nonlinear hyperbolic equations, spectral theory, and wavelet transformations: a volume of advances...

Operator Theory: Advances and Applications Vol. 145

Editor: I. Gohberg

Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel

Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Bottcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) G. Heinig (Chemnitz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam)

Subseries Advances in Partial Differential Equations

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Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations A Volume of Advances in Partial Differential Equations

Sergio Albeverio Michael Demuth Elmar Schrohe Bert-Wolfgang Schulze Editors

Springer Basel AG

Editors:

Sergio Albeverio Institut für Angewandte Mathematik

Universität Bonn

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Michael Demuth Institut für Mathematik

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Bert-Wolfgang Schulze

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2000 Mathematics Subject Classification 35Lxx, 35P15, 42C40, 47 A53

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Contents

Preface ................................................................... Vll

P.R. Popivanov Nonlinear PDE. Singularities, Propagation, Applications 1

Fumihiko Hirosawa and Michael Reissig From Wave to Klein-Gordon Type Decay Rates ..................... 95

Michael Dreher Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations ............................................... 157

F. Baldus An Approach to a Version of the S(M, g)-pseudo-differential Calculus on Manifolds ............................................... 207

F. Baldus Spectral Invariance and Submultiplicativity for the Algebras of S(M,g)-pseudo-differential Operators on Manifolds ................ 249

Andre Noll Domain Perturbations and Capacity in General Hilbert Spaces and Applications to Spectral Theory ..................................... 281

Bruno Nazaret and Matthias Holschneider An Interpolation Family between Gabor and Wavelet Transformations. Application to Differential Calculus and Construction of Anisotropic Banach Spaces

Xiaonan Ma

363

Formes de torsion analytique et fibrations singulieres ................. 395

Grigori Rozenblum Regularisation of Secondary Characteristic Classes and Unusual Index Formulas for Operator-valued Symbols.. .. .. .. .. . .. ... 419

Operator Theory: Advances and Applications, Vol. 145, 1-94 © 2003 Birkhiiuser Verlag Basel/Switzerland

Nonlinear PDE. Singularities, Propagation, Applications

P.R. Popivanov

To my wife Angela

Preface

This text deals with the singularities of the solutions of several classes of nonlinear partial differential equations and systems. Applications of the results here obtained are given for the Monge-Ampere equation, for quasi-linear systems arising in fluid mechanics, and for some nonlinear integrodifferential equations useful in solid body mechanics in media with memory. In our investigations we have used two different approaches - the classical method of characteristics in the case of systems with one space variable and the machinery of paradifferential operators in the multidimensional case.

Our text comprises three chapters divided in 10 sections. Chapter I consists of §1, §2, §3, §3 being Appendix L Chapter II consists of §4, §5, §6, §7, §8, §8 being Appendix II. Chapter III is divided into §9 and §10.

Section 1 is a survey on the propagation and interaction of singularities of onedimensional semilinear hyperbolic systems. The main theorems here were proved by Rauch-Reed, L. Micheli, L Iordanov and others. The solutions are allowed to have jump discontinuities in the initial data. A detailed study of the propagation of the jump discontinuities is given in this section. The singularities mentioned before propagate along the corresponding characteristics till their intersection (as in the linear case). After their collision new singularities can appear which will propagate along the full set of characteristics starting from the collision point. The newly created singularities are weaker than the initial ones. Often they are called anomalous singularities. Interesting effects can appear when the system possesses two tangential at one point characteristics. Thus we have nonlinear interaction between two singularities - the first one created by the jump discontinuity in the Cauchy data and the second one generated by the weak hyperbolicity.

Both in the strictly hyperbolic and the weakly hyperbolic cases, a necessary and sufficient condition for the existence of anomalous singularities is proved. Moreover, the optimal order of the anomalous singularities is also found.

Section 2 deals with the interaction of three conormal waves for the semilinear wave equation with two space variables. As it was proved by Bony and MelroseRitter, in this situation new singularities in comparison with the linear case appear.

2 P.R. Popivanov

We follow here (up to some modifications) the approach of Keller-Ting. The only new feature is the proof of convergence and of pointwise differentiation of the series, giving the solution inside and up to the light cone inscribed in the characteristic pyramid. The proof is elementary, as it involves the Gauss hypergeometric function, avoiding this way the heavy technic of second microlocalization.

In Appendix I we propose blow up results for the solutions of the Cauchy problem for some classes of quasi-linear systems in the plane. This is another type of singularities when the solutions remain bounded (eventually becoming multivalued after some time) but their gradient blows up in a finite time. Interesting books on the subject are written by Hormander [31], Li Ta-tsien [48], Alinhac [3]. We prove here a result of ours and we apply it to the nonlinear vibrating string equation.

§4 (Chapter II) deals with the singularities of the solutions of quasi-linear hyperbolic systems in the plane and "in the generic case". To do this we linearize our system via the classical hodograph transformation and we investigate the singularities of this transformation by applying the Whitney theorem for Coo normal forms in R2.

Our main results in this chapter are concentrated in §5-§7. By using the paradifferential approach we prove several theorems for microlocal singularities of the solutions of some classes of fully nonlinear systems. To be more precise, we find sufficient conditions for nonexistence of isolated singularities in a conical neighborhood of the corresponding characteristic point. Applications are given to the Monge-Ampere equation and to some systems arising in fluid mechanics. The theorems here formulated and proved are mainly due to the author.

To complete the results of this chapter and to compare them with the results from §1, Chapter I, we include Appendix II containing, with proofs, several results of Bony on interaction of conormal waves for semilinear PDE.

Chapter III is devoted to the travelling waves for several classes of nonlinear integrodifferential operators arising in solid body mechanics. We construct C 1 smooth solutions of them in the form of travelling waves of tension. Blow up results in the case of a solid body located in a medium with exponential memory are also formulated.

We have tried to give simple and clear proofs of the central results given here. They are illustrated by many examples and 28 figures. That is why we believe that our text could be interesting and useful for young mathematicians beginning their work in the domain of nonlinear partial differential equations and their applications. Probably, there are new results for the specialists in the domain of nonlinear PDE as well.

Acknowledgments are due to Prof. B.-W. Schulze and to Prof. M. Demuth for their expert co-operation. The main part of the theorems in Chapter II and Chapter III were communicated at first at the seminar of Prof. B.-W. Schulze in Potsdam, as well as at the seminars on PDE of Prof. Rodino (Torino), Prof. Parenti (Bologna), Prof. Zanghirati (Ferrara) and at the seminar of PDE in Univ. P. Sabatier - Toulouse.

Nonlinear PDE. Singularities, Propagation, Applications 3

In preparing the IbTE;Xversion of the manuscript we had the valuable help of L. Apostolova and P. Marinov. Many, many thanks to them. Acknowledgments are due also to 1. Iordanov for support in preparing §1, Chapter I and to P. Roussev for Lemma 2.2 from §2, Chapter 1.

Chapter I Interaction of singularities of the solutions of semilinear hyperbolic equations and systems

1. Singularities of the solutions of one-dimensional semilinear hyperbolic systems

1. Rather complete results have been obtained for the singularities of linear and semilinear hyperbolic systems with one space dimension. The corresponding initial data are very often assumed to have singularities of the type "finite jump in finitely many points", i.e., the jumps of all derivatives are finite and the jump discontinuities exist in a discrete set. The methods used in the proof of these results are relatively elementary and transparent. Unfortunately, this approach works only in the case of one space variable and it seems impossible to generalize it for several space variables.

It is well known that the singularities of the solutions of one space dimensionallinear hyperbolic systems generated by the singularities of their initial data propagate along the corresponding characteristics (linear effect). The paper [69] deals with a new, nonlinear effect for a class of semi linear hyperbolic equations in which anomalous singularities in comparison with the linear case appear. The new singularities are created by the interaction of already existing singularities generated by the initial data and propagating along the characteristics. The new-born singularities are weaker than the ones which create them. Moreover, they appear in the cross points of the characteristics carrying out the initial singularities and they propagate along (third) characteristics starting from these cross points. A full investigation of the problem mentioned above in the strictly hyperbolic case is contained in [70].

More precisely, a necessary and sufficient condition for the existence of anomalous singularities is found in [70].

A very important example of the appearance of anomalous singularities for one-dimensional weakly hyperbolic semilinear systems was proposed in [71] and it was studied in detail by L. Micheli in [57]. It concerns the nonlinear interaction between two singularities - the first one created by a jump discontinuity in the initial data and the second one generated by the weak hyperbolicity (linear effect). The optimal order of the anomalous singularity is found in [57]. An essential simplification of the approach of L. Micheli is given in [37]. This way it is possible to find a necessary and sufficient condition for the existence of anomalous singularities (the latter being identical to the condition from [70] for strictly hyperbolic systems). A

4 P.R. Popivanov

weakly hyperbolic system having three tangential characteristics is studied similarly in [79]. Propagation of singularities of quasi-linear strictly hyperbolic systems was considered in [56].

2. For the sake of simplicity we shall study systems written in a diagonal form

DiUi == OtUi + Ai(X, t)OxUi = fi(X, t, u), i = 1,2, ... , n, (1.1) where U = (Ul, ... , un), Ii E C=(R2+n), the real-valued function Ai E C=(R2) and

(1.2)

Thus the hyperbolic system (1.1) is investigated in the "trapezoidal" domain

CT = {(x,t) E R2 10:::; t:::; T, x(t):::; x:::; X(t)},

where having in mind the local character of our study, we assume that the curves x = x(t), x = X(t) are defined in the whole interval [0, T] and they satisfy the equations

dXd(t) = max {Adx, tn, x(O) = a, t lS;kS;n

dXd(t) = min {Ak(X, tn, X(O) = b. t lS;kS;n

Moreover, we suppose that x(T) :::; X(T). It is obvious that in the strictly hyperbolic case (1.2) we have Al(X, t) <

A2(X, t) < ... < An(X, t) and therefore the above written equations become very simple. The more general form of the equations for x(t) and X(t) will be used in the nonstrictly hyperbolic case.

The curve ~ = ~i(T; x, t) == ~i(T) given by

d~~) = Ai(~(T), T), ~(t) = x (1.3)

is called an i-characteristic of (1.1) passing through the point (x, t). The arcs ~ = ~f, where ~f == ~ ( T) for ± (T - t) > 0, are called forward (~+) and respectively backward i-characteristic through the point (x, t). Without loss of generality we shall assume that if (x, t) E CT , then all i-characteristics exist globally, i.e., they exist until leaving CT. Obviously, ~i are C= smooth with respect to their arguments.

The domain C T can be defined geometrically as follows: For each point (x,t) E CT the backward characteristics ~ = ~i(T;X,t), T < t, i = 1,2, ... ,n, are contained in CT and they cross the lower base [a, b] of CT.

The Cauchy data for the system (1.1) are given by

Ui(X,O) = u?(x), i = 1,2, ... , n, (1.4)

u? E C=([a, b] \ {x6, ... , xg}). Moreover, for each point x~ E (a, b), 1 :::; k :::; s, there exists a neighborhood

Vk 3 x~ such that u? E Ch(Vk), h = h(k, i) 2: -1, hE Z, the (h + 1) derivative of


U? is discontinuous at the point x~ and the corresponding left and right limits of u~(p) at x~ for all derivatives of order p ~ h + 1 are finite.

Thus the integer h is the order of the highest derivative of u? which is continuous at x~. Obviously, h = -1 iff the function u? has a finite jump discontinuity at x~. We shall say that the initial data u? has a (singularity of the type) finite jump of order h(i,k) at x~ and we shall denote this fact by: u? E C~(Vk \ {x~}), C~ (Vk \ {x~}) = C h (Vk) n C,;'" (Vk \ {x~}). Similar notation will be used in the case of two variables: C~(V \ C), where V c R2 is a domain and the curve C c V. Often we shall say: a function of finite order h at C.

The union S of all outgoing characteristics starting from the points

(xb, 0), ... , (xg, 0),

i.e., the outgoing characteristics starting from their cross points etc., will play a crucial role in studying the Cauchy problem (1.1), (1.2) (1.4). To do this we shall prove at first the existence of a solution having finite order at the arcs of S. In our second step we shall find the sharp order of the singularities (more precisely, of the jump of the solution).

The first problem mentioned above is solved in [70], [57] by replacing the system (1.1) with an integral system equivalent to it. The contraction mapping principle applied to the latter system in the functional class C;r (GT \ S) gives the existence result. Another approach was proposed in [36]. It combines the classical theorem for the existence and uniqueness of low regularity solution with the Coo regularity of the solution of (1.1), (1.2) (1.4) in GT \ S - proved by L. Micheli. This way the linear and nonlinear effects can be followed easily.

To realize the program mentioned above we shall remind the reader of the classical theorem for existence and uniqueness of the solution of the Cauchy problem for the linear system (1.1), (1.2) (1.4), i.e.,

n

!i(X, t, u) = L aij(x, t)Uj + bi(x, t), i = 1,2, ... , n, j=l

supposing the Cauchy data u? E Cora, b].

(1.5)

We shall say that the vector-valued function (U1 (x, t), . .. , un(x, t)) defined in GT is a solution of the problem (1.1), (1.2) (1.4) if it is continuous in GT , each component Ui is continuously differentiable along the vector field 8t + Ai (x, t )8x

and the equalities (1.1), (1.4) hold. Similar but more general results are given in the book [74]. The just proposed notion of a solution according to this book is called a solution in a broad sense. In contrast with it the notion of a classical solution for the same problem assumes C 1 differentiability of U in the interior of the domain GT .

For an arbitrary point (x, t) E GT we rewrite (1.1) this way:

6 P.R. Popivanov

and therefore into integral form

Obviously, the finding of a continuous solution of the integral system (1.7) is equivalent to the existence of a broad sense solution for (1.1), (1.2) (1.4).

From (1.7) and having in mind (1.5), we define the successive approximations for the linear case:

(s+l) (0) t ~ (8) Ui (x, t) =Ui (x, t) + in ~ aij(~i(T; x, t), T) Uj (~i(T), T)dT,

o j=1

(1.8)

(0) (0) t . where Ui (x,t) =Ui (~i(O;x,t)) + io bi(~i(T),T)dT and z = 1,2, ... ,n.

(s) (s) It is evident that U i E CO (GT ) and that each component of U i possesses

. d" 1 h d' h .. Th . D (8) a contmuous envatlve a ong t e correspon mg c aractenstic. at IS, i U i==

( >:l \ ( t) >:l ) (8). . ( 'd' h (s-I). . ) Ut + Ai x, U x U i IS contmuous provl Ing t at Ui IS continuous .

For a domain G C R2 we introduce some notation. Put

and denote

G r = GT n {(x, t)1 t:s; T}, (8) ( ) = II (s+l) _ (s) II v t U U G" t E [O,T].

One can easily see that (8) t (s-l) V:S; C io V (T)dT, (1.9)

where the constant C is estimated by max laij(x, t)l. l:O;i,j:O; n,(x,t)EGT

(s) (Ct)8 (0) Applying induction we conclude that V (t) :s; --I - V (t) and consequently

s. the sequence {C;:l (x, t)} is uniformly convergent to some continuous function u(x, t). Passing to the limit in (1.8) we see that U satisfies the integral system (1. 7). The proof of the existence of a broad-sense solution is completed.

We point out that the uniqueness result holds true in the same functional class. In fact, denote by U the difference between two broad-sense solutions of the linear system and put V(t) = IlullGt 2: O. Then we have

V(t) :s; C fat V(T)dT. (1.10)

According to the Gronwall inequality we get V(t) == 0 as V(O) = O.

Remark 1.1. In the previous proof we did not use the strict hyperbolicity of the system. Thus the same result for existence and uniqueness of a broad-sense solution is also valid in the weakly hyperbolic case.

Regularity of the broad-sense solution depends on the regularity of the initial data. More precisely, suppose that u? E C 1 [a, b]' 1 ::; i ::; n. Then obviously

(;J E C 1 , Vs E N. Differentiating (1.8) with respect to x we get

where 011 , fJ, = 1,2, stands for the derivative with respect to the fJ,-variable of the corresponding smooth function. Having in mind the uniform bounded ness of aij

and of the derivatives of ~i in GT , we get

(d1) (8) Ilox u -ax u II0t (1.11)

(t (s) (s-1) ((s) (s-l) ::; c110 II u - u Ii0rdT + c210 Ilox u -ax u Ii0rdT

with appropriate constants c1 , C2 > O. (s) (s) (s-l) (s) (s) (s-l)

Denote now W (t) = Ilox( u - u )llot' V (t) = II u - u Ilot and Co = max(C,C1 ,C2 ) where the constant C participates in (1.9).

(1) (D) (0) Applying (1.11) we obtain W (t) ::; Cot V (t) + Cot W (t). In a similar way

(2) ( (1) ((0) (0) we have W (t) ::; Co 10 V (T)dT + Co 10 (COT V (T) + COT W (T))dT.

(1) COT (0) (2) (Cot)2 (0) But according to (1.9), V (T) < I! V (T) and thus W (t)::; 2! V

() (Cot)2 (0) () (Cot? (D) ( ) t +-2-!-V t +-2-!-W t.

Applying induction to (1.11) we find that for each SEN,

(s) ( ) (CoW (0) () (Cot)S (0) ( ) W t ::; s--, - V t + --, - W t .

s. s.

Thus the sequence {ax (;J} turns out to be uniformly convergent, i.e., the broadsense solution u(x, t) is differentiable along the x-axis and therefore (1.1) implies it to be differentiable in t as well, i.e., u E C 1 (GT ). Similarly, initial data being from C k , the solution belongs to C k (Gr ), k 2: 1, kEN.

8 P.R. Popivanov

Now we are able to study the singularities of the solution of the linear Cauchy problem (1.1), (1.2) (1.4), (1.5). Let the initial data possess finitely many discontinuity points. For example, if

{ u~(x) E uO E

J

C;l([a, b] \ {xb}), C[a, b], j =I- i,

(s) then (see (1.8)) Ui is discontinuous across the i-characteristic Ci, starting from

the point (X6,O) (see Fig. 1). All the other components (:2 j , j =I- i, of the sapproximation are continuous because of the continuity of their initial data and (s) ... d h' 1 . U i partlclpatmg un er t e mtegra slgn.

t t

T T

I a o a o

Fig. 1 Fig. 2

Certainly, the boundary values of C:J i from both sides of C i are different, i.e.,

C:J i is a double-valued function on Ci .

If another initial datum also has a jump, e.g., if u~ (x) E C; 1 « a, b) \ {X6, x6} )

(see Fig. 2) for some j, we conclude inductively again that (:2 i is discontinuous

across C i , C:J j is discontinuous across Cj, Cj' and all other components are continuous.

In order to study the Cauchy problem (1.1), (1.4) with discontinuous initial data, the remarks above lead to the following generalization of the notion of broadsense solution (for short - discontinuous solution): We assume that the vectorvalued function U = (Ul, U2, ... , un) defined in GT has the properties given below:

(i) The restriction of each component Uk along each k-characteristic is continuous, and a piecewise differentiable function such as the derivative DkUk is assumed to possess a finite jump in finitely many points of the chosen characteristic

(ii) There exist at most finitely many k-characteristics Ck, (k = 1, ... , n) on which the component Uk is not single-valuedly defined, i.e., the limits of Uk

at the points of such a characteristic Ck (from both sides of Ck) are different.


Remark 1.2. A typical example of a function satisfying the conditions of that definition is WkS(X, t) given by the equality

Wk(X, t) = Vk(X, t) + lot f(f,k(r; x, t), r, u(f,k(r), r)dr,

where each component Uj of U = (U1, ... , un) is discontinuous across at most finitely many characteristics of type C j and the functions Vk are discontinuous across at most finitely many characteristics of type Ck. It is clear that for a fixed characteristic C k passing through the point (x, t) the discontinuities of the components Uj, j =I=- k, do not influence the continuity of the integral term. On the other hand, the effect of discontinuity of Uk just across the chosen characteristic Ck (along which we integrate) is the same as the effect of discontinuity of Vk across Ck, namely - the function Wk(X, t) is double-valued on Ck.

Having in mind the definition of discontinuous solution we expect for the case studied in Fig. 2 a solution with the following singularities:

- the component Ui is double-valued on C i and in G T \ C i the derivative DiUi

exists and is continuous except for the points of Cj, Cj' where it possesses a finite jump across them;

- the component Uj is continuous in each of the three closed domains into which Gr is divided by CjUCj', having Djuj discontinuous only at the points of C i ;

- the other components of u are continuous at GT .

So let the initial data for the linear system (1.1) be discontinuous in finitely many points of [a, bJ. Let So be the union of the respective characteristics starting from these points (i.e., if u? E C- 1 (( a, b) \xo), then the i-characteristic through the point (xo,O) is included in So). The curves of So divide GT into finitely many closed

(s) subsets G k . It becomes clear from the remarks made above that the sequence { U }

of successive approximations is continuous on each of the domains Gk (it is possible

for some components of ~) to have different boundary values on each of their

common boundaries). If considering a fixed component j, then the sequence {C:J j }

is continuous on each of the closed domains G-?-r, obtained through the division of Gr by the characteristics of the type j included in So. In fact, each of the domains G-tn is a union of several subsets Gk, having possibly in common only contour common points. The estimate (1.9) is obtained in a similar way and therefore

{ (s)} . ·c I u 1S unllorm y convergent. The limit function u(x, t) is a discontinuous solution of (1) with the chosen

initial data, because:

- U E C°(t:Jk ) , Vk ENdue to the continuity of the functions from the sequence (8) -

{ U } on each subset Gk .

- Djuj( = fj(x, t, u)) exists in each subdomain G k and is discontinuous only on the characteristics dividing G-tn into its component domains Gk.

10 P.R. Popivanov

Now we can generalize the result of regularity proved by the lack of discontinuities. Let us suppose that the initial data belong to the class

C- 1 ((a, b) \ {x~, ... , x~}) n CO((a, b) \ {Y~,"" yO'}), (1.12)

where some point Yo may coincide with some point xb. Let u?(x) be discontinuous at the point Xo. Then, certainly, a C- 1 singularity of the i-component (i.e., Ui E C; 1 (GT \ C i (XO, 0))) propagates along the i-characteristic through (XO, 0) denoted by Ci(xo, 0). It means that a CO singularity (even if u~(x) is smooth in a neighborhood of xo) propagates along any other characteristic Cj(xo, 0), j i- i .

Example 1. For the system (at + ox)u = 0,

OtV = u,

where uO(x) = {;: ~;: ~ ,VO(x) == 0, we have v(x, t) = it u(x, T)dT.

This function is continuous but lim oxv(x, t) = 0 and lim oxv(x, t) = 1 x---+O~ x---+o+

(see Fig. 3). These calculations are direct because we have U = 1 on the left of the characteristic t = x and U = 2 on the right of it. The same characteristic bears the singularity of U given by the initial data.

t

/ / t=x

/ /

x Fig. 3

This example explains the reason for the phenomenon formulated before. In

a more general context, in calculating OxUj(x, t) (a similar calculation for ax it] was made after formula (1.10)) we shall conclude that this derivative is discontinuous across Cj(xo,O). In fact, let us assume that the j-characteristics through the points A and B near to (xo, 0) may possibly cross a single bearing singularity characteristic, namely Ci(xo, 0) (see Fig. 4).

In order to check the continuity of ax Uj at the point D E Cj (xo, 0), we calculate that derivative at the nearby points (x, t) on the left and on the right of Cj (xo, 0) using the expression

Uj(x, t) = u~(~j(O; x, t)) + it ( .. . )dT.

For the characteristic Cj (A) the result is quite analogous to the expression men

tioned above for ax (5,;1/) after formula (1.10). For the characteristic Cj(B) in the


B Xo A

Fig. 4

(x, t)

x

11

function under the integral sign there exists a discontinuity (on the characteristic C i (xo, 0)) and therefore we use the representation

(1.13)

where tl = tl(x, t) is the value of the parameter T giving the coordinates of the cross point Ci(xo, O) x Cj(B). The x derivative of (1.13) is

it {ax( aji. ui)}dT + [ajiui]ctl) .axtl,

where {axg} denotes the derivative of g evaluated almost everywhere (i.e., everywhere except for the point of discontinuity of Ui) and [hl(tl) stands for the jump h(tl - 0) - h(tl + 0).

Having in mind that T = tl is the solution of the equation

after differentiation with respect to x we get

a ( ) _ ax~i(tl;x,t) xtl x,t - .Aj(~j(tl;B),td-.Ai(~i(tl;x,t),tlr

According to our assumption (1.2) the function axtl depends continuously on (x, t).

Remark 1.3. Suppose that the initial data u~, u?n, k =I=- m have jump discontinuities at the point Xo. Apart from the C- l singularities on the respective characteristics C k , Crn a propagation of a CO singularity along Cj , j =I=- k, m is possible (see Fig. 5); but it is also possible that the singularities compensate.

That problem has not yet been studied in detail. As a conclusion of these remarks we shall note that if some initial datum has

a singularity of 0 order at a fixed point (i.e., its first derivative is discontinuous there) then a singularity of the same order shall be propagated by it only along the respective characteristic. Only the second derivatives can be expected to be discontinued across the other characteristics starting from the same point.

12 P.R. Popivanov

Xo x Xo x Fig. 5

We shall mark two important properties.

Remark 1.4. We established that in the linear case the singularities appearing from jumps in the initial data propagate along the respective characteristics. It can be easily seen that the singularity across a certain characteristic cannot disappear (nor can a singularity appear across some characteristic if the singularity is lacking in the initial moment). For that purpose, from the integral representation (1. 7) written for characteristics near to the considered Ci , we find the following expression for the jump [Ui](X, t) of Ui across Ci :

[Ui](X, t) = [u?(x, t)](~i(O; x, t)) + lot aii(~i(T), T)[Ui](~i(T), T)dT.

Now the Grownwall inequality implies everything.

Remark 1.5. Except for the singularities described, there will not be new singularities at a later moment of time. We can easily see this even in the cases when the backward characteristic through the studied point crosses more than one bearing singularity characteristic (see Fig. 6).

Cj B

/ ./ Ci / Ie jD

c I

C f / E

"/

x Fig. 6

For that purpose it is sufficient to consider in detail the differentiability of

C;J in different cases - the appearance of jumps in points of type C, E, F but that bring about no changes in the result. Even when the studied characteristic C~ passes through the crossing point H = C i X C j of the two bearing singularities characteristics, new singularities do not appear across it (if lacking in the initial


moment). The reason is that in the function under the integral sign there are two different summands discontinuing across the respective characteristics, namely

(8-1) (s-l) akj Uj is discontinuous across Cj and aki Ui across Ci . The lack of a term discontinuous simultaneously across the two intersecting characteristics plays a key role for a typical linear phenomenon: the singularities "meet" at the point H and do not interact, i.e., at the points from C~ \ {H} the solution is smooth.

Although the approach to studying semilinear systems is ostensibly the same, the properties of the solution are principally different. The principal part of the system (1.1) being linear, the definition of "trapezoidal" domain GT does not alter. The uniqueness theorem of the broad-sense solution, i.e., continuous in GT and differentiable along the characteristics, may be proved almost the same way. If U

and v are two broad-sense solutions with the same initial data, then W == U - v satisfies the conditions:

Wi(X, t) = lot [Ji (';i (T), T, U(';i(T), T)) - li(';i(T), T, V(';i(T), T)JdT, i = 1,2, ... , n,

i.e.,

Wi(X,t) = lot [~9iS(T'U'V).W8(';i(T)'T)l dT,

i t ali where gi8 == -;:;-(';i(T), T, V(';i(T), T) + A(V - U)(';i(T), T))dA.

o UW8

Due to the assumptions made the functions gis are uniformly bounded on GT , which ensures an inequality of the type (1.10) completing the proof. The uniqueness of the discontinuous solution defined by properties (i), (ii) before Remark 1.2 is proved without modifications. Let tl, ... ,trn be the points at which the differentiability of W = v - U along Di is violated.

Therefore, by using analogous notation,

rn n jtk+l Wi(X, t) = L L gfs(T, u, V)Ws(';i(T), T)dT,

k=O s=l tk

From max Igfs I ::; const we have k

implying the uniqueness.

to = 0, tm+1 == t.

C = const > 0

In contrast with the linear case where the existence theorem is proved in an arbitrary, prescribed "trapezoidal" domain, in the semilinear case we come across a well-known difficulty - the solution exists only locally.

Hence, we shall assume the existence of a broad-sense solution, using the successive approximations only for studying its regularity (as described above).

14 P.R. Popivanov

Remark 1.2 shows that the successive approximations for a semilinear strictly hyperbolic system possess the same properties as the successive approximations in the linear case (with which the existence of a discontinuous solution was achieved with) and naturally we expect analogous results.

Unfortunately, the convergence of the successive approximations is only local (as without special restrictions for the right-hand sides Ii in (1.1) the uniform boundedness of the approximations could not be established). Even if we assume the existence of the solution in some interval [0, T] the convergence under the usual assumptions would be only local. The use of approximations in that case would lead to difficulties, with possible appearance of new singularities from the initial data on the consecutive step that do not exist de facto (see [70], p. 213).

The problem of proving the convergence of the successive approximations to the solution (existing by assumption) can be overcome by replacing them with modified successive approximations (an approach hinted at by E. Horozov).

Let u(x, t) be a discontinuous solution of (1.1) existing in GT . We denote M = 2 max lu(x, t) I and then consider the system

GT

DiUi = li(X, t, u), i = 1,2, ... , n, (1.14)

where the functions Ii are obtained from Ii after multiplying by a cut-off function belonging to Co(R~:'~u)) which is identically equal to 1 on the set {(x, t, u)l(x, t) belongs to a neighborhood of GT , lui::::; M}. As Ii are compactly supported smooth

functions, the norms of h, ~~ are uniformly bounded, the latter meaning the ex

istence of a Lipschitz constant L for the functions Ii with respect to the variable u, independent of the size of u (compare with the proof of the uniqueness theorem, where max Igis I depends on u, v). On the other hand, the very solution u of the initial problem (1.1) is an obvious solution of (1.14). The respective successive approximations defined via the integral form of (1.14) tend uniformly to it over [0, T]. To see this we should have in mind that the so-defined successive approximations are uniformly convergent over GT to a function u(x, t) which is a solution of (1.14). According to the uniqueness theorem u == u.

As in the linear case we can prove C 1 regularity of the solution under C 1_

initial data etc. One by one we investigate the different cases of finite jump type singularities

in the Cauchy data and we get the same results as in the linear case. Concluding these remarks we should point out the apparent fact that de facto the derivative with respect to x of the solution satisfies a linear system but with coefficients depending on the solution.

For the first time a new effect appears when we study a triple intersection of the characteristics, i.e., when the k-characteristic crosses two different, bearing singularities characteristics of the type i, j, i,j =I=- k (see Fig. 6 with G = (x,t)). The initial function u~(x) is assumed to be smooth in a neighborhood of the point ';i(O; x, t). This situation leads to a new effect (nonlinear) found by Rauch-Reed

in [69]. It concerns the creation of a singularity for a component having smooth Cauchy data. This is the corresponding example:

(at - ox)v = 0, vo(x) = { ~:

OtW = UV, wo(x) == 0.

x <-1 - , x> -1,

x::; 1, x> 1,

Simple calculation shows that W is a continuous function but oxw has a finite jump across the part of the characteristic x = 0, starting from the cross point (0,1) of the characteristics bearing the singularities of the components u, v (see Fig. 7).

t t

1 0 4

-1 1 x Fig. 7

t

52

0 1

X

A(xo, to)

Xo

Fig. 8

31

O2

x

Such a singularity of w does not exist in the linear case, despite the triple interaction.

We shall consider the same effect in a more general framework. Let J(x, t) E C;1(R2\51 \32), where 3 i : t = 1'i(X) are smooth curves intersecting transversally at the point A = (xo, to), to > ° and such that J possesses finite jumps across 3 i , i = 1,2. Then the derivative with respect to x of the function

u(x, t) = 1t J(x, T)dT

is continuous at the points of the characteristic of at : X = {(x, t) It :2: to, x = xo}, Ot being transversal to 3 i , if and only if

(1.15)

where Ji = lim J(x, t) and the domains Oi are the angles between (x, t) --+ (xo, to), (x, t) En i

the curves 3 1 ,32 clockwise numbered, X C 0 1 (see Fig. 8). The proof of the result mentioned above is given in [36]. Obviously, the same

assertion holds when replacing at by an operator of the type at + a(x, t)Ox.

16 P.R. Popivanov

As a consequence of this new effect different cases could appear:

Case I. There exist two systems of characteristics only. Then the underintegral functions participating in the definitions of the successive approximations contain discontinuities across only one system of the characteristics. This case is similar to the linear one.

Case II. The system (1.1) possesses more than two systems of characteristics but the height of the trapezoidal domain GT is bounded by the first cross point of the characteristics starting from the points (x(j,O), s = 1,2, ... , k, where the Cauchy data have singularities of finite order. (Such a cross point can exist for t = T.) Then the solution U E COO(GT \ So) and So stands for the union of the respective characteristics (moreover, it has across them singularities of finite order). That result combined with a Coo regularity result between the characteristics (see Lemma 1.6 below) enable us to investigate the regularity of the solution above the cross point.

Case III. The system is of type n x n, n ~ 3 and GT contains the cross point of bearing singularity characteristics. A central result in this case is the theorem for existence of an anomalous singularity of strictly hyperbolic one space dimensional system [70]. That theorem can be proved in a simple way by applying the necessary and sufficient condition (1.15).

Thus assume that the component U1 of (1.1) possesses a singularity of finite order n1 across the I-characteristic C 1 through the point (Xl, 0) and the component u2-singularity of finite order n2 across the 2-characteristic C2 starting from (X2' 0) (see Fig. 9).

X

Fig. 9

We shall show that some derivative 8~Ui of a component Ui, ii-I, 2, having Coo initial datum, is discontinuous across the arc of the i-characteristic Ci located above the intersection point A = C 1 X C2 • We remind the reader of the integral equation satisfied by Ui :


Differentiating Ui(X, t) several times in x we get integral terms and summands containing jumps of the function fi (or some of its derivatives) across G1 or G2. Assuming the discontinuity to arise across only one of these characteristics we can easily see that the expressions mentioned above are continuous functions of (x, t) in a neighborhood of the point B. The successive derivatives of fi can be written as (we apply the Faa di Bruno formula):

II These summands will contain singularities of the finite jump type if the derivatives 0~U1' P 2: n1 + 1 or 8~U2' p 2: n2 + 1 are to be part of them. According to Case I the derivative O~Ui is continuous in a neighborhood of B if that singularity (i.e., finite jump type) is across only one of the two characteristics C 1, C2 • A term containing singularities of the finite jump type across both intersecting characteristics appears for the first time in the expression for 0';:' +n2+2(f;). Obviously, this is the derivative

02fi ~n,+l ~n2+1 ~ ~ .Ux U1·Ux U2· UUI UU2

Put oU20u,jilA = F. Having in mind that the derivative oU20u,f is continuous with respect to (x, t) and denoting by Ui + hi, hi =1= 0, i = 1,2, respectively by Ui the limits at the point A of 0';:i+1Ui from above and from below the curve Gi , we can apply the condition (1.15) and conclude that an anomalous singularity of order (n1 + n2 + 3) arises at A if and only if

i.e., iff Fh1h2 =1= 0, i.e., iff

(1.16)

as hi =1= 0. The inequality (1.16) coincides with the results of [70] for existence of an anomalous singularity.

According to (1.16) the existence of an anomalous singularity in the case of double interaction does not depend on the size of the jumps hI and h 2.

Assume now that we have a triple (at least) transversal interaction of characteristics bearing finite order singularities. The previous approach shows that the necessary and sufficient condition for the existence of anomalous singularities depends on the size of the jumps at the interaction point of the respective characteristics.

We shall formulate now an interesting result of L. Micheli [57].

Lemma 1.6. Under the notation introduced before, consider the domain G T containing only one point of interaction A = (xo, to) where the crossing of at least two characteristics Ci, i = 1,2, ... , k, bearing singularities of the finite jumps type, takes place.

18

t

T

Fig. 10

P.R. Popivanov

x

t C

o Xo

x -h

Fig. 11

Let the solution belong to C':'(Gto \ UCi ). Then it belongs to C':'(GT \ S), where S is the union of all characteristics passing through the point A.

A short sketch of the proof is given further on. Thus, there are no singularities along the punctured characteristic Cj , j > k. A singularity could be born at A, propagating along the forward branch of C j passing through A.

New phenomena appear in weakly hyperbolic systems, e.g., it enables singularities of the finite jump type across one characteristic to give rise to a singularity of the infinite jump type across a characteristic tangent to the original one. The effect is linear and can be illustrated by the following example.

