

(0)

}∂x

(2)j

f (0)

?= ∂2f (0)

∂ x(1)h x

(1)k

+N2∑

j=1

∂ a(2)j,k

∂ x(1)h

(0) ∂x

(2)j

f (0).

Thus, we are left to prove that, for all j ≤ N2 and all h, k ≤ N1, it holds

∂ a(2)j,k

∂ x(1)h

(0) + ∂ a(2)j,h

∂ x(1)k

(0)?= 0.

Now, the matrix(a

(2)j,k

) = A(2)1 involves the (derivatives of the) components

of the left translation along the second layer. We know that the left translationis given by

τx(y) = x ∗ y = (x(1) + y(1), x(2) + y(2) + Q(2)(x, y), . . .)

where Q(2)(x, y) is a polynomial in x(1), y(1) of (ordinary) degree 2, mixedin x, y. Hence, for every j ∈ {1, . . . , N2}, there exists a square matrix Bj oforder N1 such that

(τx(y))(2)j = x

(2)j + y

(2)j + 〈Bj · x(1), y(1)〉.

We now use the hypothesis x−1 = −x to derive that Bj is skew-symmetric.Consequently, this gives

a(2)j,k(x) = ∂

∂ y(1)k

∣∣∣∣0

(τx(y)

)(2)

j=

N1∑

i=1

x(1)i B

ji,k.

Hence,

∂ a(2)j,k

∂ x(1)h

(0) + ∂ a(2)j,h

∂ x(1)k

(0) = Bjh,k + B

jk,h = 0,

for Bj is skew-symmetric. The assertion is proved. � Ex. 7) Suppose f, g ∈ Cn+1(G, R) are such that

(Z1, . . . , Zm)αf ≡ (Z1, . . . , Zm)αg ∀ α ∈ (N ∪ {0})m : |α| = n + 1.

Show that there exists a polynomial function p, δλ-homogeneous of degreeat most n, such that f = g + p on G. (Hint: Apply Theorem 20.3.3 to f − g

and derive that f (h)−g(h) = Pn(f −g, 0)(h) for every h ∈ G.) Derive that

(Zi1 , . . . , Zim)αf ≡ (Zi1, . . . , Zim)αg

for every α ∈ (N ∪ {0})m such that |α| = n + 1 and for every i1, . . . , im ∈{1, . . . , m}.This proves that the (Z1, . . . , Zm)αf ’s with |α| = n + 1 determine all the(Zi1 , . . . , Zim)αf ’s with |α| = n + 1.


Ex. 8) Provide a detailed proof of Theorem 20.3.3.Ex. 9) Prove the following improvement of (20.55) and (20.65). For every x ∈ G,

we have

|Rn(x, x0)| ≤n+1∑

k=1

Ck

k!×

∑

i1,...,ik≤N,σi1 +···+σik

≥n+1

d(x−10 ◦ x)σi1 +···+σik

× supd(z)≤C d(x−1

0 ◦x)

|Zi1 · · ·Ziku(x0 ◦ z)|,

where C is a constant depending only on G, d and the basis X . Here, the σi’sare the same as in (20.2).

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Index of the Basic Notation1

∂j , ∂xj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

X = ∑Nj=1 aj ∂j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Xf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4T (RN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5X ≡ XI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5γX(t, x), D(X, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6X(k), Xh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6exp(tX)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8exp(X)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8[X, Y ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 103[Zj1, · · · [Zjk−1 , Zjk

] · · ·] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11ZJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Lie{U} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11rank(Lie{U}(x)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11[V,W ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13G := (RN, ◦) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13τα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 107Jτα (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14η �→ J (η) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16H

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14{e1, . . . , eN } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19π : g → R

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21(Z1u, . . . , ZNu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22X � Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29δλ : R

N → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1 The page number refers to the first time the symbol is introduced within the text. In case ofmultiple numbering, this means that the symbol is employed in multiple contexts.