Example 2. ([71]) For the system

Xu (8t + ptp - 18x )u = 0,

8t v = u,

with initial data (see Fig. 11)

ult=-h = uO(x) = { 0, 1,

x:::; Xo 0_ ,vlt=-h = v (x) = 0, x > Xo

where p is an even positive integer, it is easy to see that in the "corner" between the X -characteristic C : x = t P and the positive t semi-axis (to which C is tangent

at the origin) the function v(x, t) == 2x i, while v == ° for x < 0. Therefore at the points {(O, t), t > O} the derivative 8x v has unbounded right limit with the

behavior of xi-I. We shall note that 8t v is bounded, i.e., not all the derivatives do blow up.

In a more general situation a similar problem has been studied by L. Micheli. The paper [57] deals with a weakly hyperbolic system of the type:

h(x,t,u), fz(x, t, u), fi(X, t, u), i 2:: 3,


where Xl = at + t p - 10;(x, t)ax , o;(x, t) 2: 0;0 > 0,0; E C=, pEN is even, X 2 = at and the characteristics of the operators Xi = at + Ai(X, t)ax , Ai E C= are transversal to the Xl and X 2 characteristics (and to each other). We choose initial data for t = -h:

(CWH) u?(x) E Ck(R) n C:;"'(R \ {xo}),

u? E C=(R), i 2: 2,

where the point Xo is defined by the condition: the Xl characteristic through the p. (xo, -h) is tangent to the t-axis at the origin (see Fig. 11).

For a similar problem the hypothesis for the order of the anomalous singularities ([71]) by the way of analogy with the strictly hyperbolic case is:

k+(k+1)+2+p-1,

p being the order of contact between the tangent characteristics. [57] achieves quite a better result, namely the order of the anomalous singularity is

[k + (k + 1) + 2]p.

The latter statement is motivated in [37] with an appropriate example but lacks a theorem with a necessary and sufficient condition of the type of the result in [70]. The method of investigation differs from the one used up to now, consisting of successive differentiation along the x-axis of the respective integral equation. While the strictly hyperbolic case always provides for bounded piecewise smooth functions by successive differentiations, Example 2 shows that there should arise unbounded derivatives - Schwartz distributions being non-integrable functions. This difficulty is overcome in [57] by using only appropriate derivatives (as not all the derivatives blow up; see Example 2). In order to make possible the use of integral equations, rather complicated changes of the variables have been done successively.

In [37] that approach is modified and the obtained simplification allows for a proof of an analogue of the condition (1.16) for existence of an anomalous singularity. The new feature of this modification is the systematic use of the same differential system and not of its integral equivalents. This enables the differentiation to be in the sense of distribution theory and afterwards, depending on the properties of the right-hand side of the equation, our coming to the conclusion of the continuity of the solution. Our arguments are based on the following

Lemma 1.7. (Lemma 1.4 from [37].) Assume that X = at + vex, t)ax , Y = at + p,(x, t)ax , v, p, E C=(R2), Iv - p,1 2: const> 0 are two real-valued vector fields. Let l : t = g(x) be a smooth curve transversal to the characteristics of Y. Denote by o the curvilinear quadrangle depicted by the arcs of the Y -characteristics passing through the points of l. The symbols 0+ (0_) stand for the open parts of 0 located above (below) l. Assume now that a,w E COCO) n C 1 (0-), Xa E COCO) and the first-order Schwartz distribution v = X w E DII (0) satisfies the linear differential equation Yv + av = YF, where F E COCO).

20 P.R. Popivanov

t

n~_-y I l

0- \ y-_ ... J

x

Fig. 12

Then v E CO(O) and therefore v turns out to be a classical X -derivative of w (see Fig. 12).

We remind the reader that the first-order distribution is a distribution in o represented in the form of a generalized first-order derivative of a continuous function in O. The set of such distributions is denoted by Dl!(O).

To give an application of Lemma 1.7, we shall sketch out the proof of Lemma 1.6, supposing for the sake of simplicity that the solution is continuous in CT. There exists in C ta at least one "angle" between two characteristics that the solution belongs to Coo (according to Case II considered above). As a starting point we can make use of a part of an "angle" located under t = to. We shall show at first the way Coo regularity is established in the adjacent angles, etc. through the final proof of the lemma. Let Ci be the characteristic through (xo, to) and 0-, 0+ be two "corner" domains bounded by C i and the two neighboring characteristics (see Fig. 13). (It does not matter whether these three characteristics are backward or forward.)

(xo,to)

Fig. 13

We assume that the solution U E Coo in 0-. Due to the strict hyperbolicity of (1) the j-characteristics Cj ' j i- i are transversal to Ci . Suppose that it is known that for some integer m 2': 1 the derivatives X["Uj, j = 1,2, ... , n, j i- i are continuous in 0- U 0+. If we prove that X;n+lUj, j i- i exist and are continuous in 0+, then it follows that Uj E cm+l(O+). In fact each (m + l)order derivative


of Uj containing at least one differentiation along Xj is continuous and the vector fields Xi, X j ' i -I- j are transversal.

To show the continuity of the derivative X;n+lUj we shall apply the following commutator identity in the sense of distribution theory:

(1.17)

where bE Coo and Pm E Co, as it is a polynomial of order m of XiUk, a:::; m, k =

1,2, ... , n. Here Xi(X;nUj) E DlI, Xi(Xr)fj E Dl!. The result (1.17) coincides with Lemma 1.9. in [37]. On the other hand, according to [69] we can use the chain rule in our situation,

obtaining this way that

Xm+1f. = ~ ofj Xm+lu + pI p' E CO t J ~ 0 t Z 171' 171 •

Z=l,Zopi Uz n

In fact, Xdj(x,t,u) = (Xdj)(x,t,u) + L(OUkfj)XiUk, XiUi = fi E CO etc. k=l

But Xi = aXj + (3Xz, l -I- i, j, where the coefficients n, (3 E Coo and therefore

Ofjxm+l ofj( X +(3X)X171 (1.18) ~ i Uz ~ n j z i Uz UUZ UUZ AXjX;nuz + BXzX;nuz Xj(AXruz) - Xj(A)Xruz + a continuous function

Xj(Fz) + a continuous function, Fz = AX;nuz.

To obtain (1.18) we have applied (1.17) to XzXruz and the fact that the commutator [Xi,Xd = e(Xi - Xz),e E Coo. Then

xZxix;n-lUZ

(XiXZ + [Xz, Xi])X;n-luz

XiXzX;n-luz + e(Xz - Xi)X;n-luz = ... X;n Xzuz + dX;nuz + Pm- I X;n fz + dX;nuz + Pm-I,

where d E Coo. The functions Xruz E CO(O), i -I- land P 171 - 1 E CO according to our assumption, Xruz E CO(O).

Certainly, A, B E Co. Put w = X;n+IUj. Then we get from (1.17), (1.18)

XjW - (b + ~fJ)w = Xj(F) + a continuous function and F = t Fz E Co. Uj Z=I,Zopi,j

One can see that the assumptions of Lemma 1. 7 hold for Y = X j , X = Xi and consequently WECo. Thus everything is proved.

Remark 1.8. In the weakly hyperbolic case we cannot write down Xj, j ~ 3 as a linear combination of the vector fields near the hyperplane t = O. Then we must use a generalization of Lemma 1.7 for 2 x 2 linear system.

22 P.R. Popivanov

We shall study now the regularity of the solution of the Cauchy problem (CWH) for the weakly hyperbolic system (WH) in a neighborhood of the jcharacteristics with j ~ 3. Let C:(Ci-), i = 1,2, ... ,n, be the outgoing (backward) characteristic through the origin. The discontinuity of a~+lUl across C1 leads to discontinuities of all components of the solution. One can find the size of these , discontinuities at the point (x, -x p ) in an explicit form as a function of x. The estimates of the size of a;:Ui' i = 1,2, ... , n, m ~ k + 2 in the zone about C1 can be prolonged in the whole domain located to the right of C 1 and afterwards we have there a direct estimate of the derivatives xla~ui' i = 1,2, ... , n by means of the expression 1 + X~+k+l-s-~.

The estimates obtained are also valid in a neighborhood of ct defined by a line l through (0,0) dividing the zero angle between ct and ct. In the remaining part of the considered trapezoidal neighborhood R of (0,0) (namely, on the left of C1 u l) we estimate similarly the derivatives X2'a~Ui' i = 1,2, ... , n.

A detailed geometrical illustration is given in Fig. 14.

, t t=o, (x)~( -x) p

"-\.

\

R

/

-h (xo,-h)

Fig. 14

x

, c;- :t=,(x)~-x p

Hypothesis HI (q), q E N. Let Os, s = 1, ... ,5 be the domains in Fig. 14. The solution U of (WH), (CWH) satisfies:

(i) U E Cq(R \ C1 \ Ct). (ii) IXla~ui I ::; const(l + x ~+k+l-s- ~) holds true in 0 1 U [22 for i = 1,2, ... ,n

provided m + s ::; q and in the domain 0 3 if in addition s ::; k + 1.

(iii) Under the assumption (ii) and ~h (0,0, u(O, 0)) -=1= 0 there exist constants U U l

cj(m, s) not all equal to 0 such that if m = Op + r, r E {I, 2, ... ,p},


e E N u {O}, then

m-(}

IX Tn!:lS ~ tjp-r l+k+l-S-e-j)1 1 U X U2 - L Cj x P

j=l

::; const (1 + x*+k+l-S-(}-~+C), C > 0 is valid in the domain [22. A similar inequality holds in [23 and the corresponding most singular part is

Tn ~ c- ·tjp-TnX-p' +k+l-s-j 'th - --I- 0 L J , WI some Cj r . j=l

Define the differential operators

Ll = Xl, L2 = OtXl, L3 = XlOtXl , ... , L 2k+2 = (OtXdk+l,

L~ = x;-(2k+2) L 2k+2 for r > 2k + 2 and i = 1 or i = 2. This is our second hypothesis.

Hypothesis H2(q). The derivatives L~uj E CO(R \ C l \ Ct) for j 2: 3 and r ::; q and satisfy the inequalities

IL~uj(x, t)l::; const(l + xk+2-~) if (x, t) E [21 U [22 U [23'

Then in [37] the validity of four statements is proved which we shall formulate as lemmas.

Lemma 1.9. Suppose Hl(q) is true. Then Hl(q + 1) holds for Ul, U2.

Lemma 1.10. The derivatives Lruj E CO(R) for all j 2: 3 and r ::; 2k + 2.

Lemma 1.11. Let q > k + 1. Then if q ::; (2k + 3)p - (k + 1), the hypothesis Hl(q) ~ H 2(q + k + 1) holds.

Lemma 1.12. HI (q), H 2(q + 1) imply HI (q + 1) for all Ui, i = 1,2, ... , n.

The maximal regularity of the component Uj, j 2: 3 near ct is given by Lemma 1.11.

In order to find a necessary and sufficient condition for the existence of anomalous singularities, we first use condition (iii) of HI (q). This way the maximal singularity of XrO~U2 is known in a neighborhood of ct ([23). Then an appropriate ODE for a (2k + 3)p + I-order derivative of Uj,j 2: 3 (the derivative being everywhere transversal to C j ) is considered and Lemma 1.7 is applied to it. The sharp estimates enable us to represent the solution of the ODE mentioned above as a sum of two functions. The first one is continuous while the second one has a jump discontinuity when crossing ct if and only if

02f o [/ (0,0, u(O, 0)) -# O. Ul U2

(1.19)

24 P.R. Popivanov

This way we can formulate the following theorem, supposing

012 ~ (0,0, u(O, 0)) =1= 0, UUl

h == 0, u~ = { ~~~~ = ~~~::~! ~ ~~~ = ~~~::~~: Cl =1= 0.

x < Xo, X -7 Xo X ~ Xo, X -7 Xo

Theorem 1.13. The [(2k+3)p+l] transversal derivatives of the components Uj,j ~

3 of the solution u of (HW), (CHW) are discontinuous across ct iff (1.19) holds.

2. New created singularities of the solutions of semilinear hyperbolic equations

2.1. Introduction

It is well known that, after the interaction of several nonlinear waves carrying some initial singularities, new singularities in comparison with the linear case can appear. Rather general results in the case of two space dimensions N = 2 were proved by Bony in [10], [11] and by Melrose-Ritter in [54]. To prove the corresponding theorems, Bony developed in the C= category the method of second microlocalization. The proof of Melrose-Ritter is also rather difficult. Our main aim in this paragraph is to prove the appearance of "new-born" singularities after the interaction of the initial travelling waves for N = 2 and under very special nonlinear conditions. Moreover, the strength of the new singularities for t > ° will be found. We shall follow here with slight modifications the paper of J. Keller-L. Ting [41]. The only new moment is the proof of the convergence of the corresponding series giving the solution inside and up to the characteristic (light) cone for the wave equation. In our opinion the proof there is simple, clear and could be useful in some other investigations. The main tool in our consideration is the explicit construction of the solution of the semilinear wave equation. To do this we introduce a special nonlinear function f(u) for which the semilinear equation

Du = f (u), 0 = 0; - ~x, x ERN, N ~ 2

becomes linear inside some simplex SN (t), ° < t < to such that Uo<t<toOSN (t) is a sum of N + 1 characteristic hyperplanes t - x.nj = 0, 1 :=; j :=; N + 1, Inj I = 1, nj, x E RN. After this we use the existence of a solution of the Goursat problem with constant data on the boundary. The theory of the hypergeometric Gauss's function enables us to find the sharp order of Holder singularity created after the collision of the initial travelling waves.

2.2. Formulation of the problem

We shall confine ourselves to the special case N = 2. Thus we consider the equation

Du = f(u),x E RN,N = 2,u = u(x,t). (2.1)

The characteristic hyperplanes of (2.1) are given by

t = x.n,n = (n 1 ,n2 ), Inl = I,N = 2. (2.2)

We assume that for t < 0 the solution u has three jump discontinuities equal to 1 on the following characteristic hyperplanes:

(2.3)

The hyperplanes (2.3) pass through the origin. We suppose that for each fixed t =1= 0 these hyperlines bound a simplex 5 3 (t) of finite volume. This simplex shrinks to o for t = 0 and then it expands for t > o. In the case N = 2 : 52 (t) is an isosceles (equilateral) triangle and this is the geometrical picture:

t t=const>O

t=const<O

Fig. 15

The unit normal vectors nj point inside 5 2 (t) for t < 0 and outside 5 2 (t) for t > 0 (see Fig. 16).

t<O t=O t>O

Fig. 16

26 P.R. Popivanov

For t < 0 we define u(x, t) by the formula

3

u(x, t) = L H(t - x.nj), j=l

where H(s) is the standard Heaviside function:

H(s) = { ~: s > 0, s < O.

(2.4)

(2.5)

Obviously, u is a distribution solution of the homogeneous wave equation for t < 0, i.e., Du = 0, t < 0 and u has unit jump discontinuities on the hyperplanes (2.3). Moreover, u = ° in 8 2(t), t < 0 and u takes values 1,2 outside 8 2(t). Geometrically, N = 2 implies the validity of the following Fig. 17, where the triangle is 8 2 (t).

u=2

u=2 u=l u=2

Fig. 17

Suppose now that t > O. Then (2.4) satisfies, in the distribution sense, Du = O. This way we conclude that the function (2.4) is a solution of (2.1) for t < 0 if

f(O) = f(l) = f(2) = 0. (2.6)

The piecewise constant function u takes values 1,2 outside 8 2 (t), t > 0, while UIS2(t) = 3, t > O. Certainly, there are two possibilities for f(u) : f(3) = 0 and f(3) =I- O. In the first case the function u, defined by (2.4) for each t, turns out to be a solution of (2.1) for all (t, x). Consequently, no new singularities in comparison with these in the linear case Du = 0 appear. So we shall assume that

f(3) =I- O. (2.7)

Then (2.4) is a solution of (2.1) outside 8 2 (t) but (2.4) is not a solution of the equation (2.1) inside S2(t), t > O. Having in mind that the function u given by (2.4) is equal to 2 on the outer boundary of 8 2 (t), t > 0 and has unit jumps across (2.3), we conclude that on the inner boundary of 8 2 (t) the function u == 3. This is the geometrical illustration:

u=l

u=2 u=2

u=l u=2 u=l

Fig. 18

This way, for the solution u of the equation (2.1) in S2(t), t > 0, we obtain the nonlinear Goursat problem

Ou = f(u), u = 3,

x E S2 ( t ), t > 0,

x E aS2 (t), t > O.

(2.8)

We seek a continuous solution of (2.8) in the domain {x E S2(t), t ::::: O}. This is the main assumption on the smooth function f:

There exists 8> 0 and such that f(u) = f(3) (2.9)

for each u for which lu - 31 < 8.

Thus, there exists to > 0 with the property lu - 31 < 8 for x E S2(t), 0 < t < to. In fact, u is continuous and u = 3 on S2(t). The nonlinear Goursat problem (2.8) is reduced for 0 < t < to to the linear Goursat problem

Ou

u

f(3), x E S2(t), 0 < t < to, 3, x E aS2 (t).

(2.10)

We shall prove the existence of a singular solution of (2.10) for t > 0, t sufficiently small. The singularities once created for 0 < t < to will propagate outward along the bicharacteristics for large t.

2.3. Singularities of the solution u of the Goursat problem on the characteristic circle

The smooth solution of the Goursat problem (2.10) will be found in the form:

t 2 t2 t2 u(x, t) = 3 + "2 f(3)[1 - w(x, t)] = 3 + "2 f(3) - "2 f(3)w.

Obviously, w will satisfy a simpler Goursat problem:

w

0, x E S2(t), t > 0,

1, x E aS2 (t), t > O.

(2.11)

(2.12)

28 P.R. Popivanov

When studying the Goursat problem with data on the characteristic (light) cone it is possible to investigate a much more general problem than (2.12), namely

w

o,x E SN(t),t > O,N 2: 2,

1,x E aSN(t), t > 0.

(2.13)

As it is well known (see [24]) the Goursat problem for the wave equation has a unique solution inside the light cone. For this reason we prefer to concentrate on (2.13) inside the characteristic cone inscribed in the simplex SN(t) and having a vertex at the origin. Certainly, in this case (2.11) takes the form

t 2 t 2 u(x, t) = N + 1 + "2 f(N + 1) - "2 f(N + l)w, (2.14)

f(O) = f(l) = ... = f(N) = 0, f(N + 1) =1= ° and the function u has N + 1 equal to 1 jump discontinuities on the characteristic hyperplanes t = x.nj, Inj I = 1, 1 ~ j ~ N + 1, nj = (nJ"'" n§"). We shall verify that w(x, t) is a homogeneous of order-zero function in (x, t), i.e., w depends on y, t > ° only. In other words,

w = w(r, <p), where r = I~I is the so-called scaled radius and <P = (<PI, ... , <PN-I) are the spherical angles defined by:

Xl = Ixl cos <PI,

X2 = Ixlsin<pl COS<P2,

... ,

XN = Ixlsin<plsin<p2'" sin<pN-l,

° ~ <Pk ~ Jr, 1 ~ k ~ N - 2, -Jr ~ <PN-l ~ Jr.

Then (2.14) becomes

t 2 Ixl u(x, t) = N + 1 + - f(N + 1)[1 - w(-, <p)],

2 t

while (2.13) can be written as:

N -1 1 (r2 - l)wrr - (2r + --)wr + 2w - "2tSNw = 0,

r r X N

rN E S (1),

) X N X w(r, <P = 1, rN E as (1), <P = N'

(2.15)

(2.16)

In the previous formula tSN stands for the Laplace-Beltrami operator on the unit sphere in R N, namely

N-l s: _ L 1 '" ( . N-j-l "!:I) - 1 UN- 'N-"-l u'P"s~n <PJU'P",ql=,

q"s~n J <po J J j=l J J

It is evident that the differential operator in (2.16) is strictly hyperbolic for r > 1 and is elliptic for 0 < r < 1. The equation is parabolic if r = 1. Thus, (2.16) is elliptic inside the characteristic cone {Ixl < t}.

This is our plan for a further treatment of the equation (2.16). We shall solve (2.16) in the case N = 2 in the hyperbolic region r > 1, i.e., outside the unit circle r = 1 and we shall study the inward propagation of the waves created by wand starting from OS2(1), where w = 1, through the simplex S2(1) till the unit circle inscribed in S2(1). This way we shall determine w(l, 'P) on the unit circle r = 1. By solving the Dirichlet problem for the elliptic equation (2.16) in the domain r < 1 and with w(l, 'P) given before, we shall find w(r, '1') inside the elliptic region. The hyperbolic-elliptic change of the type of (2.16) at the point r = 1 could give rise to some new singularities of w. Our considerations for r < 1, i.e., {Ixl < t} are valid for each dimension N ::::: 2. This way the Goursat problem with data on the characteristic cone {Ixl = t} can be investigated too. Because of this reason we give below a complete study of the Goursat problem in arbitrary space dimension N ::::: 2.

The function w(r, 'P) is smooth and even analytic for 0 < r < 1 as the equation (2.16) is elliptic there. We are looking for a solution of the equation (2.16) in the form (separation of variables):

00 kn,N

w(r, 'P) = 2..::: 2..::: an,kWn,k(r)Y~,'%('P)' (2.17) n=O k=l

where Y~~~ are the classical spherical harmonics of order n,

(n + N - 3)! y(k) = {sin n'P cos n'P}CXl kn,N = (2n + N - 2) (N _ 2)!n! ,kn,2 = 2, n,2 y'7F , y'7F n=O'

As it is known (see for example [76], [78]), the functions {y~~~} form a full orthonormal system (basis) in L2(~) and ~ is the unit sphere in RN. Moreover, y(kN) are eigenfunctions of 5 N, i.e., n,

(2.18)

An = -n(n + N - 2), n = 0, 1,2, .... In the case N = 2 : 52 = 0 2/ 0 'Pr, An = _n2. Putting (2.17) in (2.16) we obtain

(r2 - l)w~,k - (2r + N; 1) W~,k + (2 - ~~ ) Wn,k = 0, (2.19)

O<r<1. Therefore

2r2 + N - 1, 2r2 - An wn,k - r(r2 _ 1) wn,k + r2(r2 _ 1) Wn k = 0 O<r<1. (2.20)

The previous equation can be written in the form:

w" + p(r)w' + q(r)w = 0, (2.21 )

30 P.R. Popivanov

with p, q-meromorphic functions. A point at which p and q are analytic functions is called a normal one. A point is called singular if it is not normal. The points r = 0,1 are the only singular points of (2.20) in the interval [0,1]. We shall assume further on that the point r = c of the equation (2.21) is such that (r - c)p(r) and (r - c)2 q(r) are analytic near c. Then the point c is said to be regular for the equation (2.21). Suppose that c is a regular point of (2.21). Then (2.21) can be written as well as:

(r - c)2 f(r)w" + (r - c)g(r)w' + h(r)w = 0 (2.22)

and f, g, h are analytic functions near r = c. According to [80], [62] the algebraic equation

r(r - l)f(c) + rg(c) + h(c) = 0 (2.23)

is called an indicial equation of (2.22) at r = c. Let rl, r2 be the roots of (2.23) for which d = rl - r2 2': O. Then each solution

of (2.22) near r = c is given by (see [80], [62])

w = AWl + BW2, A, B = const., (2.24)

where

WI = (r - cr1 (1 + ~ ck(r - C)k) , (2.25)

the power series being convergent near r = c and

{ (r - cr2(1 + L~=l c'k(r - c)k),if d rt. z

W2 = wllog(r - c) + (r - cr2 L~=l dk(r - c)k,if d = 0 const.wllog(r - c) + (r - Cr2(-~ + L~=l dk(r - c)k),if dEN.

(2.26)

The power series above are convergent near r = c. Our equation (2.20) has the form

r2(r -1)2(r + l)w~,k - r(r - 1)(2r2 + N - l)w~,k

+(r - 1)(2r2 - An)Wn,k = O.

Its indicial equation at the regular point r = 0 is

r(r - 1) + (N - l)r + An = 0,

(2.27)

(2.28)

as (2.27) coincides with (2.22) for f(r) = (r-1)2(r+1), g(r) = -(r-1)(2r2+N-1), h(r) = (r - 1)(2r2 - An). The roots of (2.28) are rl = n, r2 = 2 - n - N; d = rl - r2 = N + 2n - 2> 0, i.e., dEN.

According to (2.25), (2.26) we have that:

WI = rn (1 + ~ ckrk) , (2.29)

( 00 ) 2-n-N 1 k W2 = CI WI log r + r - d + L dk r .

k=l (2.30)

Thus (2.27) possesses only one regular solution near r = 0, namely WI. The indicial equation of (2.27) at the regular point r = 1 is

2r(r - 1) - (N + l)r = 0

and its roots are: rl = N:i3, r2 = 0, d = N:i3 > O. Applying the asymptotic formulas (2.25), (2.26) we obtain the following solutions:

W3 = (1 - r(i3 (1 + ~ ck(l - r)k) , (2.31)

{I + L:~I cZ(l - r)k, for N - even,

W4 = const.w3Iog(1 - r) +(-2 + L:~=I dk(l- r)k),for N - odd.

(2.32)

We point out that the singularities of the solutions of the equation (2.27) at the regular point r = 1 do not depend on the summation index n.

We shall solve (2.19) by an explicit formula. To do this put

z = r2, Wn.k = zn/2 F(z) =} wn,k(r) = rn F(r2). (2.33)

After this change (2.19) is transformed into:

z(z -l)F" (z) - [n + ~ - (n -~) z] F' (z) + (n - 2~(n -1) F(z) = O. (2.34)

This is just the hypergeometric equation (see [39], [62], [77])

z(z - l)F" - [c - (a + b + l)zlF' + abF = 0 (2.35)

with parameters a = n;-2, b = n;-I, C = n + If. The hypergeometric function F(a, b, c, z) is the regular solution at z = 0 of (2.35). We remind the reader that (see [28], [62], [77]):

ab a(a + l)b(b + 1) 1 + - z + ---'---:----'---,:----:----'-

c c(c + 1)2! (2.36) F(a, b, c, z)

+ a(a + l)(a + 2)b(b + l)(b + 2) ---'------'----'--~---'----:-:-':-'------'- + ...

c(c + l)(c + 2)3!

fCc) ~f(a+j)r(b+j) j I I L...t ---=--;----:--:--- z , Z < 1 f(a)r(b) j=O r(c + j)j!

and f is the famous Euler's gamma function. The power series (2.36) is absolutely convergent for Izl < 1. Moreover, c -

a - b = N:i3 > 0 implies that (2.36) is absolutely convergent for Izl = 1. Assume that n = O. Then a = -1 and according to (2.36) and (2.33) we get

that wO,k(r) = Ao(N + r2). If n = 1, then b = 0 and therefore wI,k(r) = AIr. The assumption n = 2 =} a = 0 and then W2,k = A2r2. We observe that n 2': 3 =} a > 0, b > 0, c > O.

32 P.R. Popivanov

Our next step is to investigate the behavior of the derivatives of Wn,k near the point r = 1. Thus

dPwn,k (r) drP drP

r(c) .f-, r(a + j)r(b + j) f(a)r(b) ~ f(c + j)j!

x (n + 2j)(n + 2j - 1) ... (n + 2j - p + 1)rn +2j - p

00

~ n+2j-p ~an+2j-pr .

j=O

(2.37)

The coefficients a n +2j-p 2 0 for n 2 3, j 2 1. According to the Gauss criterion [28] the series L~o aj, aj 2 0 is convergent (divergent) if the parameter J1 in the representation ....'!:.L = 1 + i!:. + O( 12 ) is > 1« 1).

aj+l J J -

The power series (2.37) is convergent for r < 1 and is divergent for r > 1 as the d'Alembert criterion shows. On the other hand, r = 1 gives:

a n +2j-p

a n +2j+2-p

(1 + 1 )(1 + £)(1 + 2+2n-p )(1 + n-2P+1) J J J J

(1 + £)( 1 + £) (1 + 22+n) (1 + n2+ 1 ) J J J J

1-p+c-a-b 1 1 + . + O(~).

J J

In fact, l~Q = 1 - 'J + 1~2Q. /2. J J

Applying Gauss's criterion with J1 = 1 - p + c - a - b = 1 - P + Ni3, we conclude that the power series (2.37) is convergent at r = 1 for p < Ni3 and is divergent for p 2 Ni 3 . This way we have that the regular solutions Wn,k of (2.19) at r = 0, n 2 3 possess singular derivatives of order Ni 3 at the point r = 1. Going back to the old variables x, t and combining (2.15), (2.29), (2.30), (2.31), (2.32), (2.37) we come to the following theorem.

Theorem 2.1. Suppose that u(x, t) satisfies the Goursat problem:

Ou = f(N + 1), x E SN (t), t > 0,

u = N + 1,x E 8SN (t),t > o. Then this is the behavior of u(x, t) within the characteristic sphere {Ixl = t}:

u(x, t) = N + 1 + t 2 f(N + l)[h( l:J, if) + (1 - El (i 3 g( El, if)], (2.38) t t t

for N even, Ixl < t, Ixl -> t,

u(x, t) = N + 1 +t2 f(N + l)[h( l:J, if) + (1- El) Ni3 log(l- El )g( l:J, if)], (2.39) t t t t

for N odd, Ixl < t, Ixl ---> t.

The function h(r, cp) has Ni4, respectively Ni3 derivatives with respect to rat r = 1, while g(r, cp) is a bounded function.

Our next step is to determine the functions h, 9 outside the sphere Ixl = t, i.e., for Ixl > t. To do this we shall investigate u outside this sphere (circle), i.e., in the "hyperbolic region" inside S2(t) but outside {Ixl = t}. So we are working again in the case N = 2.

2.4. Behavior of u outside of the characteristic circle

Then for each fixed t > 0, S2(t) is an isosceles (equilateral) triangle (see Fig. 19).

Fig. 19

Thus S2(t) = 6.ABC and the characteristic sphere coincides with the inscribed circle in 6.ABC with center at 0] and radius t > O. We shall find the solution u of the problem (2.10) in an explicit form at each curvilinear triangle outside the inscribed circle. To fix the ideas we shall only consider the case of the curvilinear triangle T BT' in dashed lines in Fig. 19 (see Fig. 20):

We choose a new orthogonal coordinate system 0] zy and the axis 0 1 B = 0] Z

is collinear to the bisectrix O]B. Denote 0'= L.AB01 = L.CBOI and ~ = sinO' > o =} m > 1. Thus OIB = tm and therefore B = (tm,O) in the new coordinate system. From the triangle 01QT' we have that OlQ = -/';;, T' Q = -/';;vm2 - 1 and

consequently T' = (-/';;, --/';;vm2 -1) in the new coordinate system. Obviously,

T = (-/';;, --/';;vm2 - 1). Then the straight lines BT', BT are given by the following equations:

I mt - z mt - z BT : y = BT : y = - -r===;:;===:=

,.1m2 -I' vm2 -1

The points located inside the angle L.T BT' satisfy the inequality

Iyl < (m 2 - 1)-1/2(mt - z).

34

A

P.R. Popivanov

c

T

Fig. 20

B ......... z

Denote by v the unit vector collinear to 01Z, v = (Vl,V2) in the old coordinates and by ly = (l~, l~) the unit vector collinear to 01Y' Then z = v.x = VIXI + V2X2, Y = ly.x = I~Xl + l~x2' As the Laplace operator ~ is invariant under rotations, we rewrite the Goursat problem (2.10) in L.T BT' in the form:

Utt - Uzz - Uyy

U

!(3), Iyl < (m2 - 1)-1/2(mt - z),

3, Iyl = (m2 - 1)-1/2(mt - z). (2.40)

The previous Goursat problem certainly has a solution. We shall find it by a simple formula. To show this fact we make the following change of the variable: 7 = (m2 - 1)-1/2(mt - z), seeking U = U(7, y). Then the problem (2.40) becomes

U TT - U yy

U(7,y) !(3), Iyl < 7, 3,lyl =7.

(2.41)

This is again a Goursat problem in the plane. After a standard change of variables 71 = 7 - y, Yl = 7 + y, 71 > 0, Yl > 0, we come to the Goursat problem

1 "4!(3),71 > O,Yl > 0, (2.42)

u(O,yd U(71 , 0) = 3.

Its unique solution is U(71 , yd = 3 + %:!(3)71Yl and consequently U(7, y) = 3 + %:!(3)(72 _y2), Iyl < 7. Going back to the old variables (t,x) we find the solution u(x, t) of the Goursat problem (2.40) in the closed curvilinear triangle BTT':

( t) = 3 ~!(3) [(mt - v.x)2 _ 2] U X, + 4 m2 _ 1 y, (2.43)

y = ly.x, ly being a constant vector. In a similar way we can find u in the curvilinear triangles ATT", T" CT' outside and on the inscribed circle Ixl = t. Inside the inscribed circle we are seeking u in the form (2.15), namely

(2.44)

Then

t 2 3 + "2 f(3) [1 - w (I, 'P)] (2.45)

1 2 ((m-v.w)2 )2) 3 + 4f(3)lxl m 2 -1 - (ly,w ,

w (cos 'P, sin 'P), -Jr ::; 'P :s Jr.

So w(l,'P) = 1- ~(:,-;,,-wt - (ly,W)2). After this we expand w(r, 'P) in the Fourier series (2.17) and we repeat in the

case N = 2 the analysis fulfilled in Section 2.3. Certainly, an,k.Wn,k are the Fourier coefficients of the function w(l, 'P). Applying Theorem 2.1 we see that the function u(x, t) can be represented inside and up to the characteristic circle Ixl = t as:

u(x, t) = 3 + t 2 f(3) [h (I~I ,'P ) + (1 - I~I) 5/2 9 (I~I , 'P) 1 ' (2.46)

Ixl < t, Ixl -+ t,

where h(r, 'P) has three order derivatives with respect to r for r :s 1 and g(r, 'P) is bounded function.

To be more precise, we shall carefully study the behavior of the series w( r, 'P) (2.17) for r < 1 and with Dirichlet data w (1, 'P) on the boundary r = l. This way we are dealing with the Coursat type characteristic problem for the wave equation in the light cone {Ixl < t}.

At first we expand w(l, 'P) in L2('L.) via the orthonormal system (basis) Y~:%, namely:

00 kn,N

w(l, 'P) = L L Pn,kY~~Jv('P)'Pn,k = (W(l, 'P), Y~~Jv('P)) L2(~) .

n=O k=l

Certainly, w(l, 'P) E L2('L.) as it is continuous. Having in mind (2.17) we obtain that

(2.47)

where

W k (r) = r F - - 1 - - - n + - r 0 < r < 1 n (n n 1 N 2) n, 2' 2 2' 2 ' ,- - , (2.48)

wn,k(l) = FG - I, ~ - ~,n + I¥-, 1).

36 P.R. Popivanov

Estimates guaranteeing the existence of a solution of the Goursat problem in the light cone with data at {Ixl = t} are given below.

Lemma 2.2. A ssume that F (~ - 1, ~ - ~, n + 1¥- ' x), 0 ::; x ::; 1 is a hypergeometric function, n > 2, N ~ 2 and 0 ::; x ::; Xo < 1, with Xo being positive. Then there exists a constant C(N,xo), C independent ofn and such that

n n 1 N o < F("2-1'"2-2,n+2,x)

< C(N,xo).2nn l / 2 (1-~) n,

o ::; x ::; Xo < 1.

Conclusion: 0::; wn,k(r) ::; C(N,xo)2nn l /2 (l-~) n, 0::; r ::;""0 < 1,""0> O.

Having in mind the fact that if A(X) = l-V;=X, 0 < x ::; 1, we have: A(X) > 0

for 0 < x < 1, A' (x) > 0 for 0 < x < 1, A(X)x--->o ----+ 1/2 and therefore 0 < A(X) ::; A(XO) < 1, 0 ::; x ::; Xo < 1. Thus the series (2.47) is absolutely and uniformly convergent for 0 ::; r < 1. The only new feature in verifying this fact is the estimate for Wn,k (1) = F( ~ - 1, ~ - ~, n + 1¥-, 1) because IPn,k I ::; const.

As it is shown in [62]:

F b f(c)f(c - a - b) f(n + 1¥-)r( 1¥- + ~) (a, ,c,l)= r(c-a)f(c-b) = r(~+1¥-+1)r(~+1¥-+~)

and according to the famous Stirling formula rex) r-.o e-XxXe;)-1/2(1 + 0(1)), x ----+ 00 we conclude that

1 = 2-n-N-!n~+1 (1 + 0 (~)) , n ----+ 00. wn ,k(l) n

Proof of Lemma 2.2. The well-known integral representation for the hypergeometric functions [62], 0 ::; x < 1:

fCc) rl t b- l (l _ t)c-b-l F(a, b, c, x) = f(b)f(c _ b) 10 (1 _ tx)a dt,

c> b > 0,

a = ~ - 1, b = ~ - ~, c = n + 1¥-, n > 2 will be used. Put

(~ _ ~ _ ~ N) _ rl t'i-~ (1 - t)'i+~-! J 2 1, 2 2,n+ 2'x - 10 (1-tx)n/2-l dt

1 n-2

= r C l / 2 (1_ t)~+! {t(l- t) }-2 dt, n> 2. 10 1 - tx

Consider now the function <p(x;t) = ti~!), {O ::; x ::; Xo < 1,0 ::; t ::; 1}. As

<p~ = \~~1;,)f one can easily see that <p~ = 0 {=} 0 < tl(X) = l-V;=X < 1

and rp reaches its maximum at t = t1(X), i.e., 'P(x,t) ::::; 'P(x,h(x)) = e~~)2, o ::; t ::; 1.

Therefore, ;:;---:: n ~ 2

( n n 1 N) (I-vI-X) J - - 1 - - ~ n + - x < const(N) 2 '2 2' 2' - x'

n > 2,0 ::; x ::; xo < 1.

Applying again the Stirling formula we see that

r(n +!'!..) __ -----,,---'---_----"'2--'---,-,~____c__ < const ( N) 2 n n 1 / 2

r('i - ~)r('i + If +~) -which completes the proof of Lemma 2.2.

Our next step is to prove that the series (2.47) is termwise differentiable with respect to (r, 'P), 0 ::::; r < 1 infinitely many times. There are no problems in differentiation with respect to 'P [78].

Having in mind that dP(r~::(r2)) is a linear combination of terms of the type 2Prp+nF(p) (r2), ... ,n(n - 1) ... (n - p + 1)rn~PF(r2), we see that the termwise differentiability will follow from

Lemma 2.3. Denote Fn(x) = F('i - 1, 'i - ~,n + If,x), n 2: 3. Then for each pEN there exists a constant C (N, p, xo) such that

o ::; x ::; Xo < 1.

Proo]. As it is known from [62]:

dPF(a,b,c,x) () () ( ) ----'---,--------'-- = a P b pF a + p, b + p, c + p , dx p

P E N,n 2: 3,

N 2 h () - ( 1) ( 1) - r(a+p) (b) - ['(Hp) 2: ,were a p - a a + ... a + p - - ~, p - ~.

Put

So:

dPF(a,b,c,x) = r(a+p)r(c+p). [ltHP~l(l-t)C~b~ldt dxp r(a)r(b)r(c - b) Jo (1 - tx)a+p .

J(a,b,c,x) 11 tHp~l(l- t)c~b~l -----;----'-~--'--:---dt

o (1 - tx)a+p

[1 t(1 - t) ~~1. tP~1/2(1 - t)1f+~ dt Jo (l-tx) (l-tx)p ,

38 P.R. Popivanov

As in Lemma 2.2 we obtain

J < (1- vT=X)n-2 r1 t p - 1/2(1 - t)!f+~ dt x io (l-x)p

VT="X n-2 < const(N,p)(l-x)_p(l- :-x) ,n~3,

o ::; x ::; Xo < l. i.e., J::; const(N,xo,p)C- v;7a=XO)n, n ~ 3, 0::; x::; Xo < l.

According to the Stirling formula

r(n+~+p)r(~+p-1) 1 -------:--'-------==:-----'-;--:-'-''':-----;;-;c----'------;-:- ::; const(N,p)2n n 2p+z r(~ - 1)r(~ - ~)r(~ + ~ + ~)

and thus Lemma 2.3 is proved. Obviously, w(r, cp) E COO(r < 1), i.e., inside the light cone Ixl < t and for

arbitrary L2 data on {Ixl = t}. Even Coo smoothness of the boundary data on {Ixl = t} does not imply smoothness of w up to the boundary. In fact, assume that the sum defining w(l,cp) is finite. Then (2.47) is a finite sum too but the functions wn,k(r) have singularity at r = 1 of order Nt3 (see (2.37)).

As we know u(x, t) is smooth inside and up to the boundary of the three curvilinear triangles and inside the inscribed circle Ixl = t. Assume now that the three different smooth solutions of the type (2.43) could be extended inside the characteristic circle Ixl = t. They are certainly unequal in the set {Ixl < t} because they contain different combinations of the variables (x, t). Moreover, the solution inside {Ixl < t} is uniquely determined.

Conclusion: The solution u has a singularity from inside on the characteristic circle Ixl = t. So g(r, <p) :j:. 0 and the strength of this singularity is (1 - I~I )5/2, Ixl < t, Ixl ---+ t. There are no other singularities outside and inside {Ixl = t}.

2.5. A microlocal approach to the propagation of singularities. Concluding remarks

Following Beals [6] a simpler proof of the smoothness of the solution of the semilinear wave equation inside the light cone can be proposed. Thus put

cPu n- 1 82 u Du = 8 2 - L 8 2 = f(x,u), (2.49)

xn j=l Xj

x ERn, f E c oo (Rn+1)

and suppose that u E H1oc(Rn), s > ~. The radial vector field M is defined as follows:

n 8 M= LXj8x.'

j=l ]

(2.50)

Lemma 2.4. The real-valued function u E Htoc(Rn) satisfies (2.49) and Mju E HS (Rn) 'or all)' Then u E COO({x'2 < x2}nRn) where R n = {x E R n . x > loc - J" n + , ± . n «)O} and x' = (xI, ... ,xn-d.

Proof. Obviously, the commutator [0, MJ = 20. Inductively we shall prove that Mju E Htoc(Rn) , Vj E Z+. The case j = 0 is obvious. So assume that Mtu E Htoc(Rn) for some j E Z+ and each 0 ::; t ::; j and that O(Mju) = fj(x, u, Mu, ... , Mju), the functions fj being Coo smooth with respect to their arguments. It is well known that hex, u, Mu, ... ,Mju) E Htoc(Rn). Then

O(Mj+lu) = O(M.Mju) = MO(Mju) + [0, M](Mju)

= M hex, u, Mu, ... , Mju) + 20(Mju)

= Mfj(x, u,Mu, ... , Mju) + 2fj(x,u,Mu, ... ,Mju) == AHI .

Thus O(Mj+IU) E Hs-I(Rn) MHI E Hs-I(Rn) and Mj+l E HS (Rn) , loc' loc loc -' According to the linear version for propagation of singularities of the wave

equation we conclude that Mj+IU E Htoc(Rn). Moreover, then the chain rule holds, i.e.,

. _ . j+l . 00 AJ+I-fJ+I(X,u,Mu, ... ,M U),fJ+IEC. In fact, M(J(x,u,v)) = (MJ)(x,u,v)+ fu(x,u,v)Mu+ fv(x,u,v)Mv, U,v E Htoc'

Assume now that (xo,';o) E WF(u) and Xo i=- O. Then the linear form (xo, ';0) = O. To prove this fact suppose that (xo, ';0) i=- O. As (x,';) is up to a nonvanishing factor the symbol of the operator M and Mju E HtocCRn ) , Vj E Z+, we see that u E H:::cl (xo, ';0), i.e., we come to a contradiction. Define now

K = {x E R n : X'2 < x~ } ,oK = {x E R n : X'2 = x~ } .

Assume that the point 0 i=- Xo E K n R+.. Obviously, (xo,';o) tJ. WF(u) ::::} u E H:::cl (xo, ';0). Thus consider the case (xo,';o) E W F ( u). Then we know that (xo, ';0) = O. Having in mind the fact that KnR+. is a proper cone with an opening ~ we conclude that ';0 tJ. oK, i.e., (';~)2 i=- ((0)2. The wave operator 0 is microelliptic at the point (xo,';o) and therefore u E H':n~;(xo,';o) as Ou E Htoc(Rn). So u E Hs+2(xo), i.e., u E Hto~2(K n R+.). Standard bootstrap arguments prove Lemma 2.4.

Definition 2.5. The function u E Htoc is said to be conormal with respect to the smooth hyperspace ~ eRn if for each choice of smooth, tangential to ~ vector fields M I , ... , M j , j E Z+ the following relation holds:

MIM2 ... Mju E Hloc.

If the hypersurface ~ is given locally by {Xl = O}, the vector fields M j are db a a a generate Y Xl aXl ' aX2 ' ... , aXn .

Denote by N*~ the conormal bundle over~. Then WF(u) c N*"E,. In fact, V'cp(x) is the normal vector at the point x E ~ to the hypersurface ~ = {cp(x) = O}. Suppose that the covector 0 i=-'; is not collinear to V'cp(x) at x. Then there exists

40 P.R. Popivanov

a vector field M j E T(E) such that (~, Mj(x)) i= 0. The microellipticity of M j at (x,~) implies our inclusion.

In a similar way one can define conormality of the function u with respect to several hypersurfaces (see [11], [13]).

Consider now the hyperplanes Ew : Xn = w.x', x = (x', x n ), Iwl = 1. Obviously, they are characteristics for D as they are tangential to oK. Moreover, the radial vector field M(x), x i= ° is tangential to oK and to the characteristic hyperplanes Ew.

Corollary 2.6. Let u E Htoc(R3), S > ~ be a solution of (2.49) with n = 3 and u be conormal with respect to E = (EI' E 2, E 3 ) in R~ where E is a family of three characteristic hyperplanes intersecting transversally at the origin. Then

u E Coo (R3 \ (EI U E2 U E3) U {X3 = V xi + x§ } ) .

Proof. We omit the more delicate part of the proof, namely u E COO(R3 \ (EI U E 2 UE3 )U{ Jxi + x~ ::; X3}) (see [10] ). Having in mind that the radial vector field is tangential to E we have according to the definition of conormality that Mj U E

Htoc(R~), Vj E Z+. Applying Lemma 2.4 we see that U E COO(X3 > Jxi + x~). Thus everything is proved.

3. Appendix I. Blow up of the solutions of the Cauchy problem for quasi-linear hyperbolic systems in the plane

We propose in this Appendix a blow up result for the solutions of the Cauchy problem for some classes of quasi-linear systems in the plane. This is another type of singularities when the solutions remain bounded but their gradients blow up in a finite time. Certainly, there are a lot of papers on the subject but we shall mention only the fundamental papers of P. Lax [45] and F. John [33] as well as the monographs of L. Hormander [31], Li Ta-tsien [48] and S. Alinhac [1]. Interesting theorems were also proved by Klainerman-Majda [42], Colombini-del Santo [20] and R. Manfrin [52]. Below we apply our result to the nonlinear vibrating string equation and we show that in many cases an explosion of the gradient of its solutions takes place. Our main theorem generalizes some results from [48].

3.1. Statement of the problem and main results

We study here hyperbolic systems of the type:

or or ot +.A(r,s)ox os os ot +f-l(r,s)ox

rlt=o = ro(x), slt=o = so(x),

0, (3.1)

0, t 2: 0,

'\'(r,s) < Il-(r,s), '\',Il- E c1, assuming the Cauchy data ro,so E CI(RI) to possess sufficiently small CO norms. This way blow up of r x , Sx for a finite time T > 0 is possible. We shall consider the following case:

() O'\' > 0' Oil- > 0 A or - ,ro::::: 0, 8s - .

This is our main result.

Theorem 3.1. Assume that under condition (A) the next condition (B) is fulfilled:

J+= Oil-(B) os (ro(;3), so(Q)) d(3 = +00

and the point Q E RI is such that s~(Q) < O. Then Sx blows up in a finite time T > 0 supposing the uniform norms of the

Cauchy data ro, So to be sufficiently small.

Without loss of generality we can assume that '\'(0,0) < Il-(O,O) in the formulation of Theorem 3.1. The integral in (B) is taken from an arbitrary finite ;30 till +00 as the function ~~(ro(;3),so(Q)) is integrable on each compact interval ;3 E [-N,N].

Put r = ETO(X), S = ESo(x), 0 < 10 :::; Eo < 1 and consider the system (3.1) with Cauchy data r, s. Obviously: -00 < roe -(0) :::; roe +(0) < +00.

Corollary 3.2. Assume that ~(Ero(+oo),ESO(Q)) > 0, Il-~ ::::: 0, s~(Q) < 0 . Then Sx blows up for a finite time if EO is sufficiently small.

We shall illustrate the case ~(Ero( +(0), ESo(Q)) = 0 by some examples.

Corollary 3.3. Assume that Il-s = r2 L~j=o aijri sj, aij = const., L~j=o aijri sj ::::: O. Then it follows immediately that aoo ::::: 0 and if aoo = 0 we have that alO =

. , _ 2 2 n i+j i j aOI = O. Evzdently, Il-s(ErO,Eso) - 10 ro Li,j=o aijE roso-Suppose that

() ",n i+ j i ( ) j ( ) 1) ro +00 =I 0, ~i,j=O aijE ro +00 So Q > 0 for some Q. Then Sx has a blow up for a finite time.

2) ro(+oo) = O. Then condition (B) is reduced to

J+= T5 (;3)d;3 = +00

if aoo > O. Then Sx blows up in finite time.

We shall apply now Theorem 3.1 to the study of the nonlinear vibrating string equation. We remind the reader that the Cauchy problem for this equation is written in the form

Utt - (K(ux))x = 0, t::::: 0,

ult=o = uo(x), utlt=o = UI(X),

(3.2)

42 P.R. Popivanov

The functions K E C 2 (Rl), uo(x) E C 2 (Rl), Ul E Cl(Rl) and the uniform norms Ilulllco(Rl), Iluollcl(R) are supposed to be sufficiently small. Moreover, we assume that

K(O) = 0, K' > ° everywhere, K" ~ ° everywhere. (3.3)

Define now the functions

'o(x) ~ ~ (u,(X) + [~ JK'(A)dA) , (3.4)

ro(x) ~ ~ (u,(X)- t VK'(A)dA)'

Let v = G(v) E C 2 be the inverse function of the strictly monotonically increasing function v = Jov J K' ()..) d)" E C 2 (Rl ).

We formulate our second result.

Theorem 3.4. Consider the Cauchy problem (3.2) with sufficiently small uniform norms IlulllcO(Rl), Iluollcl(R) and suppose that s~ 2: 0, r~(3) < ° at some point (3 E Rl and

I J+CXl K" (B) y(G( -ro((3) + so(a)))da = -00.

Then the classical solution u(t, x) of (3.2) will develop a singularity in a finite time.

Obviously, the condition (B') is rather implicit. To simplify it we assume that ( C) K' > ° and K" ~ ° is not identically zero on any open interval of the

real line.

Corollary 3.S. Consider the Cauchy problem (3.2) in [0, +(0) x Rx and suppose that the condition (C) holds, the function so(x) given by (3.4) is monotonically increasing and ro(x) is not monotonically increasing. Then the classical solution u(t,x) of (3.2) with sufficiently small norms Ilulllco(Rl), Iluollcl(R) will develop a singularity in a finite time T > 0.

3.2. Proof of Theorem 3.1 We consider the system (3.1). We shall investigate also systems with Cauchy data r = ETO(X), s = cso(x), 0< c ~ co, co sufficiently small. The system is hyperbolic if )..(r, s) < M(r, s) everywhere. If we are to work with the small Cauchy data, then we shall assume that )..(r, s), M(r, s) are defined in a small neighborhood of (0,0). When studying the blow up we can suppose without loss of generality that )..(0,0) = -1, M(O,O) = +1. We shall argue by contradiction. Thus let r, s be C l

solution of (3.1) on [0, +(0) x Rx. We introduce the characteristic curves of the solutions r, s respectively:

I { d:t, (t, (3) = )..(ro((3), set, Xl (t, (3))), Xl (0, (3) = (3 E Rl,

(in fact, r( t, Xl (t, (3)) == ro ((3)).

II { d:it2 (t, a) = f-t(r(t, X2(t, a)), so(a)), X2(0, a) = a E Rl,

(in fact, s(t, X2(t, a)) == so(a)). So we conclude: Ilrlleo = Ilrolleo, IIslleo = Iisolleo. The curve I is called backward characteristic. The curve II is called forward characteristic. If we put r instead of Ero and 8 instead of ESo we see that IWlleo, 11811eo :::;

const.Eo, i.e., A, f-t are well defined for EO sufficiently small. Moreover, if IAI :::; EO, If-tl :::; Eo, then -3/2 :::; A(r, s) :::; -1/2, 1/2 :::; f-t(r, s) :::; 3/2. This way we can estimate for each 0 < 10 :::; EO the functions Xl (t, (3), X2(t, a) in the following way:

(3 - 3/2t :::; XI(t,(3):::; (3 -1/2t, Vt:::: 0,

a + 1/2t :::; X2(t, a) :::; a + 3/2t, Vt :::: O.

(3.5)

Evidently, Xl (t, (3) is a strictly monotonically decreasing function with respect to t :::: 0 for (3 fixed and X2 (t, a) is a strictly monotonically increasing function with respect to t :::: 0 for a fixed. For each (3 we denote by a(t, (3) the unique solution of the equation S(t,XI(t,(3)) = so(a(t,(3))). Below we propose the geometrical interpretation.

From geometrical reasons it is clear that

a(t, (3) :::; (3, Vt :::: 0, V(3 E RI,

a(O, (3) = (3, V(3 E R\

a(t',(3) < a(t,(3) if t' > t, i.e., ~~ (t, (3) :::; O. So the smooth function a(t, (3) is monotonically decreasing with respect to t for each fixed (3.

Having in mind (3.5) we have

a(t, (3) :::; Xl (t, (3) :::; (3 - 1/2t,

i.e., a(t, (3)t~+oo ....... -00, (3 fixed. The relation (3.6) implies that

0:::; t:::; 2((3 - a(t, (3)).

(3.6)

(3.7)

We shall find now an expression for ~~ (t, (3). Having in mind the fact that

(3.8)

(see Fig. 21) we get:

aX2 aX2 aa aXI at (t, a(t, (3)) + aa (t, a(t, (3)). at (t, (3) = at (t, (3),

aX2 at (t, a(t, (3)) = f-t(ro((3(t, a)), so(a)),

ax] at (t, (3) = A(ro((3), so(a(t, (3))),

44 P.R. Popivanov

t

II,s

II,s

I,r

(0, a(t', fJ)) (0, a( t, fJ)) (0, fJ) x

Fig. 21

where fJ(t, a) is the unique solution of the equation

r(t, X2(t, a)) = ro(fJ(t, a)), Vt ~ 0, Va E RI.

Thus (ro(fJ), so(a(t, fJ))) = (ro(fJ(t, a)), so(a)) and therefore ~; ~~ = .x - jJ, < 0. As ~~ ::; ° we conclude that ~~ < ° and therefore there exists a smooth inverse function t(a, fJ), :J! < ° for each fJ fixed.

In the same way we investigate the smooth function fJ(t, a), (see Fig. 22). The equality

XI(t,fJ(t,a)) = x2(t,a)

for each fixed a; t' > t implies fJ(t', a) > fJ(t,a), i.e., fJ(t,a) is monotonically increasing with respect to t ( a;: ~ 0). Moreover,

fJ(O, a) = a, Va E RI,

fJ(t, a) ~ X2(t, a) ~ a + 1j2t, fJ(t, a) ~ a, Va.

Thus fJ(t, ak-->+oo ----> +00, a fixed,

0::; t ::; 2(fJ(t, a) - a)). (3.9)


t

II,s I,r

(0, Q;) (0, ,6(t, a))

Fig. 22

From the identity xI(t,/3(t,ex)) == X2(t,ex) we get

{)xl ()xl af3 aX2 at + af3' at = at .

As in the previous identity>., f..l are evaluated at the same point we get that ~ > 0 for each fixed ex, i.e., there exists a smooth inverse function t(f3, ex), g~ > O.

As it is known from [48,51]

or I ehl (TO ((3) ,So (/3))- h, (TO ((3) ,s( t,XI (t ,(3)))

aX(t,XI(t,f3)) = ro(f3) A I (t,f3) ,

where

46 P.R. Popivanov

So: ar I eh, (ro({3),so({3»-h, (ro({3),so(a(t,{3»)

ax(t,xI(t,{3))=ro({3) AI(t,,B) ,(3.10)

t a).. Al (t, (3) 1 + r~({3) 10 ar (ro({3) , so( a( T, (3) )eh1 (ro({3),so({3»

X e-h, (ro({3),so(a(T,{3»)dT.

Remark 3.6. Al (t, (3) = 0 for some t, (3 implies r~({3) :I 0, as r~({3) O::::} Al (t, (3) = 1.

In the same way we have that as I eh2 (ro(a),so(a»-h2(ro({3(t,a»,so(a» -a (t,x2(t,a)) = So (a) A ( ) ,(3.11) x 2 t,a

ah2 _ 1 a/-l h ( ) _ r /-lr(-r, s)d"( ar - /-l-).. ar::::} 2 r, s - 10 (/-l- )..)(-r, s)'

Remark 3.7. Put in the expressions containing h2 fo instead of cro and So instead of CSo. Having in mind that Ilfolico :::; c.Co < 1, c = const. > 0, 11801lco :::; c.Co < 1 we conclude that one can find a constant k(co) > 0 such that

1 k 1 k

k>1.

(3.12)

We remind the reader that according to our condition (A) Al (t, (3) ~ 1 and therefore Irxl :::; const., Irtl :::; const. We shall impose conditions on So leading to blow up of Sx for a finite time T > O.

Consider the integral

ta/-l J(t, a) = 10 as (ro({3(T, a)), So (a)))-r({3(T, a))dT, (3.13)

where "(({3(T, a)) = eh2(ro(a),so(a»-h2(ro({3(T,a»,so(a». For a fixed a E Rl we make in (3.13) the change of the variable {3 = (3(T,a)

::::} d{3 = ~~dT ::::} dT = ~~d{3 and T({3, a) is the inverse function of {3(T, a). So we get

as (3(0, a) = a. We know that a{3 = f..L->. aT _ W-(T,{3(T,a»

aT aXl (fO( »::::} a{3 - p.->. ti73 T,,..,, T,a:

But 1 ~ /1 - A ~ 3 =}

1 aXl aT aXl :3 a(3 :=:; a(3 «(3,0:) ~ a(3 (T, (J). (3.14)

On the other hand, according to [48,51]

OXI (t <5) = A (t <5)ehl(ro(5),so(c>(t,5))-hl(ro(5),so(5)) a<5' 1,

(3.15)

and Xl (t, <5) is a solution of (I) with <5 instead of (3. Therefore aaxi (t,<5) 2: -1;:, Vt 2: 0, as A l (t,<5) 2: 1 and

1(3(t'c»a/1 ~xJ(T,(3) I(t, 0:) c> as (ro«(3), so(o:)h«(3) /1 _ A d(3 (3.16)

1 1(3(t,c» a/1 > 3k2 c> as (ro «(J) , So (o:))d(3.

Fix 0: and let t --+ +00 =}

. 1 1+00 a/1 hmt->+ox:J(t,o:) 2: 3k2 c> as (ro«(3), So (o:))d(3 = +00.

Having in mind that

a 1(3(t,a) 0/1 a(3 a/1 at a as (ro«(3) , So (o:))d(3 = at' as (ro«(3(t, 0:), so(o:)) 2: 0,

we conclude that the previous integral is monotonically increasing with respect to t.

We see that our condition (B) implies that I(t, 0:) /t->+oo +00. Consider now the continuous function A 2 (t, 0:). We have that A 2 (0, 0:) = 1,

A2(t,0:)t->+00 --+ -00. Having in mind that (3.13) is monotonically increasing we conclude that there exists a unique point to(o:) > ° such that A 2(t,0:) > 0, ° ~ t < to, A2(to, 0:) = ° and therefore there exists sx(t, X2(t, 0:)) for ° ~ t < to, while limt->to-O sx(t, X2(t, 0:)) = -00.

This completes the proof of Theorem 3.1

Remark 3.8. According to F. John, if the system (3.1) is genuinely nonlinear i.e., ~; > 0, ~ > 0, then r~ 2: 0, s~ 2: ° is the necessary and sufficient condition

for existence of global solution r, s. In our case ~; 2: 0, ~ 2: 0. Obviously, if ~ (0,0) > 0, then the condition (B) is fulfilled and we have a blow up result for Sx.

Remark 3.9. Condition (B) can be rewritten as

Joo a/1 (B) as (cro«(3), cso(o:))d(3 = +00, 0< c ~ co

for Cauchy data r, s and it is a growth condition for ro«(3) at +00. The integral in (B) is taken from arbitrary finite (30 till +00 and therefore only the divergence of the integral at +00 is important.

48 P.R. Popivanov

Fix then 0 < E ::::: Eo and 0' E Rl. Obviously ~(Ero(,8),ESO(0')) -7;3-++=

~(Ero( +00), ESo(O')). If ~(Ero( +00), ESo(O')) > 0, then (B) is satisfied.

3.3. Proof of Theorem 3.4

We reduce the second-order nonlinear hyperbolic equation (3.2) via the change v = U x , W = Ut to the system

(3.17)

Wt K' (v)vx .

This is a strictly hyperbolic quasi linear homogeneous system with two distinct characteristic roots A = -JK'(v) < 0, /-L = JK'(v) > o.

Introduce now the Riemann invariants r, s:

(3.18)

hence,

r+ S = W, (3.19)

and consider the strictly monotonically increasing function v = Jov J K' ()..) d)". As we know v = G(v) is its inverse, i.e., (3.19) implies that v = G(s - r). Then the system (3.17) reduces globally and in an equivalent way to the system

as as at - k(s - r) ox = 0, (3.20)

or or at +k(s-r)ox =0,

Slt=o = so,rlt=o = ro,

where k(v) = JK' (G(v)), ro, So are given by (3.4). Obviously, (3.20) is a system of the type (3.1) with A(r, s) = -k(s - r) < 0,

/-L(r, s) = k(r - s) > 0 =} $~ = -k' (s - r), ¥r = -k' (s - r). Having in mind that

k '( ) - 1 K" (G(v)) h . h (A) v -"2 K'(G(v)) we see t at we are m t e case . Condition (B) takes the form

- k' (so(O') - ro((3))dO' = +00, j += (3.21 )

i.e., we obtain the condition (E'). Applying Theorem 3.1 to the system (3.20) we get the desired conclusion for the Cauchy problem (3.2).

Corollary 3.5 is proved in the following way. As s~ 2: 0 and So is bounded, then sO(O')",-+= -7 so(+oo) i= 00. If k' (so(+oo) - ro((3)) i= 0, then the integral (3.21) is divergent. Assuming k' (so( +00) - ro((3)) = 0 we apply the condition (C) and we conclude that near (3 one can find a point 13 and such that r~ (13) < 0, k' (so(+oo) - ro(f3)) i= O. Thus everything is proved.


Chapter II On the regularity of the solutions for several classes of nonlinear systems of partial differential equations and some applications

Introduction

49

We consider in this chapter singularities of the solutions of several classes of nonlinear partial differential equations (PDE) and systems of PDE arising in gas dynamics and differential geometry. In Section 4 we study quasi-linear hyperbolic systems in the plane via the well-known hodograph transformation. Applying the famous Whitney theorem on the normal forms of smooth mappings from the plane into the plane, we obtain qualitative results on the geometry of the characteristic curves in the generic case. In § 5, 6 a paradifferential technique is used in order to study the propagation of singularities for some classes of nonlinear PDE. Moreover, in many cases we prove nonexistence of isolated singularities for nonlinear systems of PDE. As an application we propose several regularity results to the Meyer flow of a Tricomi gas for nozzle problem, to the Monge-Ampere equation with Gaussian curvature, changing its sign, and to hyperbolic-elliptic quasi-linear systems with characteristics degenerating at a single point. We have given a geometrical illustration of the theorems proved in this chapter. As we mentioned, a (non)linear microlocal analysis approach is used. Thus, the reader is supposed to be familiar with the classical theory of the pseudodifferential operators, paradifferential operators, microlocalized Sobolev spaces H:ncl etc.([32J, [8]).

4. Singularities of the solutions of quasi-linear hyperbolic systems in the plane

1. This paragraph deals with singularities of the solutions of several classes of nonlinear hyperbolic systems in the plane as well as with the regularity of the solutions of some second-order nonlinear partial differential equations of mixed type. In our investigations of quasi-linear hyperbolic systems we use the hodograph transformation in the plane and its normal forms via the famous Whitney theorem on singularities of mappings of the plane into the plane. A very interesting mechanical interpretation of these results was proposed in the book [23].

2. Consider now the following quasi-linear system:

where a,b,c,d are C= real-valued functions of their arguments u,v, b(u,v) i- 0

everywhere and ( ~ ) E HS(n), s > 3 is a real-valued solution of (4.1), i.e.,

according to Sobolev's embedding theorem ( ~ ) E c 2+E(n), E > O. Thus we

50 P.R. Popivanov

shall study the mapping

n :3 (x, y) ----+ (u ( x, y), v ( x, y)) E R2, (4.2)

locally near a fixed point (xo, Yo) E n. Suppose that j = [ux uy [-=I- 0 at (xo, Yo) Vx Vy

and put u(xo, Yo) = un, v(xo, Yo) = Vo· According to the inverse function theorem there exist two functions x( u, v) E C 2+0:, y( U, v) E C 2 +0: near (uo, vo) such that

J = [xu Xv [ -=I- 0 and the mapping (u, v) ----+ (x(u, v), y(u, v)) is inverse to Yu Yv CUD,VO)

(4.2) in a neighborhood of (uo,vo). Moreover,

The transformation (4.2) is called a "hodograph transformation" (see [23]). It is obvious that locally near the point (xo, Yo) the system (4.1) is equivalent

to the next system (near (uo, vo)):

(4.3)

It is interesting to note that the system (4.3) is linear with respect to the unknown functions x( u, v), y( u, v) as a, b, c, d depend only on the arguments (u, v).

Introduce now the characteristics for a linear system of partial differential operators. So, let

n a Pu = 2: Aj(x) a: + B(x)u = f(x), x Ene R n ,

j=l J

( 4.4)

where A j , B are smooth N x N matrices and u = (U1,"" UN)t, f = (il, ... , fN )t.

Definition 4.1. The smooth nondegenerate surface S : (x) = 0, x E n, V(x) -=I- 0 is called characteristic for (4.4) if for each XES,

( n 0<1»

det ~ Aj(x) aXj (x) = O.

Definition 4.2. The vector (direction) ~ E Rn \ 0 is called characteristic for P if det(2:7=1 Aj(x)~j) = o.

It is well known that characteristics are invariant under smooth nondegenerate change of the variables.

Let Ao (t, x), det Ao -=I- 0 be a smooth N x N matrix.

Definition 4.3. The system n

Aout + LAj(t,x)uXj + Bu = f j=l

is t-strictly hyperbolic if for each (t, x) E D, VC, E Rn \ 0, the algebraic equation with respect to T

n

det(TAo + LAj(t,x)c,j) = 0 j=l

has only real distinct roots T1(t,X,C,) < ... < TN(t,X,C,) (one can easily see that they are smooth in (t, x, C, -I- 0)).

This way we conclude that (4.1) is strictly hyperbolic with respect to x iff the algebraic equation

e + (a + d)c,'r/ + 'r/2(ad - be) = 0

has only real distinct roots with respect to C" i.e., iff the discriminant

D = (a + d)2 - 4(ad - be) > O. (4.5)

We shall continue our considerations for a fixed solution u(x, y), v(x, y) of (4.1) satisfying (4.5). In this case the corresponding characteristic curves depend on the fixed solution (u, v) of (4.1).

The characteristic curves of (4.1) in the (x, y) plane are given by

dy+ C+: dx = C,+, (4.6)

C_ : d;;; = C,-,

where C,+ -I- C,- are the distinct real roots of

a+d±VV )..2 - (a + d).A + (ad - be) = 0, C, ± = 2 '

C,± = c,±(u, v). Thus through each point (x, f)) E rl we have passing two transversal characteristics.

The characteristics of the linear system (4.3) are the next one:

du+ dv = (+(u, v),

du_ () -- = (_ U,v , dv

where (± are the roots of bf.L2 - (d - a)f.L - e = 0, i.e., (±(u, v) = d-a2~VD = c,\-a =} (+(u,v) -I- (_(u,v), i.e.,

du+ r + : b( u, v) dv = C,+ - a,

( du_ r _ : b u, v) - = C,- - a, dv

are the characteristics of (4.3):

(4.7)

Outside the set where j = 0, the characteristics C± are mapped onto the r ± characteristics under the hodograph transformation.

52 P.R. Popivanov

Consider the set of all Coo smooth mappings from R2 into itself equipped with the standard topology. According to Whitney [14] one can find an open and dense ~et T c Coo (R2 , R2) such that for each x E R2 and f E T the corresponding germ f is smoothly equivalent to one of the following three germs:

(i) f: R2 ----> R2 : (x, z) ----> (x, z) - regular case.

(ii) f: R2 ----> R2 : (x, z) ----> (x, Z2) = (x, y). (iii) f: R2 ----> R2 : (x, z) ----> (x, z3 - xy) = (x, y).

We point out that in the case (ii) j = 0 ¢? z = 0 and (x,O) ----> (x, 0), while in the case (iii) j = 0 ¢? x = 3z2 and j = 0 is mapped under f onto y = ± 3~X2/3, x 2: O. Thus if we put 'I for the set where j = 0 and l' for its image under the mapping f, we conclude that in the case (ii) both 'I and l' are smooth nondegenerate curves, while in the case (iii) l' has a cusp point at O. The kernel of the Jacobian of the mapping f, (ii) is of the type (0, ;3), ;3 =1= 0 and this vector will be called an exceptional direction. The vector (0, 1) is transversal to z = O. We shall concentrate our study on case (ii) as in this situation both 'I and1' are smooth curves. Moreover, we are interested in the qualitative behavior of the solutions of the system (4.1) in the generic case.

Let f : u = x, v = y2 and consider a smooth nondegenerate curve r5 in R~y passing through the point (a,O).

There are two different cases to be investigated.

1) r5 has a nonexceptional direction at (a,O).

2) r5 has an exceptional direction at (a,O).

In the case 1) the image J of r5 under the mapping f is tangential to the axis Qu at the point (a,O). In the case 2) J has a cusp point at (a,O) but both its branches are nontangential to Qu. For a geometric interpretation see Fig. 23.

Remark 4.4. Consider the analytic curve near 0:

r5 . { u = al t n1 ,

. v = bItm1 + b2 tm2 + ... ,nl ::; ml < m2 < ... EN,

and at least one mk is not of the type mk = lnl, Vl E Z. If nl, ml are even, then the situation (') appears. If nl is even, ml - odd and ml is not of the type ml = lnl' Vl E Z, then the situation (") appears.

Going back to our initial system (4.1) we assume that the following condition (A) is fulfilled:

Condition (A): j = I U x u y I = 0 along a smooth nondegenerate curve 'I in R~y, Vx Vy

j =1= 0 outside 'I and its image l' under the hodograph transformation (4.2) is also a smooth nondegenerate curve in R~v'

Theorem 4.5. In the generic case and under the condition (A) 'I is a C - characteristic of (4.1) and its image l' is r -characteristic.


y v v2:0

(0,1) f ..------.....

o x o a u

case 1).

Y v (') v (")

(0,1) 8 f -------... J

0 0 a u

case 2). Fig. 23

The C-characteristics of the other type passing through each point of'Y are, with exceptional directions, their images run through a section of a r -characteristic up to 'Y in one direction and then backward in the opposite direction. The images of all smooth curves in R;y crossing 'Y in directions different from the exceptional ones are tangential to i.

This way the image of a full neighborhood of 0 is a half neighborhood doubly covered. The image domain consists of two sheets which meet along i. The curve i is also called an edge of the image.

Proof. Let P E 'Y- As the characteristic curves C+, C_ passing through Pare nontangential each to other, then at least one of them, say C_, has nonexceptional direction at P (the exceptional direction is transversal to 'Y). If C+ has nonexceptional direction too, then the images of C+, C_ under (4.2) r +, r _ would be tangential to i at the point F. The last conclusion contradicts (4.7). So, C+ has a.:n exceptional direction which implies that "in general" r + has a cusp point at P but its branches are not tangential to i. This way we conclude that r + is a single curve passed two times - forward and backward. As it concerns i, it is an envelope of r _ characteristics having characteristic directions at each of its points. Therefore, i is a r _ characteristic as through any point there is only one curve

54

y

P.R. Popivanov

exceptional

direction

x

I':j=O

Fig. 24

v

u

with this property - the characteristic curve itself. So, I' turns out to be a C_ characteristic.

For the geometric interpretation of the previous result see Fig. 24.

3. We are going to analyse (under several restrictions) the case when double characteristic roots of the equation e + (a + d)E,'f] + 'f]2(ad - be) = 0 appear.

To simplify things we shall study the special system

b(u, v) -a(u, v)

= f (x, y) E C=.

(4.8)

Define now r = (a b ), 6.(u, v) = det r(u, v), 6.(u(x, Y), v(x, y)) e -a

6.(x, y), r = r(u, v) = r(u(x, y), v(x, y)), where ( ~ ) E Htoc(n), s > 3 are fixed

solutions of (4.8).

As the discriminant V = -46., we have three cases to study:

1) t:,.(u(x, y), v(x, y)) < 0 - strictly hyperbolic case, well studied in [8];

2) t:,.(u(x, y), v(x, y)) > 0 - elliptic case, well studied in [8];

3) t:,.(u(x,y),v(x,y)) = O.