790 Index of the Basic Notation

σ = (σ1, . . . , σN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31G = (RN, ◦, δλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31, 56|α|σ , |α|G, degG (p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33X∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35div(A · ∇T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Jτx (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42|E| (E ⊆ R

N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Q = ∑N

j=1 σj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 126g = g1 ⊕ · · · ⊕ gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45δλ : g → g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48ξ · Z, E · ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50, 133C-H(G) := (RN, ∗, δλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54x(1), . . . , x(r), x(i) ∈ R

Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56r and m = N1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 126W(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59g = W(1) ⊕ · · · ⊕ W(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59ΔG, L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62, 144∇G, ∇L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62L = div(A(x)∇T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A(x) = σ(x) σ (x)T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64qL(x, ξ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Isotr(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65u �→ J (u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67Xβ := Xi1 ◦ · · · ◦ Xik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68(X1, X2)-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70πi : R

N −→ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87ϕ : U → R

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87(Uα, ϕα) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88v(f ), Mm, T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91M , N , M ′, N ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88∂/∂ xi |m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94dmψ : Mm → M ′

ψ(m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95dψ : T (M) → T (M ′) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97X : Ω −→ T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97π(m, v) := v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97X, Y , Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Xm, X(f )(m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98f �→ Xf , Xf : M → R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98X (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98dψ : X (M) → X (M ′) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99X and X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100μ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101[X, Y ] : M → T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103G, H, F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106[·, ·] : g × g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

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g, h, f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 107◦, ∗, • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106τα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106x−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106α : g → Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108ϕ : G → H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112dϕ : g → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113expX(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Exp : g → G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24, 49, 118Log : G → g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 49, 120h = V1 ⊕ · · · ⊕ Vr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122B = (E

(1)1 , . . . , E

(r)Nr

) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128ad X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128πE : h → R

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131, 139Ψ := Exp ◦ (πE )−1 : R

N → H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131, 139Δλ : h → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132δλ := πE ◦ Δλ ◦ π−1

E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132�E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132H

∗ := (RN,�E , δλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Exp h : h∗ → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143HL(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Im(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155H

n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156ΔHn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158〈Bx, ξ 〉, B(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158(Fm,2, �) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163γi,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163H = (Rm+n, ◦, δλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169, 681ρ : N → N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173HM-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Bη, g∗

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173E = (RN,+, δλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183(B, ◦) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185B = (R1+N, ◦, δλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198h1, . . . , hk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207d , |x|G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229d0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230dX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232S(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232l(γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232


X-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Dirac0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236uε, u ∗G Jε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Jε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239B�(x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Bd(x, r), Bd(x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252, 459D(x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297�-dist(x,A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239ω� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248ωd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248KL, ΨL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252Mr , Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Mr , Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .259Φr , Φ∗

r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258βd , md , nd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255, 259ad , ad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257, 261−∫D

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275Iα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276ML(f )(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277F ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297ν⊥F at y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297AL(u)(x), AL(u)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320lim infy→x , lim supy→x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338l.s.c., u.s.c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338u, u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339(E, T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338F : V �→ F(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340HV

ϕ , HVf , HΩ

f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 361, 388

μVx , μΩ

x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 367, 388H∗(Ω), H∗(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341F ↑, F ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345(E,H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345H(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346, 388B-H∗(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348K ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349B(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353S(Ω), S(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353, 389uV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355(H-D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359UΩ

f , UΩf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

HΩ

f , HΩf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .359

R(∂Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

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R∞(∂Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361S +

c (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363sx0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370sΩx0

:= HΩf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

u0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Rf

A, RfA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

LH(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382Lε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383(G,H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388X 2u(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412V := Exp (V1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414[x ◦ h−1, x ◦ h] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .415Vx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415V -Convv(Ω), V -ConvH(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420GΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425, 427Γ ∗ μ, GΩ ∗ μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432k, K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436mr [u], Mr [u] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436λx,r (y), Λx,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436μ, μ[u], μu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441Sb(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

S +(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

I (μ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497V (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497UE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498Ru