In the last case the characteristic equation has a double root E, = o. Assume now

(a) there exists a point (xo, Yo) such that 6.(xo, Yo) = 0 but V x.y6.(xo, Yo) =J O.

As the case ox.6.(xo, Yo) =1= 0 is simpler to deal with, we shall concentrate on the more complicated case

(b) ox.6.(xo, Yo) = 0, Oy.6.(xo, Yo) =1= o. So (b) implies that one can find a smooth nondegenerate curve '"'( '3 (xo, Yo) with the properties .6.1, = 0, .6. =1= 0 outside '"'(.

Suppose, moreover

(c) oxx.6.(xo, Yo) =1= o. The last assumption means that '"'( is a strictly convex (concave) curve -

parabola type near (xo, Yo), with vertex at P = (xo, Yo). We point out that ~ = ±Y-.6.'r] and .6. has a sharp first-order zero along T

The equation of the characteristics passing through the hyperbolic region .6. < 0 is (dy)2 + .6.(dx)2 = 0, i.e.,

~: = ±v -.6.. Let (Po, qo) E '"'( \ P ( =? ox.6.(po, qo) =1= 0). It is trivial to verify that y±(x) = qo + clx - poI 3 / 2 , c(x, y) =1= 0, i.e., the

characteristics C± starting from the hyperbolic region form cusp points at '"'(. The characteristic through P degenerates at a single point. This is the corresponding figure.

y

P

o

Fig. 25

hyperbolic

region

x

Obviously, the difficulties in studying the smoothness of the solution of our system is due to the point P.

Applying methods of nonlinear microlocal analysis we can prove the next result.

Theorem 4.6. Consider a closed arc 5 of'"'( which does not contain P and assume

that ( ~ ) E Coo in a half neighborhood of 5 lying in the hyperbolic region. Then

( uv ) E Coo in a full neighborhood of 5.

56 P.R. Popivanov

In the case ( ~ ) E C= in the hyperbolic region we assume moreover that

condition (B) is satisfied: (B) if D.y(XO,yo) > 0, then D.y(xo,yo)(s -~) > !11(XO,YO)'- if D.y(XO,Yo) <

0, then lD.y(xo,Yo)l(s - ~) > -!12(XO, yo), where !11(XO,YO) > !12(XO,YO) are the eigenvalues of the matrix KiK' (xo, yo),

K(xo, yo) { -ruux - rvvx - (ruT - T (~: ~:)) u y

+ (r (~au ~av) - Tvr) vy } (xo, Yo).

Then ( ~ ) E C= everywhere.

The proof of the first part of Theorem 4.6 is a simple consequence of [8]. The proof of the second part will be given in §7.

4. We propose here several examples appearing in gas dynamics and in differential geometry.

The quasi-linear system

(4.9)

was studied by L.Bers [7] and it was said that (4.9) expresses some properties of a Meyer flow of a Tricomi gas for the nozzle problem. The symbols s, B, \}! and 

mean the speed, the inclination of velocity, the stream function and the velocity potential respectively. Rewrite (4.9) in the form

(4.10)

u = u(x, Y), v = v(x, y).

When investigating the existence of shock waves, Oleinik considered the sys-tern

(4.11)

which is hyperbolic for u < 1 and elliptic for u > 1. In their paper [49] Tay Ping Liu and Xin studied the system arising in the theory of overcompressive shock waves:

a = const > 0, b = const > 0.

The study of the following system was proposed by B. Kheifitz

(f(w))x + (g(w))y = 0,

(4.12)

(4.13)

where j = (h,h), g = (gl,g2), w = (u,v), i.e.,

A(w)axw + B(w)ayw = 0, A(w) = dj, B(w) = dg.

The Monge-Ampere equation with Gaussian curvature K(x, y),

UxxU yy - U;y = K(x, y)(l + u; + u~)2,

57

(4.14)

belongs to the same class of hyperbolic-elliptic systems if for each point (xo, Yo) for which K(xo, Yo) = 0 we have that dK(xo, Yo) i- O.

We shall study all the systems mentioned above from the point of view of microlocal analysis and shall find sufficient conditions for nonexistence of isolated singularities.

To begin with we shall consider the linearization of (4.10), namely

(4.15)

i.e., we shall obtain a system with the symbol

PI = h(i£,) - (~(x,y) ~) (iry) - (~y(X'Y) ~). For simplicity, we assume u E C 2 . So we find the principal symbol p~:

. ° (£, -ry) d . . ( 0 -zPl = -ury f, an - zpo = Z uy

Without loss of generality we suppose that our operator has the symbol -iPI ,

i.e.,

po + P = ( £, -ry ) + i ( 0 1 ° -ury £, uy (4.16)

CharpI = {detp? = e - U(X,y)ry2 = O}.

The Hamiltonian vector field of det p~ is given by the formula: 2 2 H detp? = 2£,ax - 2ryu(x, y)ay + ux(x, y)ry at, + uy(X, y)ry ary.

According to Bony's nonlinear microlocal theory, if

1) u(xo, Yo) < 0, then po = (xo, Yo, £'0, ryo) is an elliptic point for each 1£,°1 + IryO 1 >

o and the microlocal smoothness of ( ~ ) at po is studied in [8].

2) u(xo, Yo) > 0, po E Charpl, then po is a hyperbolic point and we have propagation of singularities along the integral curve £ of H detp? passing through the point po ( see [8] ).

3) u(xo, Yo) = 0; then £,0 = 0, ryO i- 0 if P E Charpl.

Let us assume that ux(xo,yo) i- O. Then Hdetp?(po) = (ryO)2(ux(xO,yo)at, + uy(xo, yo)ary) is not parallel to SCpO), where SCpO) = £'oat, + ryOary = ryOary is the radial vector field at the point pO. In that case the singularities again propagate along £.

58 P.R. Popivanov

Suppose now

4) ux(xo, Yo) = ° but uy(xo, Yo) =1= 0. Then HdetP~(pO) II 3(pO). This is the case to be studied here.

The case where u(xo, Yo) = V X,yu(xo, Yo) = ° is rather complicated because of the appearance of double characteristics and it remains out of our investigations.

Let us now multiply p~ by its cofactor matrix co p~:

( 4.17)

So we have a reduction to the scalar case but our operators (their symbols) are not C= smooth with respect to (x, y). They are only C 1 smooth.

As we shall see, some of the properties of the linearized operator will remain true for the nonlinear operator under consideration.

Let L = -Ax~ - y7] + d, 0< A < 1, d = id1 , d 1 E Rl, (4.18)

i.e., L(x, D) = -AxDx - yDy + d = i(Axox + YOy + dd, as ~ is the symbol of the operator Dx = -iox '

As it is known from [29], we can construct a Schwartz distribution v such that Lv E C= while it has a fixed singularity along the conic ray (0,1), i.e., WF(v) = (0,0,0,1) = po. In the case A = 1 there exists a function v such that Lv E C=, WF(v) = (O,O,f",) where f", = {Icp - ~I::; ~,c > O} and cp is the polar angle in the plane O~7].

Definition 4.7. We shall say that U E Hto~O iff U E Hioe' 'Vt < 8 and U <t Htoe'

One can easily see that there exists a solution v of the equation Lv = f E

C= with the properties v E H:';:eZo(pO), WF(v) = (0,0,0,1), Po = (0,0,0,1), 80 = A!l - d1 , ° < >. < 1. In the case>. = 1 we have 80 = 1 - d1 and v has an isolated singularity but in an angle: (O,O,f",).

Let us remark that 80 = - ImL~ and L~ is the subprincipal symbol of L. The definition of L~ is given below in (5.3).

In all the cases 0< >. ::; 1 the Hamiltonian vector field HL(po) = 3(po). More precisely, if ° < >. < 1, then

HL(o,o,f"r]) 113(0, o,~, 7])

iff either ~ = ° , 7] = ±1 or ~ = ±1, 7] = 0,

HL(o,o,f"r]) II 3(0, O,~, 7]), 'V(~, 7]) if >. = 1.

Remark also that the larger -d1 > ° is, the more regular in C k spaces the solution v is.

There are no difficulties in proving that there exists a solution w of the semilinear equation with analytic coefficients

(4.19)

and such that H so-o 2 H 2s0- 1- O(0 0 0 1) w E lac ,SO > , w E mel , , , ,

where w(O, 0) > 0, the analytic function 1(0,0) =I- 0, M E Rl. Thus, starting with So - 0 a smooth solution, we have an optimal microlocal

regularity 2so - ~, So > ~ + 1, n = 2. Therefore, in the nonlinear case a natural restriction on the microlocal

smoothness appears and it is of the type t ::::; 2s - ~.

It is worth studying the properties of the operator (4.18) from the point of view of nonexistence of singularities. So consider the operator P = 2:.]=1 Ajxj 8xj

in Rn and L = P + a, a = const. E Rl, Aj > O. Applying the method of characteristics one can show that: a) we have a unique classical solution u E C 1 (0), 0 :7 0 if a > 0 and 0 is

the origin in Rn. Moreover,

On singsupp Lu = 0 n singsupp u, Vu E C 1 (0);

b) we have uniqueness in C1 (0) modulo an additive constant if a = 0 and 1(0) = 0, u E C 1 (0) =} 0 n singsupp Lu = 0 n singsupp u;

c) we have nonuniqueness in C 1 (0) if a < O. Moreover, u E C 1 (0) implies that u( 0) = O.

We have mentioned above that a solution of Lv = 1 E coo with an isolated singularity at a ray exists if So = It\ - a, 0 < A < 1, a > O. Thus we get So < 1. The function v ti C 1 (0), 0 :7 0, as if v E C 1 (0), 1 E Coo(O), then v E Coo(O) near O. This is the parametrix R of the operator L in the case a): R1 = f01 t a - 1 1(tAx)dt, tAx = (tAl XI, ... , tAnXn).

5. Statement of the problem and formulation of the main results

1. Let 0 be a domain in Rn and suppose that the real-valued function U E Htac(O), S > ~ + m + 1, is a solution of the nonlinear system of partial differential equations:

Pj(X, u(x), ... , 8{3u(x), .. ')II'3I~m = 0,1 ::::; j ::::; N, (5.1 )

where Pj(x, u, ... , u{3," ')161~m are real-valued CCXJ functions of their arguments:

(x,u, ... ,u{3'···)161~rn' 8{3 = 8~: ... 8~~ if (3 = ((31, ... ,(3n), U = (U1, ... ,UN), 8{3u = (8{3u1,"" 8{3UN), u{3 = (Ul,6,"" UN,(3)'

The linearization of (5.1) contains the following two symbols:

i rn [L 8~Pj (x, 8{3u(x))I{3I~m~<> 1<>I=m k,a

(5.2)

i L 88Pj (x, 8{3U(X))I{3I~m~<>llj=I'''''N Uk <>

1<>I=rn-l ' ->k=l, ... ,N

60 P.R. Popivanov

The corresponding principal symbol is real valued,

for some E > 0, while

() "OPj ( !':l(3 ( )) a e1+0 Pm - 1 X, ~ = ~i ~ a:;;:-- x, u U X 1(3I:$rn~ E lal=m-1 k,a

is purely imaginary. This is the definition of the subprincipal symbol:

(5.3)

Consider now the point pO = (XO, ~O) E T*(Q)\O. It is proved in [8] that

det Pm(PO) i- 0 implies that u E H~c~m-'i (pO).

So we suppose

(i) det Pm(PO) = o. Assume that there exists an N x N square symbol P1-m(X, 0, positively homogeneous of order (1 - m) with respect to ~ and e= smooth for ~ i- 0, ea smooth with respect to X, (j > 1 and such that in some conical neighborhood of pO the following identity holds:

(ii) P1-m.Pm = q1IN (IN is the identity matrix in eN, q1 is a first-order realvalued scalar symbol). The notation ordt,q1 = 1 means that q1 is a positively homogeneous symbol of order 1 with respect to ~.

The matrix P1-m is assumed to be real valued, q1 E emin(Ho,a) with respect to x. Further on we denote by ~;(Q) the set of symbols positively homogeneous of order m with respect to ~, ~ i- 0, e= with respect to ~ i- 0 and belonging to CP with respect to X E Q, where p > 0 is not an integer. The notation Op(~;) stands for the set of corresponding properly supported paradifferential operators (see [8]).

Denote by pO = (xo, ~O), I~ol = 1 such a characteristic point (i.e., det Pm(PO) = 0) for which:

(iii) Hql (pO) + cS(pO) = 0 for some constant c < O.

By Hql (pO) = 2:7=1 (aq~k;O) OXj ~ aq~~:O) ot,j) we denote the Hamiltonian vector field

of the scalar function q1 at pO and SCpO) = 2:7=1 ~Jot,j is the radial vector field at the point pO. Note that det Pm (pO) = 0 implies that q1 (pO) = O.

In [26] Dencker supposed q1 to be a symbol of real principal type, i.e., Hql (pO) is not parallel to SCpO).

Unlike him we assume in (iii) that Hq , and 2: are collinear at pO E CharP = {p: det Pm(P) = O}. The subprincipal symbol of P1-m is given by the formula:

. n !:l2-_, _ ~ "'""' u P1-m P- m - 2 ~ oxo~ .

j=l J J

Introduce now the matrix-valued symbol

, i" R = -2{P1-m,Pm} + P1-mPm-1 + P-mPm

with Holder coefficients in x and ordr:, R = O. As usual, {.,.} denotes the Poisson bracket.

If A is a square N x N matrix in eN and A * is its adjoint, then 1m A = A-;: * .

(iv) Suppose that there exists a conical neighborhood r 3 pO with the next property: u E H;"cl(r\pO), t < 2s - 1 - ~ - m, s - 1 - m - ~ (j. Z and H;"cl is the microlocalized Sobolev space at r\po.

This is our main result, proved by the methods of microlocal analysis.

Theorem 5.1. Assume that the solution u E Htoc(Q), s > ~ + m + 1 of the system (5.1) satisfies (i)-(iv) and

(v) -cs + minllzll=l (1m R' (pO)z, z) > O.

Then u E H;"cl (r).

Thus we conclude that the singularity at po and under the condition (v) is not isolated, i.e., it can be removed.

The restriction t < 2s - ~ -1-m can be omitted if P is a classical scalar linear differential operator with C= coefficients. So if Pu E C=, P = Pm + Prn-1 + ... , u E Htoc' -cs + 1m P~-l (pO) > 0, then u E H:ncl(r\pO) implies that u E H;"cl(r) for each t > s. Therefore if u E Htoc' -cs + 1m P;"-l (pO) > 0, Pu E C=, W F(u) n (r\pO) = 0, then pO (j. W F(u).

The proof of Theorem 5.1 is reduced to the proof of a theorem from the theory of paradifferential operators. Our assertion is nontrivial if s < t < 2s - m - 1 - ~.

We shall verify Theorem 5.1 applying Th. 3 from [8] with d = m, p = S-E-~, 0< E « 1, (J" = P - m. Thus, there exists a paradifferential operator P E Op(~:), (J" > 1, (J" (j. Z with the symbol (5.2) and such that

P u E H s- rn+a = H s+a- 1+(1-m) U E H S (0) c CP P > m + 1 loe loe 'loc , ,

p (j. Z, u E H;"cl(r\pO). As t < s + (J" - 1 we have that Pu E Hlto-:,m+1. Without loss of generality we assume P1-m E ~;"-m. So Pu = f E Hto-:,m+1, u E Htoe' u E H;"cl(r\pO) and therefore pPu = h E H1toe ' pP E Op(~;").

Applying the paradifferential calculus we get:

_ _ _ 1 ~ 0P1-m 0Pm -pP = P1-mPm + P1-mPm-1 + i ~ ~ ox. + R,

j=l J J

62 P.R. Popivanov

where R is a (-1 + a) smoothing operator in each Sobolev space (i.e., R: H S ----+

Hs-l+o-, R being continuous for each s). According to (ii)

8 2

8x8f,. (P1-mPm) J J

So

Let us put

(5.4)

R E Op(~~). Obviously, the subprincipal symbol R~ = R'. Using standard bootstrap arguments we can see that Theorem 5.1 is a corol

lary of the following

Theorem 5.2. Let the paradifferential operator R E Op(~~), a > 1, a = 1 + t, lEN, I > 1 be given by (5.4). Suppose that u E H~;;,i;,(n), Ru E H:ncl(f), u E H:ncl(f\pO) and

-ct + min (1m R~(pO)z, z) > O. IlzlI=l

(5.5)

Then u E H:ncl (f).

Obviously, minllzll=l (1m R~(pO)z, z) is the smallest eigenvalue of the Hermit

ian matrix 1m R~(pO) and maxllzll=l(Im R~(pO)z, z) is its largest one. We propose several remarks below.

1. In many cases we multiply Pm by Pml E ~:', a tf- Z and then PmlPm = qrn+rnJN' det Prn(PO) = O. So qm+m, (pO) = 0; Hq=+=l (pO) + c3(pO) = 0, c < O.

Replacing (v) by (v)',

(v) ' (m+m1-1) , ° - c s - 2 + min (1m Ro(p )z, z) > 0, Il zll=l

where ,i f'

R o = -2"{Pm"Pm} + Pm,Pm-1 + Pm,-1Pm,

we obtain the same conclusion that u E H;"cl(r) if u E Htoc' S > -B'- + m + I, t < 2s - -B'- - m - 1.

More specially, if Pml =co Pm E I::(N-l), i.e., ml = meN - I), det Pm E I::N , we can reformulate our main result for the systems of the type (4.10), (4.11). As usual, cOp is the matrix of cofactors of the matrix p.

2. If c > 0, Theorem 5.1 remains true with (v)" instead of (v)', namely

ff ( m + ml - 1) (' ° ) (v) c S - - max 1m Ro(p )z,z > O. 2 Ilzll=l

In the scalar case we take Pml = 1 and then (v) is replaced by

Let us consider now the operator (4.18) with Lu E C=, i.e., m = I, c = -I, u E Htoc' S + 1m L~ > O. The last inequality is equivalent to s > So.

Then u E H;"cl(r\pO), pO = (0,0,0, I), po E r implies u E H;"cl(f), t > s.

3. This is a slight generalization of Theorem 5.1 for quasi-linear and semilinear systems. The same conclusion remains true under weaker conditions:

(a) u E HS, S > -B'- + m + I, t < 2s - 1 - -B'- - m for fully nonlinear systems,

(b) s > -B'- + m, t < 2s - -B'- - m in the quasi-linear system case,

(c) s > -B'- + m - 1, t < 2s - -B'- - m + 1 in the semi linear case.

Certainly, then U E H;"cl(f).

6. Some applications of the previous results

1. Let us consider at first the case 3) for the system (4.10). Having in mind that the principal symbol det p? = e - u( x, Y Jr72, we have

that if u(xo, YO) = 0, then ~o = 0, rl = ±1 if (xo, Yo, ~o, 'Il) E Char det Pl' Let us consider now the curve "y : u = 0, u(xo, Yo) = O. According to the

implicit function theorem (say ux(XO,Yo) =I=- 0) we find x = x(y) E C 3 , Xo = x(Yo). The Hamiltonian system is corresponding to det p?

X 2~, x(O) = Xo,

iJ 2u'TJ, yeO) = Yo,

~ -Ux 'TJ2 , ~(O) = 0, ij 2 -UY'TJ , 'TJ(O) = ±1.

64 P.R. Popivanov

Obviously, x(o) = 0, yeO) = 0,

x(O) -2ux(xo, Yo) i- 0, Y 2(uxx + uyiJ)TJ + 2ui] =* yeO) = 0,

y 2(uxx x2 + 2uxy xiJ + UyyiJ2)TJ +2(uxx + uyY)TJ + 2(uxx + uyiJ)i] + 2ur, + 2i1i] =*

yeO) -4u;(xo, Yo),

i.e., the characteristic curve IS in the plane Oxy is given by the equation

IS I x(t) = Xo - ux(xo, yo)t2 + O(t3 ),

yet) = Yo - ~u;,(xo, yo)t3 + O(t3 )

and has a cusp point at (xo, Yo) being located in the domain where u > 0.

Conclusion. Let us consider the system (4.10) in the case 3) (ux(xo, Yo) i- 0) and suppose that the solution u E HS(0,), 8 > 4, 0, ::7 (xo, Yo) is such that u E G=(0,+),

where 0,+ = {(x,y) : u(x,y) > O}. Then ( ~ ) E Goo in a full neighborhood of

the point (xo, Yo). If the curve 'Y: u(x, y) = 0, u(xo, Yo) = 0, 'Vu(xo, Yo) i- ° has one and only

one horizontal tangent passing through the point (xo, Yo), then u E H s, 8 > 4,

0, ::7 (XO,yo), u E GCXJ(0,+) implies ( ~ ) E G=(0,) according to the previous

remark and Theorem 5.1. Below we give the details. We shall apply now Theorem 5.1 to the system (4.10).

Assume that ( ~ ) E Htoc' 8 > 2 (i.e., u E Gl+E, C > 0) and u(xo, Yo) = 0,

ux(xo, Yo) = 0, uy(xo, Yo) i- 0, ~o = 0, TJo = ±l. So C = -8gn TJOUy(xo, Yo).

Moreover, R' (pO) = TJoiuy(xo, Yo) (~ ~1). Thus, for 2 < 8 < t < 28 - 2

solutions of (4.10) with isolated Ht singularities at (xo, Yo; 0, ±1) do not exist. A similar result is true for (4.11) with u(xo, Yo) = 1 instead of u(xo, Yo) = 0.

2. Let us consider now a slight generalization of (4.14):

I Uxx uxy I = K(x, y)f(x, y, u, 'Vu), uxy Uyy

(6.1)

where K(xo, Yo) = 0, 'V K(xo, Yo) i- 0, K E Goo, f E G=, f(x, y, U,p, q) > ° everywhere, ( p = u x, q = uy). Certainly, u, K, f are assumed to be real valued. Let u E Htoc' 8 > 4, (i.e., u E G3+E (0,)). Unlike [30] we suppose that the curve

'Y = {(x, y) En: K(x, y) = O} (6.2)

is characteristic at the point (xo, Yo) for equation (6.1) linearized on u. It is easy to see that at each point (x, y) E 'Y at least one of the tangential

and normal vectors is noncharacteristic for the linearized equation.

Proposition. Let u E GOO(0,\(xo, Yo)) and (6.2) be fulfilled. Then u E GOO (0,).

Proo]. By a rotation of the coordinates that always leaves the equation (6.1) invariant, we can assume that

K(x, y) = y + O(x2 + y2),

i.e., (xo, Yo) = 0, \7 K(O) = (0, I), the tangential vector to 'Y at 0 is (1,0). Equation (6.1) linearized on U has the following symbol:

p = Uyy t,2 - 2uxy f.TJ + Uxx TJ2 + iK jpf. + iK jqTJ - K ju, (6.3)

where

jp = jp(x, y, U, Ux, uy).

According to (6.3): uxx(O,O) = 0, Uyy(O,O) i= 0 and therefore due to (6.1) uxy(O,O) = O. The point po = (0,0, f.o, TJo) is noncharacteristic for (6.3) if f.o i= O. Then U E H;':c,3(pO). So let po = (0,0,0, TJO), TJo i= O.

Thus pg(pO) = 0, Hpg(po) = -TJ02(uxxX(O,O)o.;- + u xxy (O,O)o1))' Differentiating (6.1) with respect to x, y we find that: uxxy(O,O) i= 0,

uxxx(O,O) = 0, i.e., 0 i= Hpg(po) II S(pO), TJ5 = 1. The subprincipal symbol

p~ (pO) = 0 and therefore the conditions (v) til, (v)" are satisfied. Let r be a sufficiently small conic neighborhood of pO = (0,0,0, TJO). Then we know that U E H;':cI 3 (r\pO) 1 and therefore U E H;':cI 4(pO) which implies U E H2s-4(0) (i.e., ::lip E CD, ip == 1 near 0, ipU E H 2S- 4 ). This way we raise the smoothness of U at o with s - 4> O. An iteration of the same procedure leads to u E Coo near (0,0).

Corollary. Consider the equation (6.1) and assume that the punctured curve 'Y \ {(xo, Yo)} is noncharacteristic for equation (6.1) linearized on u. Then u E

Coo(~L), [L = {(x,y) : K(x,y) < O} implies that u E Coo in a full neighborhood of the point (xo, yo).

The geometric illustration is given at Fig. 26, where A = (xo, yo) and the characteristics in [~L are depicted by arrows.

K>O

n

A

K<O

Fig. 26

1 Here we lise the fact that u E Coo (l1\(xQ, yo».

66 P.R. Popivanov

Remark 6.1. To complete the investigation of the Monge-Ampere equation we shall formulate and give a short proof of the result of Hong-Zuily [30]. Thus, we suppose that U E COO(~L).

We shall concentrate on the characteristic points of p as the microlocal behavior of the solution near the noncharacteristic points is well known. So we assume that the curve 'Y is noncharacteristic at the point (0,0) for equation (6.1) linearized on u. This implies that uxx(O, 0) -I=- 0. So two cases must be considered:

a)uyy(O, 0) = 0, b)uyy(O, O) -I=- 0. Case a) Applying the equation (6.1) we get uxy(O,O) = 0. If 7]0 -I=- ° "* A =

(0, 0, ~o, rt°) f/- Char p. So we shall study the case 7]0 = ° only. The equation of the bicharacteristic passing through the point A with 7]0 = 0, ~o -I=- ° is as follows:

x = PE, = 2~uyy - 27]uxy , x(O) = 0,

y = Pry = 2uxx7] - 2uxy~, y(O) = 0,

~ = -Px, i] = -PY' ~(O) = ~o -I=- 0,7](0) = 0.

Thus, X(O) = 0, y(O) = 0. On the other hand,

i] = -Py = -Uyyye + 2uxyy~7] - Uxxy7]2

implies that i](0) = -Uyyy(0)~5· Differentiating (6.1) with respect to y and putting x = y = ° we obtain

Uxx(O, O)uyyy(O, 0) = f(O, u(O), V'u(O)) "*

(0 0) J(O,u(O),V'u(O» --L 0' . (0) ~c2 t --L ° Uyyy , = U xx (0,0) /, l.e., 7] = <--<"0' C = cons. / . Having in mind that

jj = Pryxx + pryyy + pryd + pryryi]

we obtain jj(O) = 2uxx (0,0)i](0) = -2f(0,u(0), V'u(0))~6 = cl~6, Cl < 0. In a similar way we get x(O) = 0. This is the equation of the characteristic

curve /3 passing through the origin:

I x(t) = o(t2),

/3 y(t) = 3t2~6 + o(t2). t ---+ 0,

The curve /3 goes to the hyperbolic region fL = {(x,y) : K(x,y) < O} as y(t) + O(x2(t) + y2(t)) = 3t2(1 + O(t4)), t ---+ 0. The fact that U E COO(fL) and the Bony propagation of singularities theorem [8] imply that u E Coo in a full neighborhood of the origin.

Case b) The equation (6.1) at (0,0) gives us that uxy(O,O) -I=- 0. One can easily see that p(O,O,~,7]) = ° ¢==::;>

(6.4)

To do this we observe that uxx(O)uyy(O) = U~y(O) =I- O. The bicharacteristic, passing through the characteristic point (O,O,~O,rl), ~o, flo given by (6.4) is uniquely determined. Moreover, its projection on the (x, y) plane is the curve

I x(t) = d1~5t2 + o(t2), y(t) = d2~5t2 + o(t2),

The details of this computation are left to the reader. Obviously, the characteristic curves go when t =I- 0 to the hyperbolic region ~L and therefore u E Coo in a full neighborhood of the origin.

There are three possible pictures of the location of the characteristic curve trough 0 in ~L. We propose them below in Fig. 27 and Fig. 28.

y y

o o x x

Fig. 27 Fig. 28

7. Proof of Theorem 5.2 and Theorem 4.6

1. Let us consider the symbol

(fJ ---> 0 further on, i.e., 0 < fJ « 1), /'\;1 E CO', /'\;1 == 1 near xo, ordE,r1 r, r11~I-T == 1 in a conic neighborhood of ~o, conesupp C1 E rIC cr.

A simple calculation shows that {Q1,cd = (1 + fJ21~12)-8AT' where AT {Q1,/'\;lId+2b/'\;1I1 x 2:.}=1 ~fJ2~j(1+fJ21~12)-1. But (l+fJ21~12)-1 is a bounded

family in ~~, V)'" > 0 and 1::1(E,1 2 is a bounded family in ~,\1, V)'" > 0, fJ E (0,1]. Thus AT E ~~-1 uniformly with respect to fJ and conesuppAT E r 1 cc r.

Define now the cut-off symbol fl, fl == 1 in a conic neighborhood of pO, fl E Sp 0'

conesuppfl E {(x,~) : /'\;1(xh1(~) =I- O}. '

68 P.R. Popivanov

So

and Bo E I:~_l uniformly with respect to J-L, conesuppBo cc r. Thus

and Do(p) = {Q1,l<n,} = {ql, I>: lid , assuming p = (x, ~), I~I = 1 in a tiny conic 1<11'1

neighborhood of pO. But

< "ilf,ql(pO), "ilx(l>:lIl(pO) > - < "ilxql(pO), "ilf,(l>:lIl(pO) > +25(1 + f.1?)-1J-L2 < "ilxql(pO),~O >

2

-c < ~o, "ilnl(~o) > +28c~ 1+J-L

-c(r-~) 1 + J-L2

-c (r + O(J-L2)) J-L ---> O.

(Here we used Euler's identity for the homogeneous function /'1, ordf,/'l = r.) Let us consider the symbol

where I>:,/" ordf,/' = 0 have the same properties as 1>:1 and /'1, i.e., 5 = 2, r = 2t (=* C!-, E I:~-2 for each fixed J-L > 0).

This way we get

(7.1)

where B o E I:~_l uniformly with respect to J-L, A2tEI:;t_l uniformly with respect to J-L, conesupp B o and conesupp A2t C c rand

Bo(pO) = -2c(t + O(J-L2)) , J-L ---> O. (7.2)

We note that U E H:ncl (conesupp (1 - ry2)A2t) and that t > s and (v) imply

1 ° ' ° -2 Bo(p ) + min (ImRo(p )z, z) > O. (7.3) Ilzll=l

(C!-,u = C!-'INu, R = qlIN + Ro + it) The bilinear form (C!-'h, C!-'u) , h = Ru is well defined in HO as C!-'h E

2 2-.l. H cornp , C!-'u E Hcor,i~ (conesuppC!-' cc r, 'l::/J-L > 0).


So we obtain

Im(Cf-lh, Cf-lu) = Im(Cf-lqlINu, Cf-lu) + Im(Cf-lRou, Cf-lu) + K, (7.4)

IKI ::; ClluI1 2 u~' and the constant C does not depend on fJ. t--2-

In fact, Cf-l E L;~ uniformly with respect to fJ and R is a smoothing operator - - 2

of order -(a - I), i.e., 1 (Cf-lRu, Cf-lU) 1 ::; 1 ICf-lRul 1 u;-,IICf-lull,;u ::; Cllull t _ u;-', a-I 1 -2- = 2['

The left-hand side of (7.4) can be easily estimated as h E H;"cl (r), conesupp Cf-l E r imply

1 (Cf-lu, Cf-lfdl ::; EII Cf-l u I16.mcl(l) + C(E)llhllz.mcl(l)' '\IE> 0 and C(E) does not depend on fJ.

The identity Cf-lu = rtCf-lu + (1 - rt)Cf-lu, (rt - l)Cf-l E L;~ uniformly with respect to fJ, the relations U E H;"cl (r\pO) and conesupp rt is concentrated near ° 0 0 ~II P ,rt == 1 near p ,show that IICf-lullo ::; 1 1 rtCf-lu 1 10 + d11I u llt.mcl(r"), p rt r .

Thus,

I(Cf-liI,Cf-lu)1 ::; EllrtCf-luI16+C(E)llhll;.mcl(r) +C1(E)llull;.mcl(r")' '\IE> O. (7.5)

Evidently, Im(Cf-lqlINu, Cf-lu) = Im(q1Cf-lu, Cf-lu) + Im([Cf-l' ql]U, Cf-lu). Put v =

Cf-lu. Then (qlV,V) = (v,qiv), ql = qlIN and according to [8] the symbol of the L2 adjoint operator qi of ql is given by the formula:

* . ""' (a) R" ql = ql - Z ~ ql(a) + , lal=1

it being a smoothing continuous operator of order - (a - 1) in the Sobolev spaces. So

(ql V,V) = (V,ql V) + i L (v,qiC'l)) + 0 (1Iull;_u;-,) lal=1

=? 2Im(ql v ,v) = L (v,qiC'l)v) +0 (1Iull;_a;-,). lal=1

Having in mind the fact that the symbol qiC'l) E L;~_1 is real valued we get, applying again Theorem 3.3 from [8]:

2Im(qlv,v) = L (qiC'l) v, v) +0 (1Iull;_,,;-,). lal=1

Therefore,

Im(q1Cf-lu, Cf-lu) = ~ L (qiC'l)Cf-l u , Cf-lu) + 0 (1Iull;-:tJ ' (7.6) lal=1

and the remainder 0(.) is independent of fJ. In a similar way we have that

Im(Cf-lRou, Cf-lu) = Im(RoCf-lu, Cf-lu) + 0 (1Iull;-f,-) .

70 P.R. Popivanov

In fact, [CfJ , Ra] E ~~-::..12' for cr > 2 as Ra E ~~-1 and [CfJ , Ra] is a (-t + cr - 1) regularizing operator, uniformly with respect to p" in the case 1 < cr < 2. Then

Im(RaCfJu, CfJu) = (ImRoCfJu, CfJu) + 0 (1Iull;-dt) , (7.7)

0(.) is independent of p" and ImRo = Ro~R~ is a Hermitian selfadjoint matrix. We know that [CfJ , q1] E ~~-1 uniformly with respect to p, and it has the

principal symbol H C fJ' qI}. So

Im([CfJ ,q1]U,CfJ U) = -Re ({CfJ , qI}u,CfJu) + 0 (1Iull;-dt) ,

i.e.,

Im([CfJ' q1]U, CfJu) = ~Re ({q1, C~}u, u) + 0 (1Iull;-ofJ . (7.8)

Combining (7.4)-(7.8) we conclude that

c:1 1 1]CfJU 1 16 + C(c:)llhIIZ,mclcr) + C1 (c:)1 lull;,mclCr ") (7.9)

2': ~Re ({ q1, C~}u, U) + (ImR~CfJu, CfJU) + 0 (1IuIIL dt ) .

The identity (7.1) shows that

Re({q1'C~}U,u) = Re(1]CfJBou,1]CfJu) +0 (1Iull;_~) +Re ((1 -1]2)(1 + p,2IDI2)-2 A2t u, u),

'" J v J

where conesupp A2t c r, and A2t E ~;t_1 uniformly with respect to p,. According to Bony([8], Corollary 3.5 a), b))

I(Ju, u)1 ::::; d2 (1Iull;_ ";' + Ilull;,mclcrll )) .

Having in mind that I 2' * 2 I (ImRaCfJu, CfJu) = (1] ImROCfJu, CfJu) + (CfJ(l -1] )ImRoCfJ u, u),

, J

v J,

(7.10)

(7.11)

J1 E ~;t_1 uniformly with respect to p" and conesupp J1 c r, J1 == 0 near pO, we can draw the conclusion that

I(J1u,u)l::::; d3 (1I u ll;_,,;, + Ilull;,mclcrIlJ

with d2 , d3 independent of p,. Obviously, 2 f f 2

(1] ImRoC,-,u, CfJu) = (ImRo1]CfJ u,1]CfJu) + O(llull t_.=.!) 2

(7.12)

as [1], ImR~] is a smoothing operator of order - (cr - 1) and 0(.) does not depend on p,.

Thus

( f f (2 2) ImRoCfJu, C,-,u) = (ImRo1]C,-,u, 1]CfJu) + 0 Ilull t_ ";' + Ilullt,mclcr") . (7.13)


According to the condition (5.5):

1) ~Bo + ImR~ E ~~-I' and 2) ~Bo(pO)IN + ImR~(pO) is a Hermitian and positively definite matrix which implies that ~Bo(p)IN + ImR~(p) is a Hermitian and positively definite matrix near pO.

Combining (7.9)-(7.13) and having in mind that

(ryCp,Bou,ryCp,u) = (BoryCp,u,ryCp,u) + 0 (1Iull;_a;-l)

we get that 'IIc: > 0,

c:1 IryCp,ul 16 + C(c:)llhll;,mcl(r) + d41Iull;,mcl(r")

~ Re ((~Bo + ImR~) ryCp,u, ryCp,u) + d51Iull;_-ir'

The constants C(c:), d4 , d5 in (7.14) are independent of /1.

(7.14)

Taking conesuppry c {p: ~B(p) + ImR~(p) > O} we can apply Garding's inequality for a positive system of paradifferential operators [8] and obtain:

Re ( (~Bo + ImR~) (ryCp,u) , ryCp,u) ~ d6 11ryC!,u116 + 0 (I IryCp,ulI=-e)

for some () = const > 0 and d6 = const > 0; 0(.) are independent of /1. The standard interpolation inequality in Sobolev spaces gives us that

1 1 ryCp,u 1 10 :S const,

i.e., for sufficiently small c: > 0 we have:

u E H;"cl(pO) =? u E H;"cl (r).

Thus everything is proved.

(7.15)

2. To prove Theorem 4.6 we shall use a microlocal approach, being in the framework of Bony's theory of paradifferential operators. The linearization of the semilinear system (4.8) has the following principal symbol:

_ ( ~ + ary bry ) and PI - CTJ ~ - ary

. (A B) Po = -z C D

where

A auuy + buvy, B avuy + bvvy, C cuuy - auvy, D cvuy - avvy.

Certainly the coefficients au, ... , Cv depend on the fixed solution (u(x, y), v(x, y)).

Obviously, detPI = 0 {::} e + ry2~ = 0, i.e., ~ = ±ryV-~.

72 P.R. Popivanov

1. In the hyperbolic region ~ < 0 the solution (u, v) E Coo.

2. In the elliptic region ~ > 0 we can use Bony's theorem for microlocal raising of smoothness of the solutions of nonlinear PDE [8].

3. Assume that 15 = (x, y,~, fj) is such that e + fj2~ = O. Then ~ = 0 :=} ~ = 0,

fj = ±1. This way we conclude that if PI = COPI = ( ~ - a1] -;b1] ), -C1] <, + a1]

then PIPI = qIh, where ql = det PI and moreover, H q1 (p) = -~xOt; - ~yoTJ while 3(p) = fjo,.,.

So, H q1 (p)113(p) iff ~x = 0, ~y #- O. But %x~(u(x,y),v(x,y)) = 0, ~(u(x,y), v(x, V)) = 0 {o} (x, y) = (xo, Yo) and P = (xo, Yo) is the vertex of the parabola "(. Therefore each point of T* ("( \ P) is either elliptic or ql is of real principal type at "(. The projection of the integral curves of Hql at R~,y, i.e., the characteristics of (4.8) form cusp points at "( \ P (Fig. 25).

Applying Bony's theorem for propagation of singularities from [8] we prove the first part of Theorem 4.6 In the case po = (xo, Yo, 0, 1]), 1]2 = 1 we shall use Theorem 5.1 Let us check its requirements having in mind that (xo, Yo, ~o #- 0,1]0)

is an elliptic point for our system. Straightforward computations show that

Putting K = -Tx - TyT + T (~ ~) we see that

I K+K* ImRo = 1] 2

The expression for K can be rewritten in the following form:

K(xo, Yo) { -TuUx - TvVx - (TUT - T (~: ~:)) u y

+ (T (~au ~av) - TV T ) V Y } (xo, Yo).

Our condition (B) is equivalent to (v) I. As the Hamiltonian vector field passing through the point (xo, Yo, 0, 1]) is parallel to the radial vector field, the corresponding bicharacteristic to Hql is a single conic ray and its projection on R~y - the characteristic curve through (xo, Yo) - degenerates at a point. Condition (B) is sufficient for nonexistence of isolated singularities of the solution (u, v) of the system (4.8).

Thus everything is proved.

8. Appendix II. Interaction of conormal waves for semilinear PDE

1. The results on creation of new singularities proposed in § 1 of Chapter I relied heavily on the fact that only the case of two variables was considered. U nfortunately, it seems impossible to generalize the proofs from §1 in the multi-dimensional case. It is interesting to note that similar results on creation (interaction) of singularities hold true in this situation, the notion of characteristics being replaced by the notion of characteristic hypersurfaces. To prove these results the machinery of paradifferential operators introduced by Bony is used. Thus, a rather different approach works in the multi-dimensional case. For the sake of completeness we shall formulate here a classical and very interesting result of Bony for interaction of simple waves. A short sketch of the proof will be given too. A refinement of the theorems to be given below was proposed by Alinhac in [1,2]. We shall confine ourselves to Bony's theorems in order to stress the main ideas, to simplify the proofs and to compare the geometrical interpretation of semilinear waves in the I-dimensional space case with waves propagating in the multi-dimensional case.

2. We shall formulate now Bony's theorem for propagation of nonlinear waves "under the shock" . So consider the fully nonlinear partial differential equation:

F(x, u, ... , ao:u, .. ')lalS;m = 0, (8.1)

where F E Coo with respect to its arguments and F, u are real-valued functions. For each sufficiently smooth solution u we can define the linearized on u

operator Pu:

Pu = L ~F (x,u, ... ,af3u(x), ... )aa UUo:

lalS;m

and its principal symbol ( up to im)

Pm(x,O = L ~F (x, u, ... , af3u(x), ... )~a. UUo:

lal=m

As usual, the characteristic set Char P = {(x, 0, ~ i= 0 : Pm (x, 0 = O}. The zero-bicharacteristic £ passing through the point pO E Char P is defined as the integral curve of the Hamilton vector field: H p = = 2:7=1 (~7 . 8~j - %7 . 8~j ). (H): Assume now that u E HS(O,), s > ~ + m, 0 E 0, and that the operator Pu

is strictly hyperbolic with respect to the direction t. Put O,± = 0, n {±t > O}. Moreover, we suppose that 0,+ is contained in the domain of influence of 0,-, i.e., each backward zero bicharacteristic, starting from a point in 0,+, meets with 0,before leaving 0,.

Theorem 8.1. (Bony) Suppose that u E HS(O,), s > ~ + m satisfies (8.1). Then

a) If pO tf- Char P =} u E H~c~m-"5: (pO).

b) If s > ~ + m + 2, pO E Char P, a < 2s - ~ - m - 1 and u E H::'ncl(pO), then u E H::'ncl(£)'

74 P.R. Popivanov

In the semi linear case the assertion b) remains true under the weaker condition s > ~ + m - 1, (J < 2s - ~ - m + 1.

Corollary 8.2. Under the assumptions b) of the previous theorem, suppose that (8.1) is hyperbolic with respect to the variable Xl and (H) holds. Then ul X1 <0 E HCT, (j> s implies that u E HCT(O,).

The symbol I: stands for a smooth variety or for the union of two smooth varieties I:i intersecting transversally.

Definition 8.3. The function u belongs to the conormal distribution space HS(I:, k), kEN U {oo} if U E Htoc and for each integer 0 ::; l ::; k M l M 2 . .. M1u E HS, where M j are arbitrary real smooth vector fields tangential to I:.

One can see that if f E Coo and U E HS(I:, k), s > ~, then f(u) E HS('B, k). Moreover, U E HS('B, k) =} U E HS+k(O, \ I:). In the special case 'B = 'Bl U 'B2 it is evident that WFs+k(u) C N*('Bd U N*('B2) U N*(I:l n I:2 ), N*('Bd being the conormal bundle over I: l .

Suppose now that I: l , ... , I:m are smooth hypersurfaces intersecting two-bytwo transversally along a variety f of co dimension 2. Put I: = Uj'B j .

Definition 8.4. The function u belongs to the conormal distribution space H S ('B, k) if u E Htoc and for each integer 0 ::; l ::; k:

Ml ... MIU E HS, where M j are first-order pseudodifferential operators whose principal symbols vanish on UN*('Bi) U N*(r).

It can be proved that if u E Htoc' s > ~ and f E Coo, then u E HS('B,k) =} f(u) E HS(I:, k).

Theorem 8.5. (Bony). Consider the semilinear equation

P(x, D)u = F(x, U, ... , f)Ci U , .. ')ICiI::;m-l (8.2)

and assume that U E Htoc(o'), s > ~ + m satisfies (8.2), the real-valued function F E Coo with respect to its arguments, P has real-valued smooth coefficients. Suppose, moreover, that the condition (H) holds, I: is a characteristic hypersurface of P and ulo- E HS('B, k). Then u E HS(I:, k) in 0,.

Theorem 8.6. (Bony). Assume that u E Htoc(o'), s > ~+m satisfies (8.2), the condition (H) holds and 'B l , I:2 are two characteristic hypersurfaces of P intersecting transversally at f, codimf = 2, f n 0,- = 0. Moreover, there are no other characteristic hypersurfaces passing through f. Then ulo- E HS('B, k), 'B = 'Bl U 'B2 implies that u E HS('B, k) in 0,.

Remark 8.7. Each second-order strictly hyperbolic operator satisfies the geometrical condition imposed on the characteristic hypersurfaces. The interaction between them will not cause anomalous singularities as it did in §1 of Chapter I, i.e., the result is the same in both dimensions.


Theorem 8.8. (Bony). Suppose that u E Htoc(o'), s> -§- + m is a solution of (8.2), the condition (H) holds, ~1,2 are two characteristic hypersurfaces of P intersecting transversally at r, r n 0,_ = 0 and ~3, ... , ~m are characteristic hypersurfaces issuing from r and intersecting transversally each other and transversal to r 1,2.

Assume that in 0,- the solution u E Hs+ k outside ~l U ~2, ub~ E HS(~i' k), i = 1,2. Then we claim that in 0,:

a) 'U E Hs+ k outside ofU'i~lL.i' b) 11 E HS(L.i, k) near ~i \ r, i = 1,2, c) 11 E HS(L.j, [s + k - t]) near L.j \ r, j .:::: 3, cr = min(s + k, 2s - -§- - m + 1).

Obviously, in case c): cr > s. Suppose that k = 00. Then u E Coo outside L.I U ~2 U UTL.j, u E H2s-~-m+I(~j,00), j':::: 3, where L.j is the forward part of L.j issuing from r and contained in 0,+. Thus when L.I, ~2 carrying conormal singularities intersect transversally at r, anomalous singularities could appear at r. They are of weaker strength in comparison with the initial singularities and propagate along the forward part of new characteristic hyperfaces starting from r. The new effect here marked is illustrated in Fig. 7 of §1 (I-space variable case).

Proof of Theorem 8.5. After a local change of variables we can assume that L. is given by the equation t.p == Xl = 0 =} \It.p = (1,0, ... ,0). According to the defini tion of characteristic hypersurface we have Pm (0, X' , \l t.p) = 0, Pm being the principal symbol of the linear hyperbolic operator P, x' = (X2' ... , xn). Thus the coefficient amo ... o in front of';l must vanish for Xl = 0 =} amo ... o(x) = xIAI(X), So the operator P takes the form

n

P = Al (X)XIO:' + L Aj(x, D)Oxj + Ao(x, D) j=2

(8.3)

and Aj(x, D), j = 2, ... , n are differential operators of order m - 1 with smooth coefficients.

Put M for the set of all tangential vectors M to ~. Evidently, M is generated by the following vector fields: Xl OXl' OX2' ... , oX n '

The well-known commutator identity [AB, C] = A[B, C] + [A, C]B enables us to conclude that

n

[aXil P] = L(oxiAj)oXj + (oxiAdxIO~ + (OxiAo), i.:::: 2, (8.4) j=2

n

[XIOXl' P] = L[XIOI, AI]Oxj + [XIOI, Ao] + (XIOXI Al - (m - I)Adxlo:', j=2

and XIO~ = O~-1(XI8xl) - (m - I)O~-l. So we have that

n

[M, P] = L Bj,oMj + Bo, j=l

where Bj,o, Bo , Ao are differential operators of order (m - 1).

(8.5)

76 P.R. Popivanov

Inductively one can show that for each multiindex I the following identity holds:

[MI,P] = L BJ,IMJ

IJI:SIII

and B J,I are differential operators of order (m - 1). Applying the chain rule which is valid in this situation we have

f3 _ of f3 of f3 aX (F(x,u, . .. ,a u)) - ;::;-(x, ... ,a u) + ... + --;:;-.oxo u,

J uXj uUf3 J

j 2: 2, 1{31 ::::; m - 1,

(8.6)

of XIOXl(F(x, u, ... , of3u )) = (xloX1F)(x, ... , of3u ) + ... + --;:;-xIOf3(OXl u).

uUf3

fore, On the other hand, xIOf3(OXl u) = 0f3(XIOIU)+C0f3-10Xl U, C = const. There-

M(F(x, u, ... , of3u )) = FI (x, U, ... , OTU, XIOXI U,

"\lOX2U, ... , oT(XIOXl u)),

(8.7)

the function FI being Coo smooth with respect to its arguments, h'l ::::; m - 1. There are no difficulties in verifying that

MI (F(x, ... , of3u )) = FI(x, u, ... , aT M/ u),

where 11'1 ::::; m - 1, if III::::; I, then II'I ::::; I and FI is Coo smooth with respect to its arguments.

To fix the idea we shall study the simplest case only. So Pu = F =} PMu = MF + [P,M]u. Put VI = (u, XIOXI U, 02U, ... ,an u). According to (8.5) and (8.7) we get

PVI + BVI = FI (x, VI, .. . , of3VI )If3I:Sm-1 (8.8)

and B is a linear differential operator of order (m - 1). As u E Htoc(0.) =} VI E Hto-;,I(0.) and moreover, VI E HS(0.-). Applying

Corollary 8.2 we conclude that VI E HS(0.), i.e., u E HS(L;, 1). Inductively we get that u E HS(L;, k).

Proof of Theorem 8.6. After a local smooth change of variables we can assume that L;l : rpl == Xl = 0, L;2 : <{J2 == X2 = 0. Each hyperplane L;3 passing through r = L;l n L;2 and different from L;1,2 is given by the equation: L;3 : (XIXI +(X2X2 = 0, (Xl (X2 i- 0. The normal vector to L;3 is ((Xl, (X2, 0). As we know L;3 is noncharacteristic for P. Having in mind the proof of Theorem 8.5 we see that

n

P=xIBI(x)o;:'; +X2B 2(X)O;' + LBj(x,D)Oxj j=3


In the previous equality B j are differential operators of order (m - 1) with smooth coefficients, while the differential operator K is of order (m-2). We rewrite P in the form:

n

(8.9) j=l

where Zl = X18Xl> Z2 = x28x2' Zj = 8xj , j 2': 3 are generators of the space of all vector fields tangential to ~.

According to our assumption Prn(X1, X2, x"', 0:1,0:2,0) =f. ° on ~3, x'" = (X3, ... ,Xn ). Taking Xl = X2 = ° we obtain K(O,x''', 0:1, 0:2,0) =f. 0. Moreover, 8prn '" _ . _ III

Of,] (O,O,x ,0:1,0:2,0) =f. ° for O:j - 0, J - 1,2. Thus, K(O,O,x ,0:1,0:2,0) =f. ° for 0:10:2 = 0. Therefore, K is a microelliptic operator on N*(f) \ {O}. Denote by H its inverse pseudodifferential operator of order 2 - m. We have

n

HP = HK8182 + LHAjZj + HAo == HK8182 j=l

n

+LRjZj+Ro, j=l

(8.10)

R j being pseudodifferential operators of order 1. Proceeding as before we conclude that

n

L CjZj + Co + D8x , 8X2 , j=l

(8.11)

and C j are differential operators of order (m - 1) while D is a differential operator of order (m - 2). Combining (8.10) and the fact that H K == 1 in a conical neighborhood of N*(f), we have

n

D8x,8x2 u = DHPu + LEjZju + Eou, j=l

where E j are pseudodifferential operators of order (m - 1), ord D H = 0.

(8.12)

Put VI = (u,x18x"X28x2,83, ... ,8n). According to (8.7) and (8.11) we obtain the system

(8.13)

in which B is a matrix-valued pseudodifferential operator of order (m - 1), L is a matrix-valued pseudodifferential operator of order ° and G I depends Coo smoothly on its arguments.

Certainly, VI E Hto-;/(O) and VI E HtoJO-). Fortunately, the result formulated in Corollary 8.2 remains true for systems with diagonal principal symbol of type (8.13) (see [9]). So we have that VI E HS(O) =} U E HS(~, 1) etc.

78 P.R. Popivanov

The proof and the above result for propagation of singularities are with local validity. Theorem 8.8 has a rather technical proof and we omit it.

3. We propose now several concluding remarks. They deal with removable singularities of the solutions of several classes of fully nonlinear systems of partial differential operators of first order. As a paradifferential approach will be used in investigating these systems, our results will be formulated via the properties of the first variation (linearization) of the corresponding system.

So we assume that Fk(X,U,P), u = (U1, ... ,UN), p = (uU, ... Uij, ... UNn), 1 ~ j ~ n, 1 ~ i, k ::::; N are real-valued Coo functions of their arguments xED, u ERN, P E RnN and consider the following fully nonlinear system with realvalued solution u:

Fk(x,u, Vu) = 0, 1 ~ k ~ N,x E D, (8.14)

D domain in Rn, u(x) = (U1(X), ... ,UN(X)) E Htoc(D), s > ~ + 2, i.e., u E

C 2+"'(D), c > 0. Then the linearization (the first variation) of (8.14) on the solution u can be

written as n 1 a

Pv = "" A(x)Dv - iB(x)v t, <c-+ D = --~ J J , J J i ox . ' j=1 J

(8.15)

where the matrix

Aj(X)=II~H(X,U,vu(X))11 ,j=l, ... ,n uU2J 1~i,k~N

and

B(x) = II ~Fk (x, u(x), Vu(x))11 . uU t 1~i,k~N

Define now the matrix

K(x) = B + B* _ ~ ~ oAj(x) 2 2~ ox ' j=1 J

B* being the adjoint matrix to B. As usual the characteristic set of (8.15) is given by

Ch", p, ~ {p ~ (x,';) E 1"«(1) \ 0, det tAl(X)';l ~ o}. In our further investigations we shall suppose that the matrices Aj (x) are symmetric, i.e.,

(i) Aj(x) = Aj(x), \:Ix E D

(i.e., 88uFk (x, u(x), Vu(x)) = 88Fi (x, u(x), Vu(x)), \:Ix ED). 'tJ Uk]

Theorem 8.9. (see[64]). Consider the nonlinear system (8.14) under the assumption (i) and the matrix Ajo(x) being positive (negative) definite in n. Suppose, moreover, that there exists a closed set W conical with respect to ~ and having compact base in n such that for each characteristic point pO E Char PI n 8W n { x jo ?: 15}, 8 = const, we have u E H;"cl(pO) for some t < 2s - 2 -~. Then u E H;"cl(pO), Vpo E Char PI n W n {Xjo ?: 15}.

It is well known that if pO tf. Char PI, then u E H 2s- 1-!lj (pO). Geometrically we can formulate Theorem 8.9 in a simpler way. If the solution

is micro locally Ht smooth in the characteristic points located at the "northern hemisphere" 8W n {Xjo ?: 15} it becomes microlocally Ht smooth in the points of the "northern massive hemisphere" W n {x jo ?: 15}. Thus u tf. H;"cl (pO), pO being a characteristic point inside W n {Xjo ?: 15}, implies that there exists at least one characteristic point p at the northern hemisphere with the property u tf. H;"cl (p).

The proof of the previous result can be found in [64].

Theorem 8.10. (see [65]). Consider the system (8.14), (i) and suppose that the realvalued solution u E Htoc' s > ~ + 3 . We shall assume that pO = (xO, ~O) E Char PI, I~OI = 1, r is a conical neighborhood of pO and such that u E H;"cl(r \ pO), t < 2s - 2 - ~. Then if

(ii)o K(xO) > 'Y1dN,'Y = const > 0

n 8A ( 0) (iii)t K(xO) + t L ; x ~2~J > 'Y1dN

k,j=1 Xk

are fulfilled we claim that u E H;"cl (pO).

Certainly, I d N denotes the identity matrix on R N, K (xO) and the matrix participating in the left-hand side of (iii)t are symmetric.

Remark 8.11. Consider the same system (8.14) on the n-dimensional torus Tn, i.e., x E Tn and suppose that g~kj (x, u,p) = t~;j (x, u,p), Vx E Tn and for each (u,p), S.t. lui + Ipl < 1. Let u == 0 satisfy (8.14) and (ii)o, (iii)t, t > 3 + ~, are fulfilled for each x E Tn, V~, I~I = 1, u = p = O. Then there exists SI > ~ + 3 with the property: for every s E (~ + 3,SI) one can find in HS(Tn) such a neighborhood W 3 0 that the system Fdx, u, ~u) = fk is solvable in HS(Tn) for arbitrary f = (II, ... , f N) in W. The proof of the assertion just formulated is published in [4]. Therefore, our conditions for removability of the singularities turn out to be a microlocalized variant of the conditions under which global existence results are valid.

80 P.R. Popivanov

Chapter III Travelling waves for several classes of nonlinear integrodifferential operators

9. Introduction

1. This chapter deals with travelling wave solutions for several classes of nonlinear partial differential equations and nonlinear integrodifferential equations arising in the applications - solid body mechanics in a medium with memory, shallow water equation, etc. In the case of solid body mechanics in a medium with memory, we construct a C 1 solution having the form of a travelling wave of tension. The solvability of the problem under consideration is reduced to the investigation of a nonlinear integral equation. Unfortunately, the contraction mapping principle does not hold in this situation. A nontrivial solution of our problem is found via the construction of successive approximations for an appropriate positive operator acting in the cone of nonnegative monotone functions. For a shallow water equation a weak solution having a corner at its peak is constructed (see [21]).

We shall begin with one-dimensional waves in a medium with memory. Following [50] we shall denote by x a co-ordinate of a point belonging to a solid body, by t - the time variable, by c - the deformation, by (J - the tension and by p - the density. For the sake of simplicity we shall assume that p is equal to 1. In a medium with memory there is a link between the tension and the deformation which depends on the history of the process. We shall confine ourselves to the next relation between c and (J:

(9.1)

In the previous equality K* is the convolution operator:

(9.2)

VI + K* stands for the development of the operator 1 + K* into a power series and the integral operator VI + K* as well as the multiplication operator a' (J) are acting on the function (J~.

It is well known from classical mechanics that the next equation holds:

82c 82 (J

8t2 - 8x2 = 0, (9.3)

supposing c and (J to be smooth functions of (t, x). Putting (9.1) into (9.3) we conclude that the tension (J(t, x) satisfies a rather

complicated nonlinear integro-differential equation. According to Theorem 7.1. from [50] the equation (9.3) with c given by (1) can be sharply factorized into two first order factors describing the propagation of two waves of tension to the leftand to the right-hand side respectively.

Here are the factors:

:t vI + K*ja'(rY) ± :x'

VI + K*ja'(rY) :t ± :x' (9.4)

We shall concentrate our attention to (9.4). Putting (VI + K*)-l = 1 - * ,

*u= [tcx) (t-T)u(T)dT

we see that each smooth solution rY of the nonlinear integro-differential equation

j a' (rY) ~; ± (1 - *) ~: = 0, rY E C 2 (x ::::: 0) (9.5)

will satisfy (9.3) with c given by (9.1). According to mechanics terminology the function is called "kernel of hered

ity". Assume that (t) = ke- kt , k > 0.

So we have that a wave of tension, propagating "to the right-hand side" by the nonlinear first-order equation

~()OrY orY _ kjt _k(t_T)OrY(T,X) d _ yaw),::, +,::, e 0 T - 0. ut uX -CX) x

We shall assume, moreover, that

a(O) = 0, a E C 2 , a' (rY) > ° and rY E C 2 (x ::::: 0),

rY = ° for x::::: 0, t::; O,rY(t,O) = rYo(t) E C 2 (R).

Obviously, rYo(t) == ° for t ::; 0.

is given

(9.6)

(9.7)

(9.8)

We shall construct a classical solution of the mixed problem (9.6), (9.8) and we shall formulate results for global existence in time t ::::: 0, x ::::: ° and for blow up of the corresponding solution. The symbol IIrYo lIeo (R') stands for the uniform

norm of the function rYo. We suppose further on that IlrYolleO(R), IlrY;)lleo(R) < 00.

This is our first result.

Theorem 9.1. Consider the mixed problem (9.6), (9.8). Then

(i) There exists a constant C(llrYo(t)lleo) such that if lia~1lco x C(llrYolleo) < 1; then the problem (9.6), (9.8) possesses a unique global classical solution rY E

C 2 (x ::::: 0, t ::::: 0). The constant C(llrYolleo) can be estimated in the following way:

C(llrYolleo) ::; max ja'((j) 10'l:::;IIO'oilco

x. 1 ,_xl/2 max la':(~)I. mmlo-l::;llO'ollco y'a'(ij5 100I::;IIO'oilco la (0')1

82 P.R. Popivanov

(ii) at blows up for a finite X > 0 if a) one can find a point (30 > 0 with the property

1 + k~;f;~) ( va'(ao((3o)) - ~) < 0,

where wo((3o) = J;o(f30 ) Ja'(A) dA, 0'0((30) > 0; b) one can find a point (30 > 0 such that 0'0 ((30) = 0,

k~~) , v

-1/2 < a~((3o)aV(O) < 0, 0'0((30) =1'= O,a (0) =1'= O.

The life span X of the corresponding solution in case b) can be estimated as

< _ _ _ 1 ( 2ka' (0) ) X _ x - ~ log 1 + '((3) v () > O.

va (0) 0'0 0 a 0

We omit the proof of Theorem 9.1 as it can be reduced by differentiation of (9.6) in t to a mixed problem for first-order quasi-linear partial differential equation containing a dissipative term.

To simplify the notation we shall write down the corresponding nonlinear integrodifferential equation which describes the wave of tension propagating "to the right-hand side" in the following form:

00' 00' jt 00' 1 b(a)ot + ox - -00 K(t-T)ox(T,x)=O,aEC, (9.9)

where the function K is called a "kernel of heredity" and a is the tension of a I-dimensional solid body in a medium with memory.

The functions b,K E C k (R1), k 2: 5, b(O) > 0, K(-z) = K(z), 'Vz, K(z) > 0 everywhere, K' (z) :s 0 for z 2: 0, limz-+oo K(z) = 0, K E £l(R+), R+ = {x E R1 : x > O}.

The equation (9.9) can be rewritten as follows under some suitable assumptions imposed on a:

00' 00' 0jt b(a)- + - - - K(t - T)a(T,x)dT = O. ot ox oX_oo

(9.10)

This is possible if a is bounded and if in a small neighborhood Uxo of each point Xo one can find a function p E £1 such that

lax(t, x)1 :S pet), 'Ix E UXo'

or if ax is bounded too. We are looking for a solution of (9.10) in the form of a travelling wave:

a(t, x) = f(t - x), where f E C 1 (z > 0) n C(z 2: 0), f = 0 for z:S 0, f =f:. O. This way we are simplifying our problem. Usually a = f(ct - x), where cis

the velocity of the wave. We do not lose generality by taking c = 1, as further on several cases will be studied and they cover the different possible values of c.

We write (9.10) for t > x as follows:

a ret,x) a a it at Jo b()"')d)'" + ox (T(t, x) - ox x K(t - T)(T(T, X)dT = 0. (9.11)

Define now the function

F(z) = 1z K(z - )...)f()"')d)"', (9.12)

where F E C 1 (R 1), F(O) = ° if f is a bounded function. The equation (9.11) implies that

a l ICt - X) a a - b()"')d)'" + -f(t - x) - -F(t - x) = 0.

at 0 ox ox

Put G(z) = f; b()"')d)'" E Ck+1(R1). Then

0- a [) at G (f (t - x» + ox f (t - x) - ox F (t - x) = ° (9.13)

for t > x. Having in mind the facts that txf(t-x) = -{(t-x) andf(+O) =G(O) = °

we obtain (Jet-x) (-X

Jo b()"')d)'" - f(t - x) + Jo K(t - x - )...)f()"')d)'" = ° for t ;:::: x.

Putting z = t - x ;:::: ° we rewrite the previous equation as

(Je z ) r Jo b()"')d)'" - fez) + Jo K(z - )...)f()"')d)'" = 0, (9.14)

f E C(z ;:::: 0) n C1(z > 0), f(O) = 0, f E LCXl(R+). Let G(z) = G(z) - z = foz b()"')d)'" - z E Ck+l(Rl). We obtain then the

following equation for the travelling wave we are looking for:

G(f(z» + 1z K(z - )...)f(>')dA = 0,

i.e.,

G(f(z» + K * fez) = 0. (9.15)

Then two different cases appear:

a) a' (0) = 0, i.e., b(O) = 1.

b) a' (0) =1= 0, i.e., b(O) =1= 1.

As the case b) is simpler to deal with we shall consider here the case a) only. These are our main assumptions:

G(z):::; O,\iz E R+,e' (z) < O,\iz > O,e' (0) = ° and there exists Gil (+00) such that ° < -e" (00) :::; 00.

84 P.R. Popivanov

Condition (*) implies that -0' (00) = -G( 00) = 00 and therefore -C is a homeomorphism from [0,00) into itself and is a diffeomorphism from (0,00) into itself.

Consider the function H(z) = - G~z) E C k(R1). Evidently H(O) = 0, H(z) > o for z > O. We suppose moreover, that

(**) H'(z) > 0 for z > O.

According to (*) the function H(z) is a diffeomorphism from (0,00) into itself and is a homeomorphism from [0,00) into itself.

We shall formulate now our last condition:

0' (0) = ... = c(r-1)(0) = 0, (-c)(r)(o) > 0,3 :S r :S k - 1.

As the case C' (0) = 0, _C" (0) > 0 is studied in a similar way we shall not give the details.

This is our main result.

Theorem 9.2. The equation (9.9) possesses a travelling wave solution a = f(t-x), f E Ck(z > 0) n C(z :::: 0) n £00, f(O) = 0, f(z) > 0 for z > 0, f-monotonically increasing if the assumptions (*), (**), (* * *) are fulfilled. Moreover, f (z) ~

1

czr=T (1 + 0(1)), z --+ +0, c = const > O.

So we conclude that the travelling wave starts with a vertical tangent at z = 0 and has a horizontal asymptote at 00.

Remark 9.3. In the case C' (0) = 0, _C" (0) > 0 there also exists a solution of (9.9) in the form of a travelling wave and belonging to the same functional class. The only difference is that f(z) has a linear growth near +0.

10. Proof of the main result and concluding remarks

At the beginning of this section we shall propose several notes to be useful further on.

1. Let f E £OO(z :::: 0). Then the convolution operator K * f has the following properties:

K * f E C(z :::: 0) n £00, (K * f)(0) = O.

2. Let 1 E £OO(z ::::: 0) and l(z) :::: 0, Vz ::::: O. Then (K * f)(z) :::: 0, Vz :::: 0 and (K * f)(z) :S 11ILoo .1, where 1 = 1000 K()")d)" > o.

So we conclude that the nonlinear operator A = (-G)-l.K * is well defined by the formula J(z) = Af(z) = (-C)-l((K * f))(z) for each function f E £oo(R+), l(z) :::: 0, z ::::: 0 and J E C(z ::::: 0) n £00, J(z) ::::: 0, Vz ::::: 0, /(0) = O. Moreover, o :S II :S h E £OO(R+) =} 0 :S AlI :S Ah.

Consider now two kernels K 1,2 having the same properties as the kernel K from (9.9). Then 0 :S K1 :S K2 and 0 :S II :S h E £OO(R+) implies that o :S K1 * II :S K2 * h and therefore, 0 :S Ad1 :S A2h, where Al = (-C)-1.K1 *, A2 = (-C)-l K 2*.


The solvability of the equation (9.15) is equivalent to the solvability of the equation fez) = Af(z), z :::: 0 in the cone of the nonnegative bounded functions.

3. Assume that the function 0 :S f (z) E L = (R +) is monotonically increasing. Then (K * 1)(z) :::: 0, Vz :::: 0 is a monotonically increasing function too and (K * 1)(00) = f(oo).I, where f(oo) = limz->= fez).

To prove 3 we use the fact that if f E LOO(R+), then K * f = f * K, i.e., the convolution is a commutative operator on L =. The Lebesgue dominated convergence theorem implies (K * 1)(00) = f(00)1.

As a corollary from 3 we see that if 0 :S f (z) E L = (R +) is a monotonically increasing function, then O:S J = Af E C(R+)nL= is a monotonically increasing function and 1(00) = (-G)-l(lf(oo)), J(z) :S (-G)-l(If(z)), Vz :::: O. It is evident that f E C(z :::: 0) n L= =} K * f E C 1(z ~ 0).

4. Let 1(z) = Af(z) and 0 :S f E L= n C(z ~ 0). Suppose that one can find a sequence Zv -+ +0 such that f(zv) > O. Then 1(0) = 0 and J E C 1 (z > 0). In fact, the main assumption in 4 implies that (K * J)(z) > 0, Vz > O.

5. Convergence property of the operator A. Let 0 :S fn(z) :S C = const., fn E L=(R+) and fn(z) -+ fez) a.e. in z :::: O. Then Afn(z) -+ Af(z), Vz :::: O. The convergence here is pointwise and not uniform.

6. Let 0 :S fez) E C(z :::: 0) n L=, f(O) = 0 be a monotonically increasing solution of the equation fez) = Af(z), z :::: o. Under the assumption (* * *) we

1 1

have that 0 :S fez) :S const.zr-l (K(O))r=T, z -+ +0, const > O. The result is interesting if f (z) does not vanish identically on each subinterval having the origin as end point.

7. We shall solve now the equation (9.15) with a constant kernel K == C > O. Obviously, C does not belong to the kernel class introduced after the equation (9.9). Thus, (-G)(f)(z) = C J~ f()...)d).... We are looking for a nontrivial solution in each one-side neighborhood of 0 : f E C(z :::: 0) n C 1 (z > 0), f(O) = O. By differentiation and after standard calculations we see that the previous integral equation is equivalent to the ODE

(-G)' (f(z))./ (z) = Cj(z). (10.1)

The nontrivial solution of (10.1) for z :::: 0 is given by the formula fez) = p-1(CZ) E C 1(z > 0) n C(z :::: 0), where p(z) = J~ - c'yl d)'" E Ck(Rl), p' (z) > 0 for z > 0 and, according to (*), p is a diffeomorphism from (0,00) into itself and is a homeomorphism from [0,00) -+ [0,00), p(O) = O. So fez) > 0 for z > 0, f E C(z :::: 0) n C 1 (z > 0), f is monotonically increasing for z :::: O. The condition (***) leads us to the conclusion that fez) = (Cz) r~' (A+o(l)), z -+ +0, A = const. In fact, w = p(z) = zT-1(M + Zk-T B(z)) for Izl «1, BE C 2 , M = const > O.

The inverse function z = p-1(W) E C(O :S z « 1) n C 1(z > 0) and z = 1 1 1 I

wr=T( 1 + O(wr-l )), w -+ +0. In a similar way we obtain that if G (0) = 0, Mr-l

-0" (0) > 0, then fez) = Cz(A + 0(1)), z -+ +0.

86 P.R. Popivanov

8. Our next step is to solve the equation (9.15) with the special kernel K I ,

where K1(z) = {K(zo), 0 S z S Zo, Zo > 0,

K(z), z 2: Zo·

Obviously, KI E Ck(RI \ {zo}). Put h = Jo= KI(Z)dz < I. Then zoK(zo) < h. Consider now the equation fez) = AIf(z), Al = (-G)-I.KI *. The operator Al roughly speaking is not a contracting mapping. Because of this reason we shall construct a sequence of successive approximations in the cone of nonnegative, bounded on R + and monotonically increasing functions. This approach is influenced by a method proposed by Krasnoselskij in [43], [44], see also [50]. If we start with a first approximation fo == 0, then all the approximations will be 0 too. So we take the following initial iteration,

OS fo(z) = { Jo~~), 0 S z S Zo, H (h), z> Zo,

and Jo(z) = p-I(K(zo).z), where H- I is the inverse function of H _ G~z) The function fo(z) is monotonically increasing and with jump at z = ZOo In fact, 0 S z S Zo =} Jo(z) S Jo(zo) S p-I(K(zo).zo). On the other hand, (-G)Jo(z) = K(zo) J; JP..)d)'" for 0 S z S zo° Thus (-G)Jo(zo) S K(zo).zoJo(zo) =} H(jo(zo)) S zoK(zo)I =} Jo(zo) S H-I(zoK(zo)) < H-I(h).

Define the second approximation h = Ado E C(z 2: 0) n L=, which is a monotonically increasing function and moreover, h (z) = Jo (z), 0 S z S Zo. Thus, h(z) > 0, z > 0, h(O) = 0 and according to property 3: h(oo) = (-G)-I(H-l(Id·h) = fo(oo) as h = H(fo(oo)) ¢=:} -G(fo(oo)) = fo(oo).h. Evidently, 0 S h(z) ::; fo(z), 'Vz 2: O. Put 12 = Alh, ... , ik = Adk-l, .... The monotonicity of Al (see property 2) shows that:

fo(z) 2: h(z) 2: 12(z) 2: ... 2: 0, fn E C(z 2: 0) n L=,n 2: 1,

fn(O) = 0, fo(z) = h(z) = ... = fn(z) = ... for 0 S z S Zo, fn(oo) = fo(oo). Certainly, the functions fn(z) are monotonically increasing.