E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Φu

E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Ru

E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501WK , VK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502μK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502C(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502, 508C∗(E), C∗(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508GΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516gΩ(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .516M, M0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489M(E), M0(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489A � B, μ|A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489νE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519〈μ, ν〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528‖μ‖ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .528E+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547M

(ρ)φ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

mφ(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558


M(ρ)

(α) , m(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558α(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558n(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562I (φ,Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564fm,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577H(m, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579eX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622F = {F1, . . . , FH } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .650z, z⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681Jz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682U(1), . . . , U(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687v : G → b, z : G → z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696σ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705u∗(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705Pk[x1, . . . , xq ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718⊕r

k=s Pk[x1, . . . , xq ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718jq , c(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721Pn = span{xα : α ∈ In} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735Zα := Z

α11 · · · ZαN

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737Pn(f, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742πn : P → Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744Pn(f, x0)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745ζ1(h), . . . , ζN (h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755Hesssymu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764

Index

B-hyperharmonic function, 348–350, 352minimum principle, 351

B-invariant set, 349–351δλ-degree (of a polynomial), 33δλ-homogeneous degree (of a polynomial),

33G-cone, 480G-degree (of a polynomial), 33G-polynomial, 734S-harmonic space, 363, 367

positivity axiom (in a), 363S∗-harmonic space, 370

on a Carnot group, 382L-maximal function, 277q-set, 477

associative algebra, 600asymptotic solid sub-Laplacian, 261asymptotic surface sub-Laplacian, 257

balayage, 376balayage (w.r.t. a sub-Laplacian), 501barrier function (in a S∗-harmonic space),

371Bôcher-type theorem, 460, 461Bouligand theorem, 371, 391, 544bounded above L-subharmonic functions,

474Brelot convergence property, 268

Caccioppoli–Weyl’s lemma for sub-Laplacians, 408

Campbell–Hausdorffformula, 128, 129, 194, 585–587, 600,

659

formula for formal power series, 600

formula for homogeneous vector fields,194, 593, 594

operation, 29, 128, 130, 131, 142, 143,585, 586, 595, 626, 652

canonical sub-Laplacian, 621, 622

of a Heisenberg-type group, 171, 692

capacitable set (w.r.t. a sub-Laplacian), 509

capacitary distribution (w.r.t. a sub-Laplacian), 502, 505

capacitary potential (w.r.t. a sub-Laplacian),502, 508

capacity of a compact set (w.r.t. asub-Laplacian), 502

and polarity, 507

characterization of the-, 503, 505

monotonicity, 506

right continuity, 506

strong subadditivity, 506

capacity (w.r.t. a sub-Laplacian), 509

and polarity, 509

exterior-, 509

interior-, 508

of Borel sets, 509

Carnot group, 56, 122, 131, 198

B-groups, 184

canonical sub-Laplacian of a-, 62

classical definition, 122

Euclidean group, 183, 280

filiform-, 207

796 Index

free-, 578, 584, 587, 588, 625, 658, 661,662

harmonic space, 381Heisenberg–Weyl group, 155HM-groups, 174homogeneous-, 121K-type groups, 186of Heisenberg type, 169, 681of Iwasawa type, 702of step r , 56, 198of step two, 158, 163, 661, 666of type HM, 174stratified change of basis on a-, 61, 165sum of-, 190with homogeneous dimension Q ≤ 3, 184

central series (lower-), 149, 207characteristic set (of a smooth open set), 710Choquet lemma, 339commutator, 103

nested-, 11of homogeneous fields, 37of length k, 11

continuity principle for potentials, 489convergence axiom, 345, 347

in a Carnot group, 382convex function

equivalence between H- and v-, 417H-, 416horizontally-, 416v-, 411

Cornea theorem, 344, 347covering lemma, 566

decomposition theorem, 303derivatives on G, 740differentiable manifold, 88differential, 95