Define now fez) = limn->= fn(z) =} f E L=(R+), f(O) = 0, f is monotonically increasing and fez) = fo(z), 0 S z S Zo =} fez) > 0 for z > O. As the convergence is pointwise it is not obvious that f (z) is a continuous function. Fortunately, the identity fn(z) = Adn-l(z), 'Vz 2: 0, 'Vn enables us to conclude that fez) = Ad(z), 'Vz 2: O. Therefore fez) E C(z 2: 0) and the convergence in limn fn(z) = fez) is uniform on each compact interval in R+. Evidently, f ( (0) = H- I (h). As the kernel K I is not smooth at z = Zo we do not claim that f is smooth for z = Zo.

And now we are ready to prove Theorem 9.2. Put

Kn(z) = { K(~), K(z),

0< z < l z;: l - n E C(z 2: 0).

- n

Thus,O::; Kl(Z) ::; K2(Z) ::; ... ::; Kn(z) ::; ... ---+ K(z), K(l) ::; ... ::; K(~) ::; ... , 0 ::; An ::; An+l' Vn. Fix the integer n. According to property 8 the equation fen) = Anf(n), An = (-G)-l.Kn* possesses a solution fen) with the following properties: 0::; fen) E C(z 2: 0) n L'=, f(n)(o) = 0, f(n)(z) > 0 for z > 0, fen) is monotonically increasing, f(n)(oo) = H-1(In), In = Jooo Kn()")d)". According to

the assumption (* * *) f(n)(z) = COnzr~' (1 + 0(1)), z ---+ +0. The "zero approximation" for fen) (z) was given by

Certainly,

O::;z::;~, z> 1..

n

1 1 p-l(K( - )z) ::; p-l(K(--)z), H-1(In ) < H-1(In+r). n n+ 1

(10.2)

We remind the reader that also Vk 2: 1 f6n) = f~n) = ... = f~n) = ... f~n) (z) 2: o fen) = A F(n) ... fen) = A fen) and f,(n) > fen) > ... > fen) > ... --+ f(n).

, 1 nJO" k n k-l 0 - 1 - - k -

Having in mind (10.2) we see that

and property 2 implies that

Conclusion: {f~rn)}~=l' kEN is a monotonically increasing sequence of nonneg

ative continuous functions for z 2: O. Evidently, 0 ::; f~n) ::; f~~l' n - fixed leads for k ---+ 00 to the inequality: 0 ::; fen) ::; f(n+l). This way we have constructed a monotonically increasing sequence of nonnegative and continuous monotonically increasing functions:

(10.3)

These functions are uniformly bounded as fen) ::; f(n) (00) = H-1(In) < H-1(I). Define now the monotonically increasing bounded function

0::; fez) = lim f(nl(z), Vz 2: O. n

Letting n --+ 00 in fen) (z) = Anf(n), z 2: 0 we apply the Lebesgue dominated convergence theorem and conclude that fez) = Af(z), Vz 2: O. Thus, f E C(z 2: 0) n LOO, f(O) = 0, f(+oo) = H-1(I), fez) 2: f(l)(z), Vz 2: 0, i.e., fez) > 0, Vz > o. A simple application of the results given by properties 4 and

k 1 1 1 6 shows that f E C (z > 0), COlzr-l (1 + 0(1)) ::; fez) ::; const.zr=T (K(O))r=T, z ---+ +0. Theorem 9.2 is proved.

88 P.R. Popivanov

Example. Put b(A) = 1 - h(A), where h(A) = L~=1 CnAn is an entire function, Cn :::: 0, h =:j O.

1. Assume that C1 = .,. = Cr -2 = 0, Cr -1 > 0, 3 ::; r. Then there exists a monotonically increasing, bounded, smooth solution of (9.9), fez) > 0, z > 0

1 and fez) ~ CZ,-l, Z ----+ +0, c> O.

2. Suppose that C1 > O. Then there exists a monotonically increasing smooth solution of (9.9), such that d2 z ::; fez) ::; d1z, 0::; z « 1, d1,2 = const > O.

Concluding remarks. Consider now the equation (9.9) with b = 1- an, 0 < a < 1. The solution a we are interested in will be nonnegative. Repeating the procedure described above we reduce the problem for finding a nontrivial solution of (9.9) in the form of a travelling wave to the solvability of the equation

1 (-G)f = a + 1 r+1 (z) = K * fez), z:::: o. (10.4)

The solution of (10.4) should be 0::; f E C(z :::: 0) nLoo, monotonically increasing, f(O) = 0, and fez) > 0 for z > O. Obviously, (10.4) is equivalent to fez) = [(a + l)K * fez)] Q~l. The solution of (10.4) (if it exists) can be estimated from above as follows:

0::; fez) ::; (a + 1)t; (K(O))t; zt;, z :::: 0

and therefore there exists / (0) = 0 ( =} f E C 1 (z :::: 0)). In the next step we shall estimate from below f, fez) > 0 for z > O. By using

the fact that K is a monotonically increasing function we obtain

fez) :::: (a + 1) Q~l (K(z)) Q~l [1 2 f(A)dA] Q~l ,z :::: o. (10.5)

Putting w(z) = J02 f(A)dA E C2(z :::: 0), w(z) > 0 for z > 0 we have w' (z) :::: AWQ~l(K(z))Q~l, A = (a+ l)Q~l. Integrating the last inequality from E to z, o < E and letting then E ----+ +0 we have

B 0'+1 1 Thus, w(z):::: "2z""Q""(K(O))a, 0::; z« 1.

According to (10.5): fez) :::: constzt;, 0 ::; z « 1, const > O. We construct the solution fez) E C1(z :::: 0) n LOO, fez) > 0, z > 0, / (0) = f(O) = 0, f monotonically increasing as in Theorem 9.2. The only difference is that in the last

case -Gil (00) = O. The observation that p(z) = J02 - c'yl dA in this case has the property p( (0) = 00 enables us to complete the proof.

In contrast with the travelling wave studied in Theorem 9.2 the travelling wave in (10.4) starts with a horizontal tangent and f(oo) = «a + 1)I)t;.

Remark 10.1. Theorem 9.2 remains true for G E CH<>, 0::; a ::; 1 with condition (*)' instead of (*):

(*)' G(z)::; O,z;::: O,o'(z) < O,'v'z > 0,0'(0) = 0,

, 100 o'().) -G (00) = 00, 0 --).-d)' = 00

and under condition (**).

Consider now the well-known R. Camassa-D. Holm equation

Ut - Uxxt + 3uux = 2uxuxx + UUxxx , t > 0, x E R. (10.6)

The function U represents the fluid velocity at time t in the x-direction in appropriate nondimensional units. Put the initial data uo(x) = u(O,x) E H 3 (R 1 )

and suppose that the Cauchy problem possesses a local-in-time solution U E

C([O, T); H 3 ) n C 1 ([0, T); H2), T > O. Introduce now the pseudodifferential operator Q(Dx) with symbol (1 + e). Certainly, Q acts in the space variable. As it is shown in [21,22]

i.e.,

Ut + UUx = -ox ~ * (U2 + ~U;)) in C([O, T); HI)

and p* f(x) = J~oo p(y)f(x - y), p(y) = ~e-IYI, f E L2(Rl) as Q-l f = p * f· We define the nonlinear operator

F(v) = ~V2 + p * (v + ~v;) and rewrite (formally) the equation (10.6) in the form of a conservation law:

Ut + (F(u))x = O,u(O,x) = uo(x) E H 3 (R1 ). (10.7)

Following [21] we give a definition of the weak solution of the Cauchy problem for (10.6).

Definition 10.2. A function U E L2([0, T); HI) is called a weak solution of the equation (10.6) with Cauchy data Uo E HI if it satisfies the integral identity

iT L (UCPt + F(u)cpx)dxdt + L uo(x)cp(x,O)dx = 0 (10.8)

for all cP E C 1([0, T) x R) being restrictions on [0, T) x R of C 1 (R2) functions compactly supported in the strip (-T, T) x R.

One can easily see that the travelling wave U c = ce-Ix-ctl is for any c> 0 a weak solution of the Cauchy problem (10.6) with initial data ce-1xl . To do this we put U c in (10.8) and compute the corresponding integrals. This example is rather interesting because U c has a corner point at its peak, i.e., this wave is not Coo smooth everywhere.

90 P.R. Popivanov

Consider the Cauchy problem for the following nonlinear nonlocal equation

Ut + UUx + I: K(x - ~)u~(t, ~)d~ = 0, t ~ 0 (10.9)

U(O, x) = uo(x),

where the kernel K E C(R) n Ll(R) is symmetric and monotonically decreasing on R+, K o;E 0 and Uo E H=(R). It is proved in [61] that for some T > 0 there exists a unique solution of (10.9) in the class C=([O, T]; H=(R)). A proof of the blow up result for (10.9) is given in [22]. More precisely, assume that Uo E H= satisfies

infu~ + supu~ :::; -2K(0) < O. R R

Then U x blows up for a finite time to > O.

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94 P.R. Popivanov

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P.R. Popivanov Institute of Mathematics and Informatics Bulgarian Academy of Sciences 1113 Sofia, Bulgaria

From Wave to Klein-Gordon Type Decay Rates

Fumihiko Hirosawa and Michael Reissig

Abstract. The goal of the paper is to derive Lp - Lq decay estimates for KleinGordon equations with time-dependent coefficients. We explain the influence of the relation between the mass term and the wave propagation speed on Lp - Lq decay estimates. Contrary to the classical Klein-Gordon case we cannot expect in each case a Klein-Gordon type decay rate. One has wave type decay rates, too. Moreover, under certain assumptions no Lp - Lq decay estimates can be proved. In these cases the solution has a Floquet behavior. More precisely, one can show that the energy cannot be estimated from above by time-dependent functions with a suitable growth order if t tends to infinity.

1. Introduction

1.1. Preliminaries on decay properties for hyperbolic equations

There exists a well-known approach to prove global existence of small data solutions for the Cauchy problem (see e.g., [11], [19]). This approach was applied to quite different nonlinear problems, we will give some comments on how to apply it to the nonlinear model problem

{ Utt - 6u = fC'Vu, Ut, \lUt, 'V2 u), U(O, x) = <p(x), Ut(O, x) = 'lfJ(x) ,

(t,x) E (0,00) x]Rn, x E ]Rn.

(1.1)

The basic tool is the so-called Strichartz decay estimate [27] for the solutions of

{ Utt - 6u = 0, u(O,x) = <p(x), Ut(O,x) = 'lfJ(x) ,

(t, x) E (0,00) x ]Rn ,

x E IRn ,

which has the following form:

IIUt(t, ')IILq + 11\l xu(t, ')IILq :::; C(1 + t) - n;-'(~-i) (11<pllw;p+ 1 + 11'lfJllw;p ),

where ~+~ = 1,1:::; p:::; 2, Np > n(~-~). This estimate expresses two important facts:

2000 Mathematics Subject Classification. 35L05, 35L15, 35Q40 . Key words and phrases. Lp - Lq decay estimates, Cauchy problem, Klein-Gordon equations, Fourier multipliers, Floquet theory, stationary phase method, oscillating integrals. This work was completed with the support of our 'lEX-pert.

96 F. Hirosawa and M. Reissig

• If p = q, then this estimate follows from conservation of energy. One can even use N 2 =0.

• But if we suppose some regularity of the data, then it is possible to estimate N

the energy based on the Lq-norm by the energy based on the Wp P -norm and one gains additionally a factor which decays if n ~ 2 and p =I- 2.

But how can such an estimate be used? It is more or less standard to prove the local existence in time of the solution for problem (1.1) in some evolution space

of low regularity CO ([0, T], Wr+1(JRn)) n C 1 ([0, T], W2(JRn)). But if we suppose

higher regularity for the data, then this regularity should be transferred to the solution. This can be obtained in principle from the energy estimate of higher order

t

::; Co exp ( C1 J II ((V, aT )2u( T, .)) t= dT) (11<pIIW;+l + 111j;11 Wi)' o

where we supposed quadratic growth of the right-hand side from (1.1). Applying Sobolev's embedding theorem and the differential equation one can conclude

with a suitable 81 = 81(n). Now due to the Strichartz decay estimate we have n-l

sup (1 + t)~4 II(V, Ot)u(t, ')lIwSl ::; C. O::;tST 4

This means that we can produce a term like (1 + t)- n::;-l in the integral appearing in higher order energy estimates. If n ~ 6, then this term is integrable over (0, CXl). This allows us to continue the solution step by step for small data. Here the autonomous character of (1.1) is very helpful because the local existence time interval of the solution with respect to t is independent of the hyperplane t = to where we prescribe the data.

One of the main observations of this approach is that one can use Lp - Lq decay estimates for suitable linear partial differential equations or systems to prove global-in-time existence results for nonlinear problems. In the case of the wave equation, the Strichartz decay estimate can be obtained from Kirchhoff's explicit representation of solution [19]. Let us demonstrate it in the three-dimensional case for the solution

of

u = u(t, x) = 4~ J 1j;(x + ty)dy 8 2

{ Utt - 6u = 0, u(O, x) = 0, Ut(O, x) = 1j;(x) ,

(t, x) E (0, CXl) X JR3 , x E JR3. (1.2)

From Wave to Klein-Gordon Type Decay Rates 97

To estimate the energy based on the Loo-norm we have to study

I J 1j;(x + ty)dyl, It J V1j;(x + ty)dyl· 8 2 82

It is useful to split the considerations for t E [1, 00) and for t E [0, 1]. If t E [1, 00), then we use

J Vk1j;(x + ty)dy = - J lyl-3Vk(V1j;(x + y)y)dy (k = 0,1).

82 IYI>t

Hence we get in this case

IIUt(t,·)IILoo + IIVxu(t,·)IILoo :::; C1111j;lIwr·

If t E [0,1]' then partial integration and transformation of variables yield

t k J Vk1j;(x + ty)dy 82

= (_1)k+1 J ~~~~:~52~: L Ya(aavk)1j;(x+y)dy (k=O,I). IYI>t lal=3-k

Hence we get in this case

IIUt(t, ·)IILoo + IIV xu(t, ·)IILoo :::; CII1j; llwt·

Both estimates together give the L1 - Loo estimate

Ilut(t, ·)IILoo + IIV xu(t, ·)IILoo :::; (1 + t)-lll1j; llwt·

Together with the energy conservation (L2 - L2 estimate), a general interpolation result from [1] and the information, that the interpolation space [Loo, L 2]0 = L qO ' q(J = 2/0, 0 E [0,1]' we conclude the Strichartz decay estimate for the solution of (1.2).

We understand that considerations for small t give the necessary regularity of the data and for large t bring the decay rate.

Another way to derive this estimate is to use the Fourier multiplier representation of the solution [1], [18]. It is sufficient to study as a model Fourier multiplier

F-1 (eitl €II';I-2IF(cp(y))(,;)) , l ~ 0. (1.3)

If we can obtain an Lp - Lq estimate for t = 1, then the transformations t'; = 'T7

and tz = y give immediately Lp - Lq estimates for each t > 0. First we localize by a suitable cut-off function X our considerations to small 1';1, that is, we study

F-1 (eill;ll';I-2Ix(IWF(cp(y))(';)) , l ~ 0.

If l > 0, then the kernel becomes singular at 1.;1 = 0. The theorem of HardyLittlewood [8] tells us, that there exists a relation between the order of singularity

2l, the dimension n, and the goal to derive Lp - Lq estimates, namely ~ - i ~ ~, 1 < P ~ 2 ~ q < 00. Under these assumptions we have

IIF-l (eil~II~I-21x(IWF(<p(y))(~)) t q ~ CII<pIILp.

To study

F- l (eil~II~I-21(1_ x(IW)F(<p(y))(~)), 1 ~ 0,

we derive at first an estimate for

IIF-l (eil~II~I-21(1 - x(I~I))k(~)) too ' where {d k>O is a dyadic decomposition of the phase space IRn. Some transformation depending on k transfers, for all k, these integrals to an integral over a compact set in the phase space away from ~ = O. This allows us to apply Littman's lemma [14], where we use that the rank of the Hessian in stationary points of the phase function x . ~ + I~I is equal to n - 1. The obtained Loo estimates can be used to derive L2 - L2 and Ll - Loo estimates and finally after interpolation Lp - Lq estimates of the following form, ~ + i = 1, 1 ~ p ~ 2:

IIF-l (eil~II~I-21(1_ X(IW)k(~)F(<p(y))(~)) tq ~ C2k(ntl(~_~)-21)11<pIILp.

From [1] it follows that

IIF- l (eil~II~I-21(1_ x(IW)F(<P(Y))(~)) tq ~ CII<pIILp

if 2l ~ ntl (~ - i), but now for ~ + i = 1, 1 < p ~ 2. Summarizing these estimates gives

IIF- l (eil~II~I-21F(<P(Y))(~)) tq ~ CII<pIILp

if ntl (~- i) ~ 2l ~ n(~ - i), ~ + ~ = 1, 1 < p ~ 2. Finally, the above-mentioned transformation to include the t-variable into the phase leads to the Lp-Lq estimate

IIF- l (eitl~II~I-21F(<p(y))(~)) tq ~ CltI21-n(~-~)II<pIILp.

If t E [0,1]' then we choose 2l = n( 1. _1.), which gives the regularity Np > n( 1. _1.) p q - p q

of the data. If t E [1, (0), then we choose 2l = n!l (~ - i), which gives the decay rate n-l (1. _ 1.).

2 p q We understand that considerations for small I~I give the necessary regularity

of the data and for large I~I bring the decay rate. Naturally, this approach allows more precise considerations and results. More

over, for more general model equations one cannot expect the existence of explicit representations of solutions, but for a wide class of model equations it is reasonable to ask for representations of its solutions by Fourier multipliers.

If we are interested in Lp - Lq decay estimates for the solutions of the Cauchy problem for the classical Klein-Gordon equation, Utt - to.n u + m 2u = 0, m is a

positive constant, then there exist at least two ways to derive those. On the one hand one can transform the Klein-Gordon equation to a wave equation with nonstandard data [29]. This can be realized via the transformation

(t ) .- -imX n +l (t X ) V ,XI, ... ,Xn,Xn+1 .-e U ,Xl,"" n,

where v solves Vtt - .6n +1 v = 0. But a new problem arises from the structure of the data

Thus one should repeat considerations for wave equations with these non-standard data.

On the other hand one can use Fourier multiplier representations with mass included into the phase [9]. Therefore one has to study the model Fourier multiplier

p-l (eit(<;)m(~);;,21P(<p(y))(~)), l;::: 0,

where (~)m = JI~12 + m 2 . We will not explain in detail how to handle such a Fourier multiplier. In Sections 6.1 and 6.2 we study more complicated Fourier multipliers appearing from Klein-Gordon type equations. We want only to emphasize, that such a representation has disadvantages; the phase function is not homogeneous, the influence of the Hardy-Littlewood lemma remains even if we have no singularities, but t cannot considered as a parameter. The advantage is that the application of Littman's lemma gives instead of n;l the factor ~ in the decay rate, because now the rank of the Hessian is n in a stationary point. The same improvement can be considered by the above described reduction from Klein-Gordon to wave equations.

We want to emphasize once more that the inclusion of the mass term into the phase gives us the possibility to derive the Klein-Gordon decay rate by applying the stationary phase method. But the influence of the Hardy-Littlewood lemma remains.

The second way seems to be more transparent and available and gives e.g., Lp - Lq decay estimates for solutions of elasticity [26], crystal optics [13], thermoelasticity [32].

Summarizing one gets the impression that most of the authors derived Lp - Lq decay estimates for model equations or systems with constant coefficients. If one is interested in studying linear equations or systems with variable coefficients, then new difficulties appear. A first step of investigation toward the solutions is to derive energy estimates (L2 - L2 estimates) taking into account the influences of mass and dissipation.

The weighted energy estimates which are adopted in [7], [16] and [17] are useful to determine some asymptotic behavior for the L2 energy to the solution. Let us consider the equation

Utt -.6.u + 2b(t,x)Ut + c(t,x)u = ° (1.4)

in [0, <Xl) X ]Rn, where non-negative functions b = b( t, x) and c = c( t, x) are dissipation and mass, respectively. Let 9 = g(t, x) and h = h(t, x) be positive weight functions. Multiplying both sides of (1.4) by gu or hUt and integrating over (0, t) x ]Rn

we arrive at the following identity:

t

£(t; g, h) + J 9(s; g, h)ds = £(0; g, h) o

under some suitable conditions to u, 9 and h as Ixl --+ <Xl, where £(t; g, h) and 9(t; g, h) are weighted energies defined by

and

9(t;g,h)

-J (2btg + 2bgt - gtt + Ct h )u2 dx, IRn

respectively. Thus the decay order of the energy should be determined by the weight function h. But we also know that the relation between the weight functions g, h and the coefficients b, C are essential for decay, in other words, the weighted energies [; and 9 should be positive. Indeed, if band c are positive constants, then one can choose the weight function h with exponential order growth. On the other hand, if b decreases like C 1 and c = 0, then one cannot choose any faster increasing weight function than those of polynomial order (see also [28], the constant dissipation case is minutely studied in [10]). Constant mass gives no effect for energy decay without dissipation, but the mass effect is essential for the decay order of the energy if the equation has a dissipative term.

Time-dependent propagation speed makes the problem more difficult, especially an oscillating coefficient. Increasing propagation speed has an effect for energy increasing. Therefore, if one proves some decay estimates, then we should consider a balance between the coefficients. Let us consider the equation

Utt - a(t)6.u + 2b(t, x)Ut + c(t, x)u = 0 (1.5)

which is a generalization of (1.4) to the variable propagation speed case, where a = a(t) > 0 . The Liouville transformation makes the equation (1.4) simpler. Setting r(t) := f~ ~ds and v(r(t), x) = u(t, x) the equation (1.5) is rewritten as

( bat) c Vrr - 6.v + 2 Va + 4aVa Vr + ~v = 0, (1.6)

where t(T) = T-l(T). If a is not oscillating, then (1.6) can be reduced to (1.4). But if a is oscillating, then such a reduction is not always applicable. Moreover, if a vanishes, such a transformation is ineffective (in [5J even some degenerate hyperbolic problems are considered).

One of the merits of the energy method is that one can prove sharp estimates, that is, estimates from above and below, whereas L 2- L2 estimates are less powerful for dealing with nonlinear problems.

1.2. Lp - Lq decay for hyperbolic equations with variable coefficients

To have the possibility to enlarge the class of problems to which one can apply the above-mentioned approach for proving global small data solutions, the second author began together with K. Yagdjian (Armenia) from Kansas State University to study the question for Lp - Lq decay estimates (not only for L2 - L2 estimates as it was done by several authors before) for wave equations with time-dependent coefficients on the conjugate line (p + q = pq). In a series of papers it was shown that this question is quite delicate. Let us explain some of the results for the wave equation with mass term

{ utt-a(t)6u+c(t)u=0, u(O,x) = cp(x), Ut(O,x) = 7/J(x) ,

(t,x) E (0,00) x ~n, X E ~n,

where a = a(t), c = c(t) E COO [0, (0), a(t) > ° and c(t) 2: 0. It turns out that it is useful to write a(t) = )..(t)2b(t?, where).. = )..(t) describes the increasing behavior (improving influence on Lp - Lq decay estimates) and b = b(t) describes the oscillating behavior (deteriorating influence on Lp - Lq decay estimates) of the coefficient a.

Then one has to explain the interplay between both parts. This can be realized by the condition

IDtb(t)1 :s; C)..(t) w

A(t)f3 (lOgA(t)) for large t, (1. 7)

where D t = -iXp A(t) = J~ )..(s)ds, w = ° if (3 E [0,1) and wE (-00, OJ if (3 = 1. If (3 = 1 and w E (-1, OJ, then in [21 J it was shown, that under some conditions the dominating influence of the increasing part guarantees an Lp - Lq decay estimate of wave-type:

IIUt(t, ')IIL q + 11)..(t)\7 xu(t, ')IILq

:s; C" (1 + A(t)) 1+,,- n;-' (~-i) (1Icpllw;+l + 117/Jllw; ), (1.8)

where c is an arbitrary positive real number and L is a positive constant depending on n,p and q. Thus the classical wave decay rate is reproduced (see [27]) up to an unessential error c. But if (3 < 1, or (3 = 1 and w < -1, then one cannot expect an Lp - Lq decay estimate for the wave equation with vanishing mass (see [22]).

From [23] one understands that a mass term can have a stabilizing effect. If e(t) = )..(t)2b(t)2, then the Lp - Lq decay estimate of Klein~Gordon type:

IIUt(t, ')IILq + 11)..(t)\? xu(t, ')IILq

:s; cfiTt)( 1 + A(t)) ~'d'(~~i) (11<pIIW,f+l + 11?f'!II w,f ) (1.9)

holds if f3 E (~, 1] and w E (-00,0). Thus the classical Klein-Gordon decay rate is reproduced (see [9], [29]). By studying a special example it was shown that the Klein-Gordon decay rate is sharp (see [4]).

The case of wave equations with a dissipation, that is,

{ Utt - aCt) 6, U + d(t)Ut = 0, u(O,x) = <p(x), Ut(O,x) = ?f'!(x) ,

(t,x) E (0,00) x]Rn,

x E ]Rn,

is studied in [20]. It is shown that there are similar results as in the Klein-Gordon case, only the methods to prove these are different.

The goal of the present paper is to study the Cauchy problem for

)..(t)2b(t)2 Utt - )..(t?b(t)2 6 U + 28 U = 0 (1.10)

(e + A(t))2'Y (log(e + A(t)))

under the condition (1.7) for f3 E [0,1], "( 2 0 and <5 -I- 0 only for "( = 1. For "( = 0 we have (1.9); for "( = +00, due to our arrangement this means no mass, we have (1.8) if wE (-1,0]. It seems to be reasonable and we will follow this strategy that the oscillating behavior in the mass term coincides with that from the main part.

1.3. Some preparations to the approach

What kind of results do the readers expect?

Cases f3 E (~, 1); f3 = 1, W < -1:

• If "( is small, this means we have a bit smaller mass than in the case "( = 0, then one should expect a Klein-Gordon type decay rate.

• If "( is large, this means we have a small mass, then one cannot expect an Lp - Lq decay estimate; maybe one has a non-decay result of Floquet-type which for a special case was studied in [22].

It should be interesting to describe the change-over from Klein~Gordon to Floquet, to find the critical mass "(0 = "(0(f3) and to study the influence of the mass on decay estimates (see Table 3 in Section 5 for Ol,i)'

Case f3 = 1, w E (-1, 0]:

• If "( is small, then one should expect a Klein-Gordon type decay rate. • If "( is large, then one should expect a wave decay rate.

It should be interesting to describe the change-over from Klein~Gordon to wave decay rates, to find the critical mass "(0 = "(0(f3) and to study the influence of the mass on decay estimates (see Table 3 in Section 5 for 02,i and 03,i).

Case (3 E [0, ~):

• If we have no mass ('Y = (0), then one cannot expect an Lp - Lq decay estimate.

It should be interesting to study what happens during the change-over from 'Y = 00 to 'Y = 0. Does a Floquet-type effect appear? What is the influence of the mass term (see Table 3 in Section 5 for stI,i)?

We try to answer these questions and to describe the change-over to the critical cases. One of the results is the following theorem.

Theorem 1.1: Let us consider the next Cauchy problem

{ Utt(t, x) - (1 + t) 21 b(t? ( 6. -(1 + t)-2k )u(t, x) = 0,

u(O, x) = cp(x) , Ut(O, x) = 'lfJ(x) ,

(t,x) E (0,00) x 1Ftn ,

x E 1Ftn ,

where l 2 ° and k 2 ° are real numbers and b = b(t) is a Coo, positive and periodic function on [0,(0). Then the following Klein-Gordon type decay estimate holds:

Ilut(t, ·)IIL q + 11(1 + tltV7 xu(t, ·)IILq

:s; CJ(1 + t)l ((1 + t)l+l) T-~(~-i) (1Icpllw;P+l + 11'lfJllw;p ),

where~+i=l, 1<p:S;2, Np= [n(~-i)] +1 and

• r = ° if l > 1 and ° :s; k < l - 1, • r is large if l 2 1 and k = l - 1.

If l 2 0 and k 2 l, then there exist no Lp - Lq decay estimates. In this case one can show that the energy has to increase in a special way and that the energy estimate which follows from Gronwall's inequality is near to an optimal one.

Remark 1.1: If l 2 ° and k 2 l the increasing behavior of the energy is described by Theorem 8.1. We will call this behavior classical Floquet behavior (see Definition 8.1). In the exceptional case: l - 1 < k < l, we have no result of Lp - Lq decay. However, we know that the Lq energy of the solution is estimated to a better order from above than the classical Floquet estimate from below (see Remark 8.2). We expect an increasing behavior of the energy in the case l - 1 < k < l, too. For this reason we shall call such a behavior a modified Floquet behavior.

Now let us focus on the following Cauchy problem for second-order hyperbolic equations:

{ Utt(t, x) - )..(t)2b(t)2 (6. - m(t?) u(t, x) = 0, (t, x) ~ (0, (0) x 1Ftn ,

u(O, x) = cp(x), Ut(O, x) = 'lfJ(x) , x E 1Ft , (1.11)

where b = b(t),).. = )..(t) and m = m(t) satisfy the following conditions:


b • I-periodic, non-constant, smooth and strictly positive. A • smooth and strictly positive,

• ,\' (t) 2': 0 and there exist positive constants Ak such that

ID~ A(t)1 ~ Ak (~~~~) k A(t)

for large t and positive integer k. m • smooth and strictly positive,

• m' (t) ~ 0 and there exist positive constants mk such that

ID~m(t)1 ~ mk (~~~D k met)

for large t and positive integer k.

Moreover, we introduce the following assumptions for relations between the coefficients:

(A.1) There exist positive constants bk such that

(i) ID~b(t)1 ~ bk (:(~1(3 ) k , (3 E [0,1) ,

or

(ii) ID~b(t)1 ~ bk ((eo + A(t)) (~~~eo + A(t))t ) k , w E [-1,0] , for

large t and every positive integer k, where eo is a real number satisfying eo 2': e.

(A.2) The function m = met) is defined by A = A(t) as follows:

(i) met) = (eo + A(t)) -I' , ')' 2': 0 , ')' =1= 1 ,

or

(ii) met) = (eo + A(t)) -1 (log(eo + A(t))) -0 ,

Remark 1.2: These functions m = met) satisfy the above assumptions. But it is not necessary that m = met) is explicitly described as in (A.2). However, we shall consider only this case for simplicity.

Remark 1.3: The restriction in (A.1) to the parameters (3 and w is natural from the point of view of Lp - Lq non-decay ([22]). But the periodicity of b is necessary only for the application of Floquet theory. To derive Lp - Lq decay estimates it is sufficient to suppose (A.1).

We shall introduce two typical examples for A and m which satisfy the preceding conditions.

Example 1.1: Let A(t) = (1 + t)l (l 2': 0). Then

met) rv G(l + t)-I'(l+l) for (A.2)-(i),

met) rv G(l + t)-(l+l) (log(eo + t))-8 for (A.2)-(ii).

Noting that A~W(3 = (l + 1)/3(1 + t)I-/3(l+l) the assumption (A.l)-(i) holds for

13 ::; 1~1 ' that is, l ~ 1~/3 .

Example 1.2: Let A(t) = etn , a E (0,1]. Then

Noting that

met) '" G(1 + t)-r(l-a)e-rtn for (A.2)-(i),

met) '" G(1 + t)-(1-a)-a8 e-tn for (A.2)-(ii).

A(t) '" G(eo + t)a-1-aw, (eo + A(t)) (log(eo + A(t))) w

the assumption (A.l)-(ii) holds for w ::; a;:-l, that is, a ~ 12w'

Let us separate the following areas of

D:= {(f3,')',w,0) E [0,1] x [0,00] x (-00,0] x (-00,00) C ]R4}:

(I) 13 < 1, w = 0 = ° : 01.1 = {(f3,')',O,O) ED; ')' < 2f3-1}, 01.2 = {(f3, ')', 0, 0) ED; ')' = 213 - I}, 01.3 = {(f3, ')', 0, 0) EO; 213 - 1 < ')' < f3}, 0l.4 = {(f3, ')', 0, 0) EO; 13 = ')'}, 01.5 = {(f3, ')', 0, 0) ED; 13 < ')'}.

(II) (3 = 1, ')' =I- 1,0 = ° : O2.1 = {(I,')',w,O) ED; ')' > 1, w = O}, O2 .2 = {(I,')',w,O) ED; ')' > 1, -1 < w < O}, O2 .3 = {( 1, ')', w, 0) ED; ')' > 1, w = -I}, O2 .4 = {(I,')',w,O) ED; ')' < 1, w::; a}, O2.5 = {(I,')',w,O) ED; ')' > 1, w < -I}.

(III) 13 = 1, ')' = 1 : O:u = {(I, l,w,o) ED; 0 < 2w -I}, 0 3 .2 = {(I, l,w,o) ED; 2w -1::; 0 < 2w}, 0 3 .. 3 = {(I, l,w,o) ED; 0 = 2w, 0 < w}, 0 3 .4 = {(I, l,w,o) ED; 2w < 0 < w}, 0 3 .5 = {(I, l,w,o) ED; w = 0,0> O}, O:u={(I,I,w,O)ED; -1<w<O, 0>0}, 03.7={(I,I,w,0)ED; w=-I, o~Oorw>-I, O=O}, 0 3 .8 = {(I, 1, w, 0) ED; w = 0 < O}, 0 3 .9 = {(I, l,w,o) ED; w < -1, w < o}, 0 3 .10 = {(I, l,w,o) ED; -1::; w < 0, w < 0 < O}.

Let us explain, as the end of our Introduction, the character of the problem deriving Lp - Lq decay estimates for solutions of (1.10). From the point of view of the authors this is a degenerate hyperbolic problem although the Cauchy problem itself is of strictly hyperbolic type. But there exists an effect similar to the loss of


, :(/3",0,0)

derivatives which is characteristic for such degenerate hyperbolic problems. The time-dependent coefficients don't allow us to consider t as a parameter as in the case of Lp - Lq decay estimates for solutions of Cauchy problems with constant coefficients. Thus t should be included into the phase space, definition of zones, symbol classes etc. The characterization of oscillating behavior (1.7) and main tools of approach are similar to those used in [25] or [30] for weakly hyperbolic Cauchy problems or in [2], [6] or [12] for strictly hyperbolic problems with non-Lipschitz coefficients.

One observation from Table 3 (Section 5) is that in the case of fast oscillations the decay rate differs from the classical decay rate by a constant. This corresponds to the finite loss of derivatives for weakly hyperbolic Cauchy problems and strictly hyperbolic Cauchy problems with non-Lipschitz coefficients. But if the oscillations are slow, then the decay rate differs from the classical one only by an arbitrary small constant c.

Especially the last explanations emphasize that all these degenerate hyperbolic problems have a common feature, but the tools should be chosen according to the special problem under consideration.

From Wave to Klein~Gordon Type Decay Rates 107

We will represent the solution and its first derivatives by Fourier multipliers of the form

F~l (ei'P(t'~)a(t, ~)F(?/J)),

where F denotes the partial Fourier transformation with respect to space variables. Sections 2 to 5 are devoted to deriving WKB-solutions of the form v = v( t,~) =

ei'P(t'~)a(t, ~)va(O + .. '. In Section 2 we introduce several parts of the phase space, so-called pseudo differential zone Zpd(N) and hyperbolic zone Zhyp(N). Appropriate classes of amplitudes having symbol character only in Zhyp(N) are introduced. A diagonalization procedure in Zhyp(N) will be presented in Section 3. It gives the structure of t.p and of the amplitude a. For Klein~Gordon type equations we include the mass into the phase. A hierarchy of symbol classes allows us to carry out an arbitrary number of steps of perfect diagonalization. This implies relations of the amplitude a and a finite number of its derivatives with respect to ~ to classical symbols. Careful estimates for the amplitude yield decay functions on the one hand and differences to the classical wave- or Klein~Gordon decay rate on the other hand (see Section 4 and Table 3 in Section 5). In Section 6 a special Littman type lemma (Theorem 6.1) allows us to deriving Lp - Lq decay estimates of Klein~Gordon type. The main difficulties appear from the non-standard phase function

t

t.p(t,~) = J )..(s)b(s)(Om(s)ds. a

We have fewer problems with the phase function than we have in the wave-type case. Fourier multipliers with amplitudes supported in Zpd(N) or in Zhyp(N) are studied in Sections 6.3.1 and 6.3.2. The main tools are the Hardy~Littlewood lemma, dyadic decompositions of Zhyp(N), a Littman-type lemma and interpolation arguments. Combining all the results from Sections 2 to 6 gives the desired Lp - Lq decay estimates in Section 7. Finally, counterexamples based on Floquet theory are considered in Section 8.

2. Tools of the approach

The approach to prove Lp - Lq decay (or non-decay) estimates is related to the asymptotic behavior of the product between the increasing part).. and the mass m. In the case of constant mass lim )..(t)m(t) = 0 when m == 0 and lim )..(t)m(t) = 00

t---+(.X) t---+oo

when m = ma > O. In the first case we have a wave-type equation and in the second case a

Klein~Gordon type equation. The assumptions (A2)-(i) or (ii) guarantee that the limit lim )..(t)m(t) exists. Thus it seems to be natural to call the equation from

t-+oo

(1.11) a wave-type equation if lim )..(t)m(t) = 0 and a Klein-Gordon type equation t-+oo

if lim )..(t)m(t) = 00. In particular, we shall call the equation of critical type if t-+CXJ

lim )..(t)m(t) = C > O. Now let us illustrate this classification for our Examples t->oo

1.1 and 1.2 if the assumptions for b from Section 1 are satisfied. In the case of Example 1.1 we see that if

({3,'"'(,w, J) E Ol.l U 01.2 U 01.3,

then the equation is of Klein-Gordon type, if

({3,,",(,w,J) E 01.5,

then the equation is of wave type, and if

({3,,",(,w,J) E 01.4,

then the equation is of critical type. In the case of Example 1.2 we see that if

({3, ,",(, w, J) E O2 .4 U O:u U 0 3 .2 U 0 3 .3 U 0 3 .4,

then the equation is of Klein-Gordon type, if

({3, ,",(, w, J) E O2.1 U O2.2 U O2.3 U O2.5 U 0 3 .5 U 0 3 .6 U 0 3 .7 U 0 3 .9 U 0 3 .10 ,

then the equation is of wave type, and if

({3,,",(,w,J) E 0 3 .8 ,

then the equation is of critical type. From now on we suppose that ({3, ,",(, w, J) E 0 \ (01.4 U 015 U O2.5 U {03.8 :

J < -I} U 0 3 .9 ), because we expect a classical Floquet behavior (see Section 8) in these excluded parts of O.

For a given real number N > 1 and a positive constant eo > e we define t.; = t(I';I; N) by the following implicit formula:

A(t.;) (log(eo + A(t.;))) w 1';1 = N

for wave-type equations and

A(t.;)f3 (lOg(eo + A(t.;))) W (';)rn(td = N

for Klein-Gordon type equations, respectively, where (';)rn(td = JI~12+m(t.;)2. Here we note that the definition of tf, is equivalent to

A(t.;)f3(';)rn(tIJ = N

in the case (A.1)-(i) and

A(t.;) (log(eo + A(t.;))) W (';)rn(t<) = N

in the case (A.1)-(ii), respectively. Then we easily see that limlf,I--->o t(I~I; N) 00 for wave-type equations and limlf,I--->o t(I';I; N) < 00 for Klein-Gordon type equations.

Remark 2.1: We want to emphasize that limlf,I--->o t(I';I; N) = 00 for wave-type equations if we adopt the same definition for tf, as for Klein-Gordon type equations.

From Wave to Klein~Gordon Type Decay Rates 109

Remark 2.2: The asymptotic behavior ofte = t(I~I; N) for I~I ---70 gives information about possible results for concrete Cauchy problems of type (1.11). The relation limlel--+o t(I~I; N) < 00 leads in a lot of cases to better decay rates than wave-decay rates if /3 E [~, 1] or to a modified Floquet behavior if /3 E [0, ~). The relation limlel--+o t(I~I; N) = 00 leads in a lot of cases to a wave-decay rate if /3 = 1, w E [-1,0] or to a classical Floquet behavior if /3 E [0,1) or /3 = 1, w E (-00, -1).

Moreover, we can prove the following property for te:

Lemma 2.1: There exist positive constants No, eo and Ck such that

I dkte I < { Ck(~):;;"~tf.) ~g:j, for Klein-Gordon type equations,

dl~lk - Ckl~l~k ~~::j, for wave-type equations,

for all k :::: 1, ~ E jRn \ {O} and N :::: No. Moreover, we have ~ < 0 for any ~ E jRn \ {O}, that is, te is monotone decreasing with respect to I~I.

Proof. We shall prove the statement of the lemma only for Klein~Gordon type equations. Let us consider the following identity:

N 2 = A(te)2,6 (log(eo + A(te)) rw (~);"(td ' (2.1)

where we note that /3 E [0,1), w = 0 or /3 = 1, w E (-00,0]. By differentiating both sides of (2.1) with respect to I~I, noting /3 > , and w = J = 0 or w > J we have

2,6 1 ( )2W dte 2 I~I 0= 2>.(te)A(te) - log(eo + A(te)) dl~1 eo(~) + 2N (~);"(tf.) ,

where

2 ( wA(te) ) eo(~) I~I /3 + (eo + A(te)) log(eo + A(te))

2 (/3 ,A(te) (w - J)A(te) ) + m(tt;) - eo + A(te) + (eo + A(te)) log(eo + A(te))

Choosing eo > e~W yields eo(~) :::: Cll(~);"(tf.) with a positive constant C1

C1 (/3",w,J). This implies the conclusion of the lemma for k = 1. By induction we can prove the statement for all k > 1. 0

We split the set [0,00) x (jRn \ {O}) by te in subdomains which will be called zones. For a real number N > 1 we define the pseudodifferential zone Zpd(N) and the hyperbolic zone Zhyp(N) by

Zpd(N) := ((t,~) E [0,00) x (jRn \ {O}) ; 0::; t::; td and

Zhyp(N):= ((t,~) E [0,00) x (JRn \ {O}) ; t:::: td, respectively.


For further considerations we need suitable classes of amplitudes which have a symbol character only in the hyperbolic zone Zhyp(N).

Definition 2.1: i) For real numbers r1,r2,r3,(3 E (0,1]' W E [-1,0]' we denote for Klein-Gordon type equations by Sp,w {r1' r2, r3} the set of all amplitudes a = a(t,~) E COO((O,(0) x]Rn \ {O}) satisfying

ID~DE'a(t,~)I::; Ca,I(~)~Ctial'x(tr2 ( ,X(t) w)T3+1

A(t)P (log(eo + A(t)))

for all (t,~) E Zhyp(N), all multi-indices a and anl with constants Ca,l independent of N. ii) By 5\'w{r1,r2,r3} we denote for wave-type equations the set of all amplitudes a = a(t,~) E COO((O, (0) x ]Rn \ {O}) satisfying

ID~DE'a(t, ~)I ::; Ca,II~ITI-lal'x(tr2 ( ,X(t) w) T3+ 1

A(t) ( log(eo + A(t)))

for all (t,~) E Zhyp(N), all multi-indices a and anl with constants Ca,l independent of N.

Lemma 2.2: The following rules of the symbolic calculus hold:

(i) Sp,w{r1,r2,r3} C Sp,w{r1 + k,r2 + k,r3 - k} for k 2: 0; (ii) if a E Sp,w{r1, r2, r3} and b E S6,w{ Sl, S2, S3},

then ab E Sp,w{r1 + Sl, r2 + S2, r3 + S3}; (iii) if a E Sp,w{r1, r2, r3}, then Dta E Sp,w{r1' r2, r3 + 1}; (iv) if a E Sp,w{r1, r2, r3}, then DE'a E Sp,w{r1 - lal, r2, r3};

(v) the same rules hold for Sl,w{r1, r2, r3}' Proof. We restrict ourselves to the case of Klein-Gordon type equations. The rules (ii) rv (iv) are evident by Definition 2.1 and the Leibniz rule. To show (i) we use the following estimates in Zhyp(N):


Consequently, Sj3,w{rl,r2,r3} C S;3,w{rl + k,r2 + k,r3 ~ k}, where we take into consideration, that the higher order derivatives can be estimated in a similar way.

D

It is clear that

A E Sj3,w{O, 1, o}, S\w{O, 1, O} and b, mE Sj3,w{O, 0, O}, SI,w{O, 0, O}

from the conditions to b, m and (A.l).

3. Consideration in the pseudo differential zone Zpd (N)

We shall derive an estimate for the fundamental solution. After partial Fourier transformation with respect to x, the problem (1.11) is rewritten as the following Cauchy problem for an ordinary differential equation with the parameter .;:

3.1. Estimates for Klein-Gordon type equations in Zpd(N)

Setting U := (A(t)(';;rn(t)V, DtV)T, equation (3.1) can be transformed to the system of first order

Dt U(t,';) ~ A(t, ';)U(t,';) = 0, (3.2)

where

We are interested in the fundamental solution to the Cauchy problem for that system; this is the matrix-valued solution U = U(t, s,';) to the Cauchy problem:

{ DtU(t,s,';) ~ A(t,OU(t,s,~) = 0,

U(s, s,';) = I, (3.3)

where I is the identity matrix. Using the matrizant we obtain for U = U(t, s,';) the explicit representation

CXJ t T1 'Tk-l

U(t,s,';) = 1+ 2:. J A(TI,';) J A(T2,';)'" J A(Tk,.;)dTk·· ·dTI . k=1 s s

Hence we have

IIU(t,o,<)II < 1 + ~ ~! (!IIA("OlldS) k

( j t jt N(s) jt -m'(s)m(s) ) < exp C )..(s)(~)m(s)ds + )..(s) ds + (~)2 ds

o 0 0 m(s)

< exp ( C A( to) (Om(to) :::(~~\ + log)..( to) + 109(~)m(to))

< N )..(to)A(to)-f3 ( log(eo + A(to))) -w

x exp (CNm(O)A(to)l- f3 -" (log(eo + A(to))) -W-O) < CO,O,N

for any t ::; tf; ::; to with a positive constant CO,O,N, where Nand m' are the first-order derivatives of ).. and m with respect to t and to denotes tf; for ~ = 0. The higher-order derivatives of U = U(t, O,~) can be estimated in a similar way. Thus we obtain the following proposition:

Proposition 3.1: The fundamental solution U = U(t, O,~) of (3.3) satisfies in the case of Klein-Gordon type equations, that is, in 01.1 U 01.2 U 01.3 U O2 .4 U 0 3 . 1 U 0 3 .2 U 0 3 .3 U 0 3 .4 , the following estimates:

IID~ DfU(t, 0, ~)II ::; Cc>,k,N(~):I(~l ()"(t)(~)rn(t)) k

for every k, a and (t,O E Zpd(N).

3.2. Estimates for wave-type equations in Zpd(N)

Setting U(t,O:= ()"(t)I~lv,DtV,)"(t)m(t)v)T equation (3.1) can be transformed

to the system of first order

D t U- A(t,~) U = 0,

where

-)"(t)b~t)2m(t) ) . DtA(t) + Dtrn(t)

A(t) m(t)

_ (DltS) )..(t)I~1 A(t,~) = )..(t)b(t)21~1 °

° )..(t)m(t)

Let U = U(t, s,~) be the matrix-valued fundamental solution to the Cauchy problem

{ ~tU(t, s,~) - A(t, ~)U(t, s,~) = 0,

U(s,s,~) = I. (3.4)

Similar to the con~iderations for Klein-Gordon type equations we have the following estimates for U:

IIU(t,O,~)11

<: exp ( C (j >'(,)liOld, + ! >.(,)m(')d') + ! ~(~? d, -! :(~? d')

< A(t)m(O) exp C A(t)I~1 + A(s)ds 8 ( (

t ))

- A(O)m(t) ! (eo + A(s))l' (log(eo + A(s)))

in O2 .1 U O2 .2 U 0 3 .5 U 0 3 .6 ;

in O2 .3 U 0 3 .7 ;

in {03.8 : 15 E [-1, O)} U O:uo;

for any given E > 0, a positive constant EO, large t and any N = N c , where we use the next estimates in the case w ::::: -1:

A(t)I~1 :s; N( log(eo + A(t))) -w = N l+w log (eo + A(t)) (log(eo + A(t)))

and

t

J ____ A_(-'-S-'-) d_s ___ ----;c8 :s; _1_ ( loge eo + A (t) )) - 8 + 1 .

(eo + A(s))l' (log(eo + A(s))) -15 + 1 o

The higher-order derivatives of U = U(t, 0, 0 can be estimated in a similar way. Thus we obtain the following proposition:

Proposition 3.2: The fundamental solution U = U(t, 0, 0 of (3.4) satisfies in the case of wave-type equations the following estimates for every k, C\' and (t,~) E

Zpd(N):

IID~ DfU(t, 0, ~)II :s; C c ,a,k,NA(t)m(t)-l (eo + A(t))cl~I-lal (A(t)I~I) k

with a positive number Eo in fh3 U 0 3.7 ,

IID~ DfU(t, O,~) II :s; Ca,k,N A(t)m(t)-le (log(eo+A(t)) r 8 +1 I~I-Ial ( A(t) I~I) k

in {03.8 : 15 E [-1, O)} U 0 3 .10 .


4. Consideration in the hyperbolic zone Zhyp(N)

4.1. Estimates for Klein-Gordon type equations in Zhyp(N)

We carry out a diagonalization process to get estimates for the solutions to (3.2). Let us define a diagonalizer for the principal part by

1 [i:(t) (b(t) -1) M(t) = "2V b(i) bet) 1 '

and set Va = Vo(t, s,~) = M(t)U(t, s,~) for the solution of (3.3). Noting the identities:

( ) ( 0 'x(t)(~)m(t)) ()-l _ () ( )/ ) (-1 0) M t 'x(t)b(t?(~)m(t) 0 M t - ,X t b t \~ met) 0 1 '

M(t) (1 0) M(t)-l = ~ (1 0) + ~ (0 1) o 0 201 210 and

[M(t) D ]M(t)-l = _ Dt(,X(t)b(t)) (0 1) Dt,X(t) (-1 1) , t 2,X(t)b(t) 1 0 + 2,X(t) 1 -1 '

we get the following first-order system for Va:

( 4.1)

where

with

Dt,X(t) m(t)Dtm(t) T±(t,O := ±'x(t)b(t)(~)m(t) + ~ + 2(~);"(t)

and

B = B( ~).= _~ (Dt('x(t)b(t)) m(t)Dtm(t)) (0 1) t, . 2 ,X(t)b(t) + (~);"(t) 1 0

Here we note that IT+(t,~) - L(t, ~)I 2 To'x(t)(~)m(t) with a positive constant TO, DE S,a,w{l, 1, O} and B E S{3,w{O, 0,1} by (A.l) and (A.2).

We will carry out further steps of perfect diagonalization, namely, diagonalization modulo S,a,w{ -M, -M, M + I} for some given non-negative integer M.

Let us consider the first step (the further steps can be considered inductively).

We define Ml = Ml(t,~), M(1) = M(l)(t,~) and Rl = Rl(t,O by

Ml = I + M(l) := 1+ ( B~' 7+-7_

~) T_-T+

o


and

R1 = M I 1(Dt -V + B)M1 - (Dt -V) = MIl (Dt M 1 - [V, M1l + BM1)

respectively, where Bij = B(t,f,)ij (1 :-::; i,j :-::; 2) denotes the (i,j)-th element of the matrix B. Then we have

) ( ~ B ) -B12 + 7+-7_ 12 ° B B12B21 21 7_-7+

_~ ) E S{3,w{ -1, -1, 2} .

Taking account of MIl E S{3,w{O, 0, O} (a sufficiently large N will imply this), D tM 1 E S{3,w{ -1, -1, 2}, we have R1 E S{3,w{ -1, -1, 2}. In general, we can prove the following proposition:

Proposition 4.1: For any fixed non-negative integer k there exist matrix-valued functions Mk = Mk(t,f,) E S{3,w{O,O,O}, H = Fk(t,f,) E S{3,w{-1,-1,2}, Rk = Rk(t,f,) E S{3,w{ -k, -k, k + I} such that the following operator-valued equality holds in Zhyp(N):

M;;l(Dt -V + B)Mk = D t - V + Fk + R k,

where Fk is diagonal while the matrices Mk, M;;l E S{3,w{O, 0, O}, provided that the parameter N in (2.1) is sufficiently large.

Proof. We already discussed the case k = 1. Suppose that Mk and Fk have the following representation:

k

Mk = LM(j) , M(O) =1, j=O

k-1

(j :::: 0) ,

Fk = LF(j) , F(O) = 0, F(j) = diag (MjRj ) (j:::: 1) . j=O

Suppose that k:::: 1 , Fk E S{3,w{ -1, -1, 2} and Rk E S{3,w{ -k, -k, k + I}. Noting the properties:

E

and

_ (M(l) + ... + M(k)) F(k)

S", {-k - 1 -k - 1 k + 2} p,W , ,

116

we get

F. Hirosawa and M. Reissig

(Dt - V + B)Mk+l - M k+1 (Dt - D + Fk+d MkRk - MkF(k) - [D, M(k+l)] + B M(k+l) + DtM(k+ 1)

_M(k+l) Fk+1

E S/3,w{ -k - 1, -k - 1, k + 2}.

Finally, we have M(j) E S/3,w{ -j, -j,j}, that is, there exists a positive constant Cj independent of N such that

for any (t,~) E Zhyp(N) and j = 1, ... , k + 1. This implies

A sufficiently large N provides IIMk+l - 111 :::; ~ in Zhyp(N) and, in consequence, the statements concerning Mk+l,Mk~l,Rk+l and Fk+l. 0

Let us consider the system (Dt - V + Fk + Rk) Vk = 0 . Let us define the

matrix-valued functions E2 = E 2(t, r,~) and Rk = Rk(t, r,~) by

and

A(t) A(r)

respectively. Then, we have the following lemma:

Lemma 4.1: The function Rk = Rk(t, r,i:') satisfies the following estimates:

118i8~8fRk(t, r, 011 :::; Ck,l,p,a ('\(t)(Orn(t)) I ('\(r)(~)rn(r) r (~):I(~j ,\( t)

X 2

N1A(t)2!3(log(eo +A(t))) W(Orn(t)

x met) + 1 [ A(t)lal (~)Ial 1

(A(t)!3 (log(eo + A(t))) W (~)m(t)) k-l

in 01.1 U 01.2 U 01.3 U O2.4 U 03.1 U 0 3.2 U 0 3.3 U 0 3.4 with constants Ck,l,p,a independent of N.

P 1 S · ( ) - .\(t) roo. ettmg E2 t, r, ~ - .\(r) (0=(,) E- (t C) . -(e) 2, r,." glves

.." m(r)

Taking note of

(~)~~t) :::; N-1 A(t)!3 (lOg(eo + A(t))) W :::; A(t),

118i8~ 8f £2 (t, r, 0 II :::; Cl,p,a (,\( t) (~)m(t))l (,\( r) (~)rn(r))P A( t) lal

and Proposition 4.1 we obtain

118~8~8nRk(t, r, 0 + Fk(t, 0) II :::; Ck,l,p,a'\(t)A(t)la l

X ('\(t)(~)rn(t)) I ('\(r) (Om(r) r (O~~t) (A(t)!3 (log(eo + A(t))) W) -k-l

for any l, p and ct, where the constants Ck,l,p,a are independent of N. By Proposition 4.1 we have

I -Ial '\(t) < CUa('\(t)(Om(t)) (~)m(t) ( )2W

NIA(t)26 log(eo + A(t)) (~)rn(t)

C (,\( )() )l( )-Ia l '\(t) :::; k.l.a t ~ met) ~ met) ( ) 2w

A(t)2!3 log(eo + A(t)) (~)rn(t)

for any N 2': 1. Noting 118i8~8fRkll :::; 118i8~8f(Rk + Fk)11 + 118i8fFkli the lemma is proved. 0

By the aid of Rk we define for t 2': t~ the matrix-valued function

t t, tJ-l

Qk(t,t~,~):= ~ij J Rdtl,t~,Odtl J Rk(t2,t~,~)dt2'" J Rk(tj,t~,~)dtj. ]- tf, tf. tf.

Then the matrix-valued function Vk = Vk(t,tE,,';) = E 2(t,tE,,';)(I + Qk(t,tE,,';)) solves the Cauchy problem

{ (Dt-V+Fk+Rk)Vk(t,tE,,';) =0,

Vk(tE" tE".;) = I (t 2': tE,).

Therefore, we can conclude that the fundamental solution to (3.3) with s = 0 has the following representation:

U(t, 0,';) = M(t)-l Mk(t, ';)E2(t, tE" ';)(1 + Qk(t, tE" .;)) XMk(tE,,';)-l M(tE,)U(tE" 0, ';).

To estimate the fundamental solution U(t, 0,';) we note that

where besides a careful calculation shows that

118fE2(0,tE".;)II::; Ca(.;)~I(~~)"

It remains to estimate Qk = Qk(t,tE".;)'

( 4.2)

Lemma 4.2: The function 8 k,a = 8 k,a(t,.;) := 118fQk(t, tE,,';) II satisfies the fol

lowing estimates for any (t,.;) E Zhyp(N) and a satisfying 10'1 < k(!3-1~~!3-1 in

01.lU01.2U01.3U02.4 and 10'1 < k(w-~:W-l in 03.1U03.2U03.3U03.4! respectively:

(i) if ((3",w,6) E 0l.1 U O2 .4 U 0 3 .1 , then there exists a positive constant Ck,Oi

such that 8 k,0i(t,';) ::; Ck,Oi(.;)~I(~~; (ii) if ({3, " w, 6) E 0 3 .2 , then for any positive constant s there exists a positive

constant Ck,Oi,c such that 8 k,0i(t,.;) ::; Ck,Q,c(O~I(~~ (eo + A(tW; (iii) if ({3, " w, 6) E 01.2 U03:>! then there exist positive constants So 2': 1 and Ck,Q

such that 8 k,o,(t,';) ::; Ck,Q(O~I(~~(eo + A(tWo; (iv) if ({3, " w, 6) E 0 3.4, then there exists a positive constant Ck,Q such that

(v) if ({3, " w, 6) E 01.3, then there exists a positive constant Ck,a such that

8 k,o:(t,';) ::; Ck,a(O~I(~j exp ((eo + A(t)),-2!3+1) .

Proof. Let us prove the statements in the case 10'1 = ° with k = 1. In this case it is sufficient to estimate the integral (see Lemma 4.1)

Then E>k,O(t,O :s; Co exp((h (t, ti,;, ~))(~);;,I(~;. Noting limli,;l->o ti,; = to « 00) we have

to 1 j ..\(s) A(ti,;)-2i3+1

--------....,;2,---w ds :s; :---:---=--~-:-:----

(~)rn(to) tf. A(s)2i3 (lOg(eo + A(s))) (2(3 - l)(~)m(to)

< (~)m(td A(t )-13+ 1 < (~)m(O) A(t )-13+1 - N(2(3 - l)(~)m(to) i,; - N(2(3 - l)(~)m(to) 0

m(0)A(to)-i3+1 < ----,':""":""--'---'--,---,----,-- N(2(3 - 1) m(to)

for (3 > 1/2, w = 6" = 0 and m satisfies (A.2) (i) or (ii) with <5 = 0,

to

1 j ..\(s) d

( )2W A(S)2 s

log(eo + A(to)) (Om(to) te

1 1 <

(log(eo + A(to))fw (Om(to) A(ti,;)

< m(O)

for eo > e, (3 = 1, w :s; 0 and m satisfies (A.2) (ii). Furthermore, we obtain

t j 1 ..\(s) ds

to A(s)' (log(eo + A(s))) S m(s) A(S)2i3-, (log(eo + A(s))) 2w-S

t

< ( eo+A(to))'j ..\(s)ds

A(to) to A(sj2i3-,( log(eo + A(s))fw-

s '

here we note that e01(~(;~0) is estimated by a positive constant independent of N 2: l.

If ((3",w,6") E 0 11 U O2 .4 U 03.1, then e1(t,to,O :s; el(OO,to,~) < 00. Thus, we obtain (i).

If ((3", w, <5) E 0 3 .2 , then for any positive constant c there exists a positive constant Cc such that

eo + A(to) ( eo + A(t) ) el(t,to,~):S; A(to) log log eo+A(to) :S;Cc+log(eo+A(t)r·

Thus we obtain (ii).

If ({3",w,6) E n12 U n3 .:1, then there are positive constants C and Eo such that

(h(t,to,~):S C+log(eo+A(t))"o.

Thus we obtain (iii). If ({3", w, 6) E n3 .4, then we have

t

< 1 J >.(s) ds

A(to) (log(eo + A(to))) 0 m(to) to A(s) (log(eo + A(s))fw-

o

( ) -2W+O+1 :S C log(eo + A(t))

Thus we obtain (iv). If ({3", w, 6) E nl.3, then we have

Thus we obtain (v). In the case that \0:\ 2: 1 and k 2: 1 we have to estimate among other things

the following integral (see Lemma 4.1):

t J _______ >._('---s'---) ---w-'-;-Ck--'--+"'l)--- ds =: 02(t,t~,~). t" A(s)!3Ck+1)-la l (log(eo + A(s))) (O~(~)I

If, = 0, that is, m is a positive constant, then O2 (t, t~,~) is estimated by 01 (t, t~,~) since \0:\ :S (3(k - 1) as in Corollary 5.1 from [23].

In the case that ({3",w,6) belongs to nl.l U n1.2 U nl.3 U n2.4 we have

t J >.(s) d A(s)!3Ck+1)-la l (~)k-Ial S

t" mCs)

t

< 1 J >.(s) d A(t ),Ck-1a l) (~)k-Ial A( s )!3Ck+1)-la l-,Ck-1al) s

~ mCt,,) t"

< C N-k+laIA(t )(1-!3lCH1a l) < C A(t )(1-!3)CH1a l) !3",k,a ~ _ !3",k,a 0

for k 2: \0:\, where we used the inequality (3(k + 1) - \0:\ -,(k - \0:\) > 1.

In the case that ((3", w, 6) belongs to n:u u n3 .2 u n3 .3 u n3 .4 we consider the integrals O2 (to, t~, 0 and O2 (t, to,~).

For the first integral we have

Ih(to, tt,,';)

Jt o ( log( eo + A( s))) -wlnl ).( s )ds

- tf. {A(s) (log(eo + A(s))) W (';)m(s)}k-Inl-l A(s)2ln2W(eo + A(s))(Om(s)

( log (eo + A(to))) -wlnl

::; Nk-Ial-l (h(to,tt,,';)

for k 2: lal + 1. Moreover, taking account of the inequality w(k + 1) - 5(k -Ial) > 1, we have

1

( ( ) 8 )k-Ial A(to) log(eo + A(to)) (';)m(to)

t

X J ).(s) ds < C ( )

w(k+l )-8(k-lal) - w,8,k,a· to A(s) log(eo + A(s))

By the previous estimates for fh and O2 , Lemma 2.1, Lemma 4.1, the Leibniz formula and monotonicity for (';)~~t)' eo+A(t) with respect to t, we can also prove the other integrals to get the desired estimates for the higher-order derivatives of Qk = Qk(t, tt,,';) with respect to .;. 0

Remark 4.1: The statements of Lemma 4.2 yield estimates for the amplitudes (appearing in the representations of solutions for (1.10)) in the hyperbolic zone. All these estimates allow Lp - Lq decay estimates besides those for 01.3U03.4 which are better than the classical Floquet behavior from Section 8, but worse than the Lp - Lq decay estimates derived in Section 6. Thus it seems to be reasonable to expect a modified Floquet behavior in 01.3 U 0 3.4.

4.2. Estimates for wave type equations in Zhyp(N) Let v = v(t,';) be a solution of (3.1) in the case of wave type equations. Setting U:= ().(t))I';lv,Dtv)T, then (3.1) can be transformed to the following first-order system:

Let [; = U(t, s,.;) be the matrix-valued solution to the Cauchy problem

{ D, U(t, "I;) (

U(s,s,';) = I. (4.3)


Using the same diagonalizer M as in the Klein-Gordon case, we have

(Dt - D(t,~) + 13(t,~)) Va(t, s,~) = 0,

where Va := MU, D = D(t,~) := (TO T:)'

13 = 13(t C) '= _~ (A(t)b(t)2m(t)2 Dt(A(t)b(t))) ( 0 ,.". 2 I~I + A(t)b(t) 1 ~ )

and - '= ±A( )b( )ICI DtA(t) A(t)b(t)2m(t)2 T± . t t." + A(t) + 21~1

Here we note that T _ -T+ = -2Abl~1 E S\,w{l, 1, a}, (T _ _ T+)-l E Sl,w{ -1, -1, o}

and Ab2m 2 /I~I :::; bi A/ (N A(log(eo + A))w), thus Ab2m 2 /I~I E Sl,w{O, 0, I}. In analogy to the diagonalization procedure for Klein-Gordon type equations,

we obtain the following proposition:

Proposition 4.2: For any fixed non-negative integer k there exist matrix-valued functions Ah = l'Vh(t,~) E Sl,w{O, 0, O}, A = .h(t,~) E Sl,w{ -1, -1, 2}, Rk = Rk (t,~) E Sl,w { - k, -k, k+ I} such that the following operator-valued equality holds in Zhyp(N):

-1 - -- - --M;; (Dt - V + B)Mk = D t - V + Fk + Rk,

where A is diagonal while the matrices Nh, Mk 1 E Sl,w{O, 0, O} provided that the parameter N in (2.1) is sufficiently large.

Let us define the matrix-valued function

Rdt, r,~) = -Fk(t,~) - E2(r, t, ~)Rk(t, ~)E2(t, r,~),

where

) Lemma 4.3: The function Rk = Rk(t, r, 0 satisfies the following estimates:

lIa~afafRk(t, r,~) II :::; Ck,l,p,ex()..(t) IW I ()..(r)I~I)p I~I-Iexl

A(t) ( A(t)lexll~llexl )

x A(t)2(log(eo + A(t)))2w I~I (A(t) ( log(eo + A(t))) w 1~lf-1 + 1

with constants Ck,l,p,ex independent of N.

Proof For a proof we refer to the proof of Lemma 4.1. D

By the aid of Rk we define for t 2': tf, the matrix-valued function

t 71 7j_1

Ch(t,tf,,i:,):= ~ij J Rk(T1,tf,,~) J RdT2,tf,,~)'" J Rk(Tj,tf,,~)dT1···dTj. J- t< t< t<

Lemma 4.4: The function 8k,o: = 8k,o:(t, 0 := 118fQdt, tf" ~)II satisfies the following estimates for any (t,O E Zhyp(N) and a satisfying lal s k - 1:

(i) if ((3",w,15) E O2.1 U O2.2 U [h5 U 0:i.6, then for any positive constant E

there exists a positive constantCk,o:,E such that8k.o:(t,~) S Ck,o:,EI~I-lo:l(eo+ A(t))c;

(ii) if((3",w,15) E 02.3U03.7U{03.8: 5 E [-I,O)}U03.1O, then there exist positive constants EO 2': 1 and Ck,o: such that 8 k,o: (t,~) S Ck,o: 1~1-lo:l (eo + A( t) )co .

Proof. Let us consider the case lal = ° and k = 1. We define B1 = B1 (t 2 , t1, 0 by

t2 - J A(S) B1 (t2,t 1 ,O:= 2w ds. t, A(s)2 (log(eo + A(s))) I~I

Then 8 k,0(t, 0 S Co exp(Bl (t, tf,,~)) 1~1-lo:l. Let To be a positive constant satisfying tf, < To < t. We shall separate our considerations to the intervals (tf" To) and (To, t).

The proof in the cases O2.1 , O2.2 , O2 .3, 0 3.5 , 0 3.7 , {03.8 : 5 E [-1, O)}, 0 3.10

is similar as that for 0 3 .6 . The function B1 (To, tf" 0 is estimated by

< A(tf,) (log(eo + A(To)) rw I~I < 00.

Integration by parts and monotonicity of A gives

Bl(t,To,~) < ~(lOg(eo+A(t)))-W t J -2WA(S)

+ 2 +1 ds . To A(s)(eo + A(s)) (log(eo + A(s))) w I~I

Using t J -2WA(S) -2w --------'--'-------;2'w--;+'1- ds S N log(eo + A(t))

To A(s)(eo + A(s)) (log(eo + A(s))) I~I

we obtain to any given positive constant e with a suitable large N = N(e) the estimate

jj1(t, tf,,~) :::; CN + log(eo + A(t))O: .

Thus, the case S13 .6 is proved. In the case lal 2:: 1 and k 2: 1 it remains to estimate the integral (see Lemma

4.3) t J ),(s) w(k+1) ds =: jj2(t,tf"O·

t" A(s)k+1- Ja J (lOg(eo + A(s))) 1~lk-JaJ

Using the same steps as in estimating jj2(to, tf" 0 in the Klein-Gordon case we get for jj2(t, To,~) the estimate

t

J {( 1 r r-1a, ' To A(s) log (eo + A(s)) l~l

),( s) ( ) -w(JaJ+2) X A(s)21~1 log (eo + A(s)) ds

< CN,To(eO + A(tW

for any positive real number e > 0 with a sufficiently large N = N(e). By the previous estimates for jj1 and jj2, Lemma 2.1, Corollary 4.2, the Leibniz

formula and monotonicity for eo + A(t) with respect to t, we can also estimate the other integrals which are necessary to derive the desired statements for higherorder derivatives of Qk = Qk(t, tf,,~) with respect to ~. D

5. Estimates for the amplitudes

By using the approach which was described after the proof of Lemma 4.1 we get WKB-representations for the solutions to (3.1) containing Fourier multipliers of

the form F- 1 (ei'P(t,f,) aCt, ~)F( 7/')) , where <p( t,~) = J~ ),( s )b( s) (~)m(s) ds for Klein-

Gordon type equations and <p(t,~) = J~ ),(s)b(s)dsl~l for wave-type equations. Here

we applied Lemma 2.1 in the wave case to derive llot exp(i J~" ),(s)b(s)dsl~l)ll :::; Ca,o:l~lo:-JaJ.

The following tables express the obtained estimates for the amplitudes. The number "0" in these tables means that the amplitude is uniformly bounded with respect to t, the letter "e" means that the amplitude is estimated by Co:(eo+A(t))c for any given positive real number e, the letter "eo" means that the amplitude is estimated by C(eo + A(t))co for a positive constant eo (it may depend on n), the

term "eiogS " means that the amplitude is estimated by C exp (( log(A(t))r), the

term "e s " means that the amplitude is estimated by C exp(A(t)S) and finally KG,

TABLE l. Estimates of U(t,O,O in Zpd

\i 1 2 3 4 5 6 7 8(0<-1) 8(-1<0<0) 9 10

D1.i 0 0 0 - -D2.i c c co 0 -

D3.i 0 0 0 0 c c co - e 10g 8+1 - e 10g 8+1

TABLE 2. Estimates of 8k,0(t,~) (or 8k,0(t, ~)) in Zhyp

\i 1 2 3 4 5 6 7 8(0<-1) 8(-1<0<0) 9 10

D1.i 0 co el'-2i3+ 1 - -

D2 .i c c co 0 -

D3.i 0 e 1og8 2w+l

C co c c co - co - co

TABLE 3. Estimates in Zpd U Zhyp

\i 1 2 3 4 5 6 7

D1.i O/KG co/KG el'-2i3+ 1 F F

D2 . i c/W c/W co/W O/KG F

D3.i O/KG c/KG co/KG e 1og8 2w+l

c/W c/W co/W

8(0<-1) 8(-1<0<0) 9 10

F e 10g -6+ F elogO+1

Wand F mean Klein-Gordon type decay, wave-type decay and classical Floquet behavior, respectively.

What can we expect from these tables, especially from Table 37 In Dl.l UD1.2U D2.4 UD3.1 UD3 .2 UD3 .3 we expect an Lp - Lq decay estimate of Klein-Gordon type. But therefore we have to prove a Littman-type lemma for oscillating integrals with phase functions depending on t and ~ but in a more complicated way than for those studied in [23]. This will be done in Section 6.1. In Ih 1 UD2.2UD2.3UD3.5UD3.6UD3. 7

we expect an Lp- Lq decay estimate of wave type. Here the question for a Littmantype lemma is easy to answer, but Lp - Lq decay estimates for Fourier multipliers should be proved. Some model Fourier multipliers will be studied in Section 6.2. In D1.4 U D1.5 U D2.5 U {D3.8 : r5 < -I} U D3.9 the solutions show a classical Floquet behavior. Such a behavior was observed and studied in [22] for a special case. The proof, which is based on the considerations of [22], will be given in Section 8.


Between these qualitative quite different behaviors for the solutions we can find a modified Floquet behavior in some parts of n1.3Un3.4u{[hs : 6 E [-1,0)}Un3.10 .

Up to now we are not able to show that the behavior suggested by Table 3 is optimal in these four parts (see Remark 8.2).