of a homomorphism, 113, 114, 121of a smooth map, 95of the exponential map, 119

dilation, 31, 48, 56, 121, 132, 191, 593, 627,638, 649, 656, 669

-invariance of a sub-Laplacian, 63differential operator homogeneous with

respect to-, 32, 34function homogeneous with respect to-,

32, 34invariance of the Lebesgue measure with

respect to-, 44on the Lie algebra g, 46

Dini–Cartan theorem, 343Dirichlet problem (in a harmonic space),

359, 361, 371doubling measure, 277down directed (family of functions), 342,

343, 349, 356, 357down directed (family of sets), 349

eikonal equation, 466energy

w.r.t. a sub-Laplacian, 528equilibrium distribution (w.r.t. a sub-

Laplacian), 497, 505equilibrium potential (w.r.t. a sub-Laplacian),

497as barrier function, 514fundamental theorem on-, 498uniqueness of the-, 508

equilibrium value (w.r.t. a sub-Laplacian),497

characterization of the-, 505exponential

function generated by a system of vectorfields, 194

map, 24, 49, 118, 129, 131, 626inverse function of-, 27, 49, 626of a homogeneous group of step 2, 166

of a vector field, 8, 23, 117extended L-Green function, 516, 518extended maximum principle for L-

subharmonic functions, 493extended Poisson–Jensen’s formula, 518exterior L-capacity, 509

filiform Carnot group, 207, 209, 285fine topology (w.r.t. a sub-Laplacian), 537fractional integral (in a Carnot group), 277free Carnot group, 578free homogeneous Carnot group, 584,

586–588, 625, 629, 656, 658, 659,661, 662

free nilpotent Lie algebra, 577, 579, 583,586, 624, 650, 656

Hall basis for the-, 579fundamental solution, 236, 425, 456, 477,

649, 661–663, 665H-type group, 696

fundamental theorem on L-equilibriumpotentials, 498

Index 797

gauge function (w.r.t. a sub-Laplacian), 247generalized solution (in the sense of

Perron–Wiener–Brelot), 359, 361, 390generators of a homogeneous Carnot group,

56geodesics (for a homogeneous Carnot

group), 309, 314gradient (canonical G-), 62Grayson–Grossman theorem, 582Green function

extended-, 516, 518of a general domain, 427

approximation of the-, 429symmetry of the-, 431

of an L-regular set, 425, 426, 445, 446,448

potential of a measure related to a-, 432

H-groups (in the sense of Métivier), 174H-inversion map, 705H-Kelvin transform, 705H-type algebra, 681H-type group, 681

canonical sub-Laplacian, 692fundamental solution, 696H-inversion map, 705H-Kelvin transform, 705Iwasawa group, 702prototype-, 169, 687

Hall basis, 579, 581, 583, 586, 587Hardy–Littlewood–Sobolev theorem, 277harmonic function (w.r.t. a harmonic sheaf),

340, 346, 347, 353, 357harmonic function (w.r.t. a sub-Laplacian),

146, 381, 388–390, 408, 433, 441–445,458–461

Brelot convergence property for a-, 268decomposition theorem for a-, 303removable (or isolated) singularity, 458

harmonic majorant (least-), 358harmonic measure (w.r.t. a harmonic sheaf),

341, 367harmonic measure w.r.t. a sub-Laplacian,

388, 391, 426, 445, 518of a polar set, 515

harmonic minorant (greatest-), 358, 427harmonic sheaf, 340harmonic space, 345, 347, 348, 353

in a Carnot group, 381

Harnack inequality, 265–267on rings, 267, 460

Hausdorff dimension w.r.t. d, 558, 568Hausdorff measure w.r.t. d, 557height

δλ-height of a multi-index, 32G-height of a multi-index, 33

Heisenberg grouppolarized, 180

Heisenberg–Weyl group, 155, 156, 172, 580,625, 630, 632, 659, 702

Heisenberg-type algebra, 681Heisenberg-type group, 681

canonical sub-Laplacian, 692fundamental solution, 696H-inversion map, 705H-Kelvin transform, 705Iwasawa group, 702prototype-, 687

Hessian (horizontal-), 412higher-order derivatives on G, 738HM-groups, 174homogeneous Carnot group, 56, 121, 131,

198, 206, 586, 630, 637, 649, 656, 659generators of a-, 56

homogeneous dimension, 44, 126, 128, 184,303, 477, 559, 663

homogeneous Lie group on G

homogeneous dimension of a-, 477homogeneous Lie group on R

N , 31, 196homogeneous dimension of a-, 44of step two, 158, 163

homogeneous norm, 229pseudo-triangle inequality for a-, 231, 484

homomorphism, 114, 130of Lie algebras, 112of Lie groups, 112, 121

Hopf-type lemma, 297horizontal Hessian, 412horizontal segment, 415horizontal Taylor formula, 763horizontally convex function, 416

equivalence with v-convex function, 417Hörmander condition, 12, 69, 185, 193, 202,

210, 281hyperharmonic function (w.r.t. a harmonic

sheaf), 341, 348, 352, 353, 355, 356

798 Index

hypoelliptic-(ity), 188, 193, 280, 434, 441,633

analytic, 280hypoharmonic function (w.r.t. a harmonic

sheaf), 342

interior L-capacity, 508isolated singularity, 458–461Iwasawa-type group, 702

Jacobi identity, 11, 103, 107Jacobian basis, 19–22, 26, 45, 50, 58, 59,

69, 115, 143, 144, 156, 159, 185, 187,588, 622, 625, 627–629, 638, 639,643, 649, 653, 656, 658–662, 666

of a homogeneous Lie group, 42, 43

Kelvin transformation, 704

l.s.c. function, 338Lagrange mean-value theorem (on a Carnot

group), 746Laplace operator, 100, 183, 445, 621, 622left-translation, 106length

δλ-length of a multi-index, 32G-length of a multi-index, 33

Levi–Cartan theorem, 343, 347, 348Lie algebra, 11, 14, 107, 197

filiform-, 207free nilpotent-, 577generated by a set, 11Jacobian basis, 19of a Carnot group, 59of a Lie group, 108

Lie group, 106, 195Carnot group, 56composition law of a homogeneous-, 39,

41, 50, 58homogeneous on R

N , 31on R

N , 13structure on a Lie algebra, 130

Lie polynomial, 600Lifting, 653–656, 659, 661, 665, 666Liouville-type theorems, 269, 270, 274, 461,

633asymptotic-, 274–276

lower functions, 359lower regularization, 339, 501

lower solution, 359Lusin-type theorem, 495

Mac Laurin polynomial (on a Carnot group),742

Maria–Frostman domination principle, 495,499, 503

maximum principle, 474-set, 474for L-subharmonic functions, 409

extended-, 493on unbounded open sets, 474strong version, 426weak version, 388, 389, 474

mean value formula, 391, 447mean value operator, 397, 456

solid-, 399, 404, 405, 432, 441, 447, 458superposition formula, 457, 458surface-, 401, 404, 410, 447, 456

measure function, 557minimum principle for B-hyperharmonic

functions, 351, 360mollifier, 239, 240, 401, 455MP set, 474mutual L-energy, 528

non-characteristic exterior ball condition,384, 385, 387

peaking function at a point, 542Perron family, 356–358, 362Perron–Wiener–Brelot operator, 359Perron–Wiener–Brelot solution (related to a

sub-Laplacian), 390Perron-regularization, 355Poisson–Jensen’s formula, 445, 448

extended-, 518polar set (w.r.t. a sub-Laplacian), 491, 493,

495, 508, 544, 559, 568and harmonic measure, 515characterization in terms of capacity, 507,

509polar∗ set (w.r.t. a sub-Laplacian), 498, 499,

508the L-irregular points are a-, 514

polarized Heisenberg group, 180polynomial functions on G, 734positivity axiom, 345, 351, 354

in a S-harmonic space, 363in a Carnot group, 382

Index 799

potential of a measure (related to thefundamental solution Γ ), 445, 451,458

continuity principle (for the), 489potential of a measure (related to the

L-Green function GΩ ), 432, 433, 441,443, 444

prototype H-type group, 169, 687pseudo-triangle inequality, 231, 400, 484

improved-, 306PWB function, 361, 428

quasi-continuity of L-superharmonicfunctions, 528

reduced function, 376reduced function (w.r.t. a sub-Laplacian),

501regular point, 371regular point w.r.t. a sub-Laplacian, 518,

542, 544and L-polarity∗, 514

regular set (w.r.t. a harmonic sheaf), 341,345, 346, 348, 353, 355

regular set w.r.t. a sub-Laplacian, 385, 387,388, 391

approximation by-, 430Green function of a-, 425

regularity axiom, 345, 346in a Carnot group, 383

removable singularity, 458–461resolutive function, 361

characterization of the-, 367Riesz measure (of an L-subharmonic

function), 441–443, 445, 447, 451,502, 518

Riesz-type representation theorem, 441,443–445, 451, 458

Riesz-type representation theorem, 484

segment (horizontal-), 415separation axiom, 345, 352

in a Carnot group, 382sheaf (of functions), 340, 353

harmonic-, 340Sobolev–Stein embedding theorem, 279Stone–Weierstrass theorem, 366stratification, 45, 122, 131, 583

of a Carnot group, 60, 309, 314

stratified change of basis, 61, 165stratified group, 122, 131

harmonic function, 146sub-Laplacian of a-, 144

stratified Lagrange mean-value theorem, 746stratified Taylor formula, 750stratified Taylor inequality, 749strong maximum principle, 296sub-Laplacian, 62, 66, 198, 623, 625, 637,

641, 650arising in control theory, 205Caccioppoli–Weyl’s lemma (for-), 408canonical-, 62, 198, 625, 637, 641, 650,

663on a Heisenberg-type group, 692

characteristic form of a-, 65degenerate-ellipticity of a-, 66harmonic measure related to a-, 388harmonic space related to a-, 381invariance with respect to the dilation, 63of a stratified group, 144of Bony-type, 202, 223, 285of Kolmogorov-type, 204Perron–Wiener–Brelot solution related to

a-, 390q-set w.r.t. a-, 477regular set w.r.t a-, 388Riesz measure of a function subharmonic

w.r.t. a-, 441subharmonic function w.r.t. a-, 389superharmonic function w.r.t. a-, 389thin set w.r.t. a-, 474

sub-mean function, 397–399, 401sub-mean properties, 399subharmonic function (w.r.t. a harmonic

sheaf), 353criterion (for subharmonicity), 354

subharmonic function (w.r.t. a sub-Laplacian), 389, 401, 402, 404, 405,411, 441, 442, 445, 447, 451, 456,494, 516, 518

bounded above in G, 451bounded above-, 474

extended maximum principle for-, 493Riesz-type representation theorem for a-,

441, 443subharmonic smoothing for a-, 456

subharmonic minorant, 356, 358

800 Index

superharmonic function (w.r.t. a harmonicsheaf), 353

characterization of the-, 353superharmonic function (w.r.t. a sub-

Laplacian), 389, 410, 432, 457, 458,491

superharmonic majorant, 358surface mean value theorem, 391, 404symmetrized horizontal Hessian, 764

tangent bundle, 91tangent space, 91, 109tangent vector, 91Taylor formula (on a Carnot group), 750Taylor inequality (on a Carnot group), 749Taylor polynomial (on a Carnot group), 745thin set w.r.t. a sub-Laplacian, 474thinness of a set at a point (w.r.t. a

sub-Laplacian), 538, 550, 553total gradient, 22total set in C+

0 (G), 529

u.s.c. function, 338up directed (family of functions), 342–344,

347, 348, 354

upper functions, 359upper regularization, 339upper solution, 359

v-convex function, 411equivalence with H-convex function, 417

vector field, 4, 97complete-, 102, 116generating a Carnot group, 191integral curve (of a), 6, 101left-invariant, 14, 17, 107, 197

completeness of the-, 116Lie-bracket, 10related-, 99, 105, 106, 114smooth-, 98

weak maximum principle, 295Wiener resolutivity theorem, 364, 390Wiener’s criterion for sub-Laplacians, 547,

550Wiener’s regularity test for sub-Laplacians,

553

Zorn’s lemma, 350

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