6. Littman-type lemma and Lp - Lq decay estimates for Fourier multipliers

6.1. Littman-type lemma for Klein-Gordon type equations

Here we prove a Littman-type lemma (see [14]) for oscillating integrals having as one part of the phase function J~ )..(s)b(s)(~)m(s)ds. The time dependence of m raises new difficulties. In [23] the case in which m is a positive constant is considered.

Theorem 6.1: Let us suppose that 18fa( t,~) I :::; C (~) :I(~l for lal :::; n + 2 and (t,~) E [to, 00) x lRn \ {O}. Let r be a positive constant and let the function 'lj; = 'lj;( T) be given from COO(lR.) with'lj; == 1 for ITI :::; r , 'lj;(T) == 0 for ITI ~ 2r and 0:::; 'lj; :::; 1. The mass term is chosen as m = m(t) = (eo + A(t))-"Y with"Y E [0, ;3). Then there exists a constant C such that

-1 ( if >.(s)b(s)(i;)m(s)ds ( Ii;I )) ()-~(1+"Y) F e 0 a(t,O'lj; met) :::; C 1 + A(t) .

Loo(JR:n)

Proof. For the stationary points of the phase function we get the relation

t

X + J )..(s)b(s) ~ds = O. (~)m(s)

o

Case 1: Let us suppose that to given x and t we have no ~ E lRn \ {O} such that this equality is satisfied. Using spherical polar coordinates it is clear that there exists no ~: I~I E lR+ fulfilling

t J )..(s)b(s)I~1 Ixl = (1~12 + m(s)2)1/2 ds .

o

Th f t · f f() rt >.(s)b(s)y d . ... (0) e unc lOn = y = Jo (y2+m(s)2)'/2 S 1S monotone 1ncreas1ng III y E ,00.

Consequently, Ixl ~ sup f(y). We estimate the function yE(O,CXJ)

t

q> = q>(x, t,~) = x + J )..(s)b(s) ~ds (~)m(s)

o

in the following way:

1cI>1 2:: Ixl - f(IW 2:: f(l~ol) - f(l~I),

where ~o E IRTI \ {O} will be chosen later. To estimate f(l~ol) - f(I~I) we proceed as follows:

t J ( I~ol I~I) f(l~ol) - f(IW = ).(s)b(s) (~O)m(s) - (Om(s) ds o

t

> Cb J ).(s) ( I~ol - I~I ) ds. - (~O)m(s) (~)m(s)

o

Taking account of Lemma A.l from the Appendix we get

t

J ).(s) ( I~ol - I~I ) > _1_A(t) ( I~ol - I~I ) (~O)m(s) (~)m(s) - 1 + "I (~O)m(t) (~)m(t)

o

Thus,

For a given t we choose ~o such that !(;) = 3r. The conditions with respect to

'lj; guarantee rl1~) s: 2r. Hence, 1cI>1 2:: Cb."rA(t). This allows us to construct the n

operator L = 1cI>1-2 2: cI>rDf,r satisfying r=l

t t ix·E+i J ).,(s)b(s)(Om(s)ds ix·f,+i J ).,(s)b(s)(f,)m(s)ds

Le 0 =e 0 •

Integration by parts yields

_1<_1 <2r m(t)-

for all t 2:: to. Here we need N derivatives of a = a(t,~) with respect to~. Moreover we use that lat 1:121 s: Cl A(t)-l(l-,); 1cI>1 2:: Cb",r A(t) and supp 'lj;' belongs to

{~ E IRn : rl1~) E (r, 2r)}.

Using the assumptions of the theorem we have shown in the first case

t

I J ix·HiJ '\(s)b(s)(e)>n(s)ds ( lei) I e 0 a(t,f,)'l/J m(t) dt;,

( ) -n-2(l-'Y)

~ CN,b,'Y,r 1 + A(t) .

(6.1)

Case 2: Let us suppose that for given x and t we have a stationary point t;,o E

IRn \ {O} satisfying !Ctl) 2: 3r. Then we can follow the ideas from the Case 1 and get an estimate of type (6.1).

Case 3: For given x and t there exists a stationary point t;,o E IRn \ {O} fulfilling J;Ctl) ~ 3r. The stationary point is uniquely determined. We develop the phase function

'P = 'P(x, t, t;,) = 'P(x, t, t;,o) + (t;, - t;,O)T H<p(p)(t;, - t;,o) ,

where H n(t)-

The exponent is a positive quadratic form, thus

n t ( ) A(S) pjPk 2: Cb :L J ( ) 8jk - ( )2 ds(t;, - t;,O)j(t;, - t;,O)k

j,k=lO P m(s) P m(s)

t () n ( ...J!.L~) _ A s m(s) m(s) - Cb J ( ) L O;k - ( , ) (~ - ~");(~ - ~O)k ds .

P m(s) ·k=l I~I + 1 0]' m(s)


We know that I rr!'w I ~ I rr!ttJ I ~ 3r. Thus we have a family of positive definite quadratic forms on a compact set with respect to the matrix of coefficients

n

Cll~ - ~012 ~ L Ajk(~ - ~o)j(~ - ~O)k ~ C21~ - ~012 . j,k=l

Similar to Lemma A.l one can prove the second inequality in

t

A(t) > j ..\(s) ds> _1_ A(t) (Plm(t) - (Plm(s) - 1 + 'Y (Plm(t)

o

The first inequality is trivial. Summarizing brings

iX.~+iJ A(s)b(s)(~)",(s)ds (~) I e 0 a(t, ~)'lj; met) d~

The above calculations show that there exists for each t 2:: to a diffeomorphism with a Jacobian m(t)n for all t such that

t m(t)2 jt ..\(s)b(s) (Sjk _ Pj:k ) ds (~ - ~o) (~ -~o) j,k=l 0 (Plm(s) (Plm(s) m(t) j m(t) k

will be mapped to m(t)A(t) 11712. It is enough to mention that the integration is over a uniform neighborhood U of 0 with respect to 17. By using the assumptions concerning the amplitude a = a(t,~) and by [14), [18] (here we need lal ~ n + 2) the decay function of

is

j eim (t)A(t)I'11 2 a(t,~(17))'lj;(17)d17 U,,(O)

( ) -~(1+1)

1 + A(t) , (6.2)

where 8~a(t, ~(7])) brings no difficulties due to the assumptions for 18fa(t, ~)I, see for example

1

187)a(t, ~(7]))1 :::; C1 (~)~~t) 187)~(7])1 :::; C 1m(t) (1~12 + m(t)2) -2 :::; C1 .

We are integrating over a compact set with respect to m~t). If we fix a point m1t)

belonging to this set, then we can find a uniform neighborhood such that the Morse lemma is applicable and leads to the decay function from (6.2). Consequently, a finite number of applications allows us to study our starting integral in the Case 3. Thus we have studied all possible cases. From the estimate (6.1) together with the decay function (6.2) we conclude the statement of the theorem. 0

Remark 6.1: In the case of m(t) = (eo +A(t))-l (log(eo +A(t))) -8 with <5 < w < 0

we can prove by following the same approach under the same assumptions

-1 (iJA(S)b(S)W=(S)dS ( ) (1';1)) (log(eo+A(t)))~ F e 0 a t ~ 'Ij; - < C -'------;-------:-;;;-'--'m(t) - (I+A(t)r

L=(lRn )

6.2. Lp - Lq decay estimates for Fourier multipliers in the Klein-Gordon case In this subsection we will study Lp - Lq decay estimates for model Fourier multipliers. From these results we immediately obtain the main results of this paper about Lp - Lq decay estimates for solutions of (1.10) in the Klein-Gordon case which are formulated in the next section (see Theorems 7.1 and 7.2). As we explained in the Introduction the inclusion of the mass into the phase allows us to derive Klein-Gordon decay rates. But the general time-dependent mass term gives the phase function a more complicated structure. Thus new difficulties appear for application of the stationary phase method.

Proposition 6.1: Let us suppose that 18fa(t,~)1 :::; C(~):I(~~ for 10:1 :::; n + 2 and (t,~) E [to, 00) x IRn \ {O}. Then under the assumptions of Theorem 6.1 we have

-1 ( iJ A(s)b(s)(';)=(s)ds -21 (J5.L)) ()-~ F e 0 a(t, ~)(~)m(t)X met) :::; C 1 + A(t)

L= (IRn)

if 2l = :g., where X = 1 - 'Ij; and 'Ij; is taken from the previous theorem.

Proof. We choose a non-negative COO-function cP = cP(T) having compact support in {T E IR : 2mo :::; 7 :::; 2mo+2}. We set cPk(7) := cP(2- kT) while cPO(T) := 1 -2:~=1 cPk(T). One can find such a function that 2:;:'-00 cP(2- kT) = I, 7 =1= O. Hence, supp cPo C {7 E IR : 171 :::; 2m o+2}. We consider for k 2: 0 all Fourier multipliers

( iJ A(s)b(s)W=(s)ds () ( )) F- 1 e 0 a(t,~)(~)~(~)X rl1L cPk A(t)!3(~)m(t) F(zp) .

From Wave to Klein-Gordon Type Decay Rates 131 But it is enough to study these multipliers for k ::::: ko, ko large. The reason is that from 2mo+k ::=; A(t)!3(c,;m(t) ::=; 2mo+k+2, t::::: to, it follows that

1

A(t){3 (~)m(t) = A(t){3m(t) ( (~~~)) 2 + 1) 2 ::=; 2mo+k+2, J{L <C m(t) - k,

respectively. Thus we can apply the previous theorem. Let us consider the Fourier multipliers for k ::::: ko. We use the transformation

The function X vanishes for small ~~~)' Theorem 6.1 allows us to include this function into the amplitude. The function X helps us to understand that the transformation ~ = ~(rJ)rJ in radial direction has a Jacobian which can be estimated by

C( A~~(3)n. Thus, setting

t

'_Jixl;+i[)..(S)b(S)(I;)m(s)dSa(t,~) (11;1) ({3 ) h·- e (62lX met) cPk A(t) (~)m(t) d~

IIl:n met)

we have

_ I J i x'l;+i J )..(s)b(s)(I;)m(s)ds a(t,~) (11;1) ({3 ) I Ihl- sup e 0 -(C)21 X met) cPk A(t) (~)m(t) d~ xEIIl:n IIl:n ." met)

1

_ I J i Y'7J+il )..(s)b(s) ( 2:~t;;~i +m(s)2_m(t)2) 2 dsa(t,~(rJ))A(t)2{31 - sup e 22kl ( )21

yEIIl:n rJ 1 IIl:n

XX( 1.~:(2)1 )cP( (rJ)dl det J 7J (OldrJ l

k( -21) (3(21-) I J iY·7J+i J ),,(S)b(s)(2:;t;;~r +m(s)2_m(t)2) ~ ds ::=; C 2 n A(t) n sup e 0

yEIIl:n IIl:n

xa(t'~(rJ))X(II;(7J)I)cP((rJ) )d I (rJ)i l met) 1 rJ·

Let us study the function X( I~~»I). The support of the derivatives lies in the set

{(t,~): C 1 ::=; I~I/m(t)::=; C2 } with suitable positive constants C 1 and C2 . Hence,

C 2 < 22k (rJ)i _ 1 < c2 1 - A(t)2{3m(t)2 - 2'

(6.3)

that is, all derivatives o~X are bounded. As in the proof of the previous theorem we see that the supposed estimates for ofa are transferred to o~a(t,~(TJ)). Setting

it remains to derive an estimate for

where K denotes a compact set with respect to TJ, which is away from zero and the same for all t 2 to.

Case 1: Let us suppose that the phase function has a stationary point TJo, this means, for given y and t there exists TJo with

where

jt 22k

Y + A(t)2f3 ).,(s)b(s)g(s, t, (TJoh)-lds a

TJo = 0,

g(s, t, (TJ)d := 22k (TJ)i 2 2 A(t)2f3 + m(s) - met) .

Moreover we suppose that TJo belongs to a large ball BR(O). For the Hessian of the phase function we obtain

H (p) = (jt ~ ).,(s)b(s) (15 k _ ~prif;)i~Pk ) dS) n

'P A(t)2f3 g(s, t, (ph) J 22k(~)r + m(s)2 _ m(t)2 a ACt) {3 j,k=l

The bounded ness for possible TJo implies the bounded ness for possible p. The monotonicity of m = met) brings the uniformly positive definiteness of the following quadratic form in TJ - TJo:

<

<

with suitable positive constants C1 and C2 . The integral

t

j ).,( s) ds is equivalent to g(s, t, (p)d

a

A(t) get, t, (ph)


Thus we get as a large parameter A~tk)#. As in the proof of Theorem 6.1 we apply the Morse Lemma, [14], [18] (here we need a :::; n + 2) and obtain

I J iY·7J+i} A(s)b(s)(22kA(t)-21'l(7J)~+m(s)2-m(t)2)!ds I sup e 0 v(t, TJ)dTJ

yEIRn K

:::; C(2kA(t)1-,B)-~.

But this gives for h the estimate

Ihl :::; C A(t),B(21-n)2k(n-21) (2k A(t)1-,B) -~ :::; C( 1 + A(t))-~

if 2l = :g:.

Case 2: Let us suppose that the phase function has a stationary point TJo outside the large ball BR(O). Then we have to estimate

22k A(s)ds A(s)ds I (

t t )

A (t)2,B ! g(s, t, (TJoh) ITJol - / g(s, t, (TJh) TJI .

For this difference we get by using Lemma A.2 the estimate from below

~ A(s)ds ITJol _ A(s)ds ITJ > 2kA(t)1-,B ~_1IL ( t t )

A(t)2,B ! g(s, t, (TJoh) / g(s, t, (TJh) I - ((TJoh (TJh) ,

where TJo ~ BR(O) and TJ E K. After choosing the operator L as in the proof of Theorem 6.1 and partial integration leads to

It (22k('I)2 2 2)! I J i Y·7J+ i J A(s)b(s) ~+m(s) -met) ds

sup e 0 A(t) v(t, TJ)dTJ yEIRn

K

:::; C 2k(n-N) A(t)-N-,B(n-N) :::; C A(t)-n

for N = n.

Case 3: If the phase function has for given y and t no stationary point, then we use the monotonicity of

t J A(s)dsITJol g(s, t, (TJoh)

o

with respect to ITJol and proceed as in the previous case. From the estimates for Ik and with [1] we conclude the desired L1 - Loo estimate. 0

Remark 6.2: In the case of m(t) = (eo +A(t))-l (log(eo + A(t))) -6 with 6 < w < 0

we can prove under the same assumptions by following the same approach t

II F- 1 (ei£ .\(s)b(s)(Om(s)ds a(t ~)(~)-21 X (JS.L)) II ~ c (1 + A(t))-~ , met) met) Loo(JRn)

if 2l = ~, where X = 1 - 'IjJ and 'IjJ is taken from the previous theorem.

Straightforwardly, one can derive corresponding L2 - L2 estimates. These estimates and the L1 - Loo estimates from Theorem 6.1, Proposition 6.1 and Remarks 6.1, 6.2 give, after applying an interpolation argument [1], the next results.

Corollary 6.1: We assume for the mass term m(t) = (eo + A(t))-'Y with I E [0, ,6), ,6 E (0,1]' where,6 is taken from (1.7). Let us suppose that 18fa(t, ~)I ~

C(~):I(~J for 1001 ~ n+2 and (t,O E [to, (0) xIRn\{o}. Then we have the Lp-Lq decay estimate

t

IIF- 1 (ei£ .\(s)b(s)(~)m(s)ds a(t,~)F(CP)(~)) II Lq(JRn)

~ C (1 + A(t)) -~(~-i) IlcpIIWpNp(JRn)

with Np = [!!. (1. - 1.)] + 1 1. + 1. = 1 and 1 < p < 2. 2p q 'p q -

Corollary 6.2: We assume for the mass term m(t) = (eo + A(t))-l (log(eo +

A (t) ) ) - 6 with 6 < w < 0, where w is taken from (1. 7). Let us suppose that

18fa(t, ~)I ~ C(~):I(~l for 1001 ~ n + 2 and (t,~) E [to, (0) x IRn \ {O}. Then we have the Lp - Lq decay estimate

t

IIF- 1 (ei£ .\(s)b(s)(~)m(s)ds a(t, ~)F(CP)(O) II Lq(JRn)

~ C( 1 + A(t)) -~(~-i) Ilcpll w,;"p (JRn)

with N = [!!. (1. - 1.)] + 1 1. + 1. = 1 and 1 < p < 2. p 2p q 'p q -

6.3. Lp - Lq decay estimates for Fourier multipliers in the wave case

One strategy is to generalize the proof of [18] to the Fourier multiplier t

( i J .\(s)b(s)dsl~1 )

F- 1 e 0 a(t, OI~I-21 F(cp)(~) .

Here we understand under a generalization that it is sufficient to divide the phase space into two zones, the pseudodifferential and the hyperbolic one, contrary to the considerations for Fourier multipliers appearing in the case of Klein-Gordon

type equations (cf. with proof of Theorem 7.1). The considerations in the pseudodifferential zone, hyperbolic zone, correspond to the study of (1.3) for small I~I, large I~I, respectively. The content of this section is based on the considerations of Section 3.3 of [24]. For the considerations of this section see also [21].

6.3.1. Lp - Lq DECAY ESTIMATES FOR FOURIER MULTIPLIERS WITH AMPLITUDES

SUPPORTED IN THE PSEUDODIFFERENTIAL ZONE

Proposition 6.2: Let the function 1/J = 1/J( 7]) E CO' (Rn) be the same as in Theorem 6.1 with r = 1. Suppose that la(t,~)1 ::; C for all (t,~) E Zpd(N). Setting K(t) :=

N A(t)-l ( loge eo + A(t))) -w, then we have the following Lp - Lq estimate fort> 0:

t

IIF- 1 (e i[ A(Slb(SldSIEII~I_211/J( i~L )a(t, OF(<p)(O) tq(IRnl

::; CN A(t)21-n(i- il ll<pIIL p (IRn)

provided that l :2': 0, 1 < p ::; 2, -k + i- = 1, 2l ::; nO - i-) . Proof. We consider

Using the transformations 7] = K1t), z = K(t)x and setting

a(t,7]) = 1771- 2I a(t, K(t)7])

we see that

10 K(t)nq -2Iq -n IIF- 1 (1/J(I7]I)I7]I- 2I a(t, K(t)7])F(<p)(K(t)7])) II~q(IRnl

K(t)nq-21q-n IIF-1 (1/J(I77I)a(t, 7])F(<p)(K(t)7])) 1[/(IRnl . The point (t, K(t)7]) = (t,~) with 17]1 = N- 1 A(t) (log(eo + A(t))) W I~I ::; 1, that is

7] E supp 1/J, belongs to Zpd(N). Therefore

a(t,7]) ::; CI7]I-21 for all (t,7]) E [0,00) x (Rn \ {O}).

Thus

Io~ = K(t)n-21-';j IIF-1 (1/J(I7]I)a(t, 7])F(<p) (K(t)7]) ) tq(IRn) .

Now let us denote

We have

Due to Theorem 1.11 [8] we have F(Ct ) E M$ for all nO - ~) ::::: 2l. Here M$ denotes the set of Fourier transforms F(f) of distributions f E Lj" and Lj, denotes the space of distributions f E S' satisfying

Ilf * uIILq(lRn) S CIIUIiLp(lRn), U E S,

where the constant C is independent of u. Hence, C t E Lj, and

IIF- 1 (F(cp)(K(t)7])) tq(IRn ) < CpK(t)-n \\cp (K~t)) tp (lR n )

< CpK(t)-n+'i; IlcpIILp(lRn)

for all t E (0,00). Thus we have proved

10"0 S CK(t)-21+n(';--"o)llcpIILp(IRn) S CNA(t)21-n(';--"o)llcpIILp (IRn)

by using the assumption 2l S n (~ - ~). Summarizing we get the Lp - Lq decay

estimate t

IIF- 1 (e i [ A(S)b(S)dSI~II~I_21~(J~~) )a(t, ~)F(CP)(~)) IILq

(lRn

)

S CNA(t)21-n(';--"o)llcpIILp(IRn).

Thus the proposition is proved. o

6.3.2. Lp - Lq DECAY ESTIMATES FOR FOURIER MULTIPLIERS WITH AMPLITUDES

SUPPORTED IN THE HYPERBOLIC ZONE

Proposition 6.3: Let us choose ~ E Co (JRn) as in Proposition 6.2. Suppose that 18fa(t,~)1 S CQI~I-IQI for allial S n + 1, and (t,~) E Zhyp(N). Setting K(t) :=

N A( t) -1 ( loge eo + A( t)) ) -w, then to each E > 0 there exists a positive constant Co,I,N such that the following Lp - Lq decay estimate holds for t ::::: 1:

t

IIF- 1 (e i [ A(S)b(S)dSI~II~I_21 (1 - ~ (J~~)) ) aCt, ~)F( cp )(~)) tq

(lRn

)

< C A(t)21+o-n(,;--"o) II II n _ o,I,N cP Lp(1R )

provided that l ::::: 0, 1 < p S 2, ~ + ~ = 1 and (nt 1) (~ - ~) S 2l :s n (~ - n. If t E (0,1]' then the same estimate holds with E = O.

Proof. The point (t,~) with J~L ::::: 1, that is, (t,~) E supp (1 - ~(K~t)))' belongs

to Zhyp(N). Furthermore, we choose a non-negative function ¢ = ¢(T) having compact support in {T E JR : ~ S ITI s 2}. We set ¢k(T) := </J(2-kT) while ¢O(T) := 1- L:~1 ¢k(T). One can find (see [1]) the function ¢ in such a way that

supp ¢o C {7 E IR : 171 S; 2} and that 1 = L:~-oo ¢(2-k7), 7 =I- O. This partition of unity allows us to estimate

t

IIF- 1 (ei [ A(S)b(S)dsl~II~I_2l (1 - ~ (l~L) ) aCt, ~)F( 'P )(0) tq(JRn)

00 t

'" II -1 ( if A(s)b(s)dsl~1 ( ( I~I)) (I~I) a(t,~) ) II :s:: 6 F e 0 1 - ~ K(t) ¢k K(t) 1~1-21 F('P)(~) Lq(JRn)'

Similar to the consideration for Fourier multipliers with amplitudes supported in Zpd(N) we get

t

II F- 1 (ei[ A(S)b(S)dsl~II~I_2l (1 - ~ (J5.L)) ¢o (J5.L)a(t, OF('P)(~)) II K(t) K(t) Lq(JRn)

:s:: CNA(t)2l-n(~-i)II'PIILp(JRn)

for 21 :s:: nO -n. Here we use the inequality 1~1-1 :::: N- 1 A(t) (log(eo +A(t))) -w.

Let us set ry:= 2- k $' According to Lemma 3 [1] and assumptions to a we have

t

II F- 1 (ei[ A(S)b(S)dSI~II~I_2l (1 - ~ (J5.L)) ¢k (J5.L)a(t,~)) II K(t) K(t) Loo(JRn)

:s:: C2k(n-2l) K(t)n-2l

II ( i2k K(t) J A(s)b(s)dsl1)1 ) II x F- 1 e 0 Iryl-2l (1 - ~(2klryl)) <f;(lryl)a(t, 2k K(t)ry) Loo(JRn)

n-1

:; C2'(n-") K(t)n " (1 + 2' K(t) ! >'(S)b(S)dS) --2-

X I: IID~ (1Tt1-2l (1- ~(2klryl)) ¢(Iryl)a(t, 2kK(t)ry)) too (JRn)

lal~n+1

by using [14], [18] (here we need lad :s:: n + 1). Taking into consideration

ID~ (Iryl-2l (1 - ~(2klryl)) ¢(Iryl)a(t, 2k K(t)ry)) I

< I: Ca ID~'lryl-2lD~2(1 - ~(2klryl))D~3¢(lryI)D~4a(t, 2k K(t)ry) I lal=lal

< I: Calryl-2l-lalIID~2(1 - ~(2klryl))1 lal=lal

x 1(8~3¢)(lryl)l (2kK(t))la41IDt4a(t,2kK(t)ry)l,

IDt 4a(t, 2k K(t)ry) I :s:: Ca4 (2kK(t)lryl) -la 41

and

we have

ID~ (11]1- 21 (1 ~ 'If'!(2kl1]l)) ¢(2kl1]l)a(t, 2k K(t)1])) I :::: Ca.

Therefore, it follows that t

IIF- 1 (ei [.\(S)b(S)dsl~II~I_21 (1 ~ 'If'! (~)) ¢k (~)a(t, 0) II K(t) K(t) L=(IRn)

n-1

<; C2k(n-" ",') K(t)n-" (1 + K(t) I )'(S)b(S)dS) --2-

:::: C2 k(-21+ nt ' )A(t)-n+21.

Finally, this allows us to conclude the L1 ~ Loo estimate

t

IIF-1 (ei[ .\(S)b(S)dsl~II~I_21 (1 ~ 'If'! (~)) ¢k (~)a(t, OF('P)(O) II K(t) K(t) L=(IRn)

:::: C2k(-21+ nt') A(t)-n+2111'PIIL1(IRn).

Taking account of t

II F- 1 (ei [ .\(S)b(S)dsl~II~I_21 (1 ~ 'If'! (~)) ¢k (~)a(t, ~)) II K(t) K(t) L2 (IRn)

< sup 11~1-21a(t,~)1 :::: C221-2klK(t)-21 :::: CE.N221-2klA(t)21+E. 2k-1«::1~1/ K(t)«::2k+1

gives the L2 ~ L2 estimates

t

II F- 1 (e i [ .\(S)b(S)dsl~II~I_21 ( 1 ~ 'If'! (1~~) ) ) ¢k (1~~») a( t, ~)F( 'P) (~)) t2(IRn)

:::: Cc,N221-2kl A(t)21+E II'PII L2 (IRn).

Using the previous estimates an interpolation argument [1] yields the Lp ~ Lq decay estimates

t

IIF-1 (e i [ .\(S)b(S)dsl~II~I_21 (1 ~ 'If'! (~) ) ¢k (K~t») aCt, ~)F( 'P)(~)) tq(IRn)

21+k(n+1(1_1)_21) 21 (' ') :::: C E •N 2 2 P q A(t) +c-n p-q 114?IILp(IRn)

provided that 1 < p:::: 2, ~ + i = 1, 2l:::: nO ~ i)· Using (n~1) 0 ~ i) :::: 2l::::

n(~ ~ i) the application of Lemma 2 [1] proves the proposition. D

6.3.3. Lp-Lq DECAY ESTIMATES FOR FOURIER MULTIPLIERS IN THE WAVE CASE

Propositions 6.2 and 6.3 bring together the following result for Fourier multipliers with prescribed behavior of the amplitude functions.

Theorem 6.2: Suppose that for all t 2': ° the amplitude function satisfies

and

18fa(t,OI S; Call;I-lal, ICYI S; n + 1, (t,l;) E Zhyp(N).

Then to each E > ° there exists a positive constant CE,N such that the following Lp - Lq decay estimate holds for t 2': 0:

t

II F- 1 (e i [ A(s)b(s)dsl~1 a( t, .;)F( ip) (l;)) tq

(JRn)

S; CE,N (1 + A(t)) E- n;-' (~-i) lIipll w:p (JRn)

with Np = [n(l - l)] + 1 l + l = 1 and 1 < p < 2. p q 'p q -

t

( i J A(s)b(s)dsl~1 )

Proof. Instead of F- 1 e 0 a(t,.;)F(ip)(.;) we study

t

F- 1 ( e i [ A(S)b(S)dsl~IIl;I_2l a( t,.;) 1~12l F( ip )(l;)).

If t E (0, 1], then we apply Propositions 6.2 and 6.3 with 2l = nO - i). This

gives the regularity of the datum ip. If t > 1, then we apply Propositions 6.2 and

6.3 with 2l = (nt1) 0 - i)· This gives the decay rate.

7. Lp - Lq decay estimates for solutions of Klein-Gordon equations

Summarizing all the results from the previous sections we have proved the following Lp - Lq decay estimates for the solutions of our model Cauchy problem

Utt - A(t)2b(t)2(~ - m(t)2)u = 0, u(O,x) = ip(x), Ut(O,x) = '!f;(x), (7.1)

under the next assumptions:

• The function A = A(t) is smooth, strictly positive and strictly monotone increasing. There exist positive constants Ak such that

k (A(t))k IDt A(t)1 S; Ak A(t) A(t)

for large t and every positive integer k. Finally there exist positive constants /3 E [1/2,1) and C such that

for large t.

A(t) 0< C S:. A(t)f3

• The function b = b(t) is I-periodic, non-constant, smooth and strictly positive.

• The function m = m(t) is defined by m(t) = (eo+~(t))." where'"Y 2: 0, eo > e.

First we focus on the case 01.1 U 01.2 U O2.4.

Theorem 7.1: The solution of (7.1) satisfies the Lp - Lq decay estimate of Klein~ Gordon type

IIUt(t, ')IILq + IIA(t)V' xu(t, ')IILq

S:. cfi(t)( 1 + A(t) r~~(~~%) (1I<pllw;v+ 1 + 117jJllw;p),

where~+i=l, 1<pS:.2,Np= [nO-i)J+1 and

• r = 0 if 0 S:. '"Y < 2/3 - 1, that is, in 01.1 U O2.4 ;

• r is a positive constant if'"Y = 2/3 - 1, that is, in 01.2.

Proof. The representation (4.2) gives us representations for Ut,)..(t)V'xu by the aid of Fourier multipliers. Using Lemma 2.1, Proposition 3.1, Proposition 4.1 and Lemma 4.2 we get for each multiplier the corresponding estimates of the amplitudes in Zpd(N) as well as in Zhyp(N). Now we divide the phase space [0,00) x (JRn \ {O}) into three parts. From Corollary 6.1 we obtain a decay estimate of Klein-Gordon type for all Fourier multipliers and for all t 2: to, where the constant r in the above estimates appears from the estimates of the amplitudes (d. with Table 2 from Section 5). It remains to estimate the Fourier multipliers for t S:. to. Here we take into consideration that the influence of the Hardy-Littlewood lemma remains. Therefore one can follow the proofs of Propositions 6.2 and 6.3 with K(t) = A[:)13'

If we choose 2l = n( ~ - i) in the derived estimates and t S:. to, then the statements of this theorem follow immediately. Thus, Corollary 6.1 guarantees the KleinGordon decay rate and the application of Propositions 6.2 and 6.3 for t S:. to yields the necessary regularity of the data. 0

The statement ofthis theorem was proved in [23) for the case /3 E [1/2,1)' '"Y = O. Now let us suppose that the above conditions are satisfied with

(log(eo + A(t))) ~o m(t) = A( ) , eo + t

0< C S:. ~~~~ (logA(t))~W, where w S:. 0, eo > e and 8 E JR. Then we can prove in the case 0 3 .1 U 0 3 .2 U 0 3 .3

the next result.

Theorem 7.2: The solution of (7.1) satisfies the Lp - Lq decay estimate of KleinGordon type

lI ut(t, ·)IILq + IIA(t)\7 xu(t, ·)IILq

::; G~( 1 + A(t)r-~(~-~) (1I<pllw:v+1 + 1I1fJllw:v) ,

where ~ + ~ = 1, 1 ° arbitrary, if'Y = 1 and 2w - 1 ::; 15 < 2w, that is, in

0 3 .2 ;

• r is a positive constant if'Y = 1 and 15 = 2w, 15 < w, that is, in 0 3 .3 •

Proof. The proof coincides with that of Theorem 7.1. o Finally we are able to derive Lp - Lq decay estimates of wave type in the

case O2.1 U O2.2 U O2.3 U 0 3.5 U 0 3.6 U 0 3.7, where in this case f3 = 1.

Theorem 7.3: The solution of (7.1) satisfies the Lp - Lq decay estimate of wave type

lI ut(t, ·)IIL. + IIA(t)\7 xu(t, ·)IILq

::; GA(t)m(t)-1 (1 + A(t) r- n;-" (~-~) (11<pllw:V+1 + 1I1fJllw:v) ,

where ~ + ~ = 1, 1 < p::; 2, Np = [n(~ - ~)] + 1 and

• r = e, G = G(e), e > ° arbitrary, ifb > 1 ,W E (-1,0]} or {'Y = 1, W = 0,15 > O} orb = 1 , wE (-1,0) ,15 > O}, that is, in 02.1U02.2U03.5U03.6;

• r is a positive constant if b > 1 , W = -I} or b = 1 , W = -1 , 15 2: O} or b = 1, W > -1,15 = O}, that is, in O2.3 U 03.7.

Proof. The representation (4.2) gives us representations for Ut, A(t)\7 xU by the aid of Fourier multipliers. Using Lemma 2.1, Proposition 3.2, Proposition 4.2 and Lemma 4.4 we derive for every Fourier multiplier the corresponding estimates of the amplitudes in Zpd(N) as well as in Zhyp(N). From Theorem 6.2 we obtain a decay estimate of wave type for all Fourier multipliers, where the constant r in the above estimates appears from the estimates of the amplitudes (cf. with Tables 1,2 from Section 5). 0

Remark 7.1: For the study of Lp - Lq decay estimates for (7.1) with more general oscillating coefficients b satisfying assumption (A.1) from the Introduction, classifications of oscillations as in special cases described in [20], [21], [22], [23] are useful. That this classification depends heavily on the stabilizing effect of the mass term follows from the Theorems 7.1 to 7.3. In 01.1 U02.1 U02.2U02.4U03.1 U03.2U03.5U 0 3.6 we have slow oscillations, this means, we have almost the classical Strichartz decay estimate of wave- or Klein-Gordon type. In 01.2 U O2.3 U 0 3.3 U 0 3.7 we have fast oscillations, this means, the decay rates differ by an (eventually large)

constant from the classical ones. This observation for fast oscillations coincides with the well-known loss of derivatives appearing for solutions of weakly hyperbolic Cauchy problems or for solutions of strictly hyperbolic Cauchy problems with nonLipschitz coefficients. In all other parts from 0 we expect very fast oscillations, this means we cannot expect Lp - Lq decay estimates. That in most of these parts we have really very fast oscillations will be shown in the next section.

8. Classical Floquet behavior

In the previous section we formulated results about Lp - Lq decay estimates for solutions of (7.1). We posed the Cauchy conditions on t = 0, but without new difficulties we can derive Lp - Lq decay estimates if we pose the Cauchy conditions on t = to, to -I- 0. In this section we shall introduce counter-examples which show that under certain assumptions one cannot expect Lp - Lq decay estimates for (7.1) with Cauchy conditions on an arbitrary t = to. The proof of the idea for nondecay is based on the paper [22]. The authors recommend this paper to readers to find some basic ideas for the proof and for applications to more general equations.

Let us consider the following functions ,X = ,X( t) and areas 01.4 U 01.5 U O2 .5 U {fh8 : 15 < -1} U 0 3.9 :

,X(t) {(eo + t)l exp(ta )

({3 E [0, l),l = 1~j3)' ({3 = 1, 0' = 12w' w ::::: 0),

01.4 {({3,')', 0,0) EO; {3 = ')'}, 01.5 {({3,')', 0,0) EO; {3 < ')'}, O2 .5 {(l,,),,w,O) EO; ')' > 1, W < -1},

0 3.8 {(I, 1,w,t5) EO; w = 15 < O},

0 3 .9 {(1, 1,w,t5) EO; w < -1, w < t5}.

Let us consider the Cauchy problem

Utt - 'x(t?b(t)2(~ - m(t)2)u = 0, u(to,x) = <p(x), Ut(to,x) = 'l/J(x) , (8.1)

where A(t) = J~ 'x(s)ds, m(t) = (eo + A(t))-I' ( log(eo + A(t))) -6 and b = b(t) is a

I-periodic, non-constant, smooth and strictly positive function. For the solutions of (8.1) we can prove the next theorem.

Theorem 8.1: There are no constants p, q, Co, C 1 , L and co < 2 such that for every initial time to E [T, 00) with a positive constant T and for every initial data <p, 'l/J E Co (JR. n) the estimate

Il Ut(t, ')IIL q + 11\7 xu(t, ')IIL q ::::: Co'x(t)C1a(W'Q (11<pllw;+l + 11'l/Jllw; ) (8.2)

is fulfilled for all t E [to + 1,00) and ({3, ,)" w, 15) E 01.4 U 01.5 U O2.5 U {03.8 : 15 < -1} U 0 3.9 , where a(t) = logJt(t).

Definition 8.1: The behavior of the solution as described in the previous theorem will be called classical Floquet behavior.

The notion of classical Floquet behavior will be motivated by the fact that one can understand this behavior of solutions for (8.1) after application of the Floquet theorem (see Lemma 8.1 for a special explanation). Contrary to this behavior we are not able to understand the modified Floquet behavior (see Remark 1.1) of solutions for (8.1) by Lemma 8.1.

Corollary 8.1: There are no constants p, q, Co, C I , L and co < 2 such that for every initial time to E [T, (0) with a positive constant T and for every initial data r.p, 'IjJ E COO (I~n) the following estimate holds:

IIUt(t, ·)IILq + IIV'xu(t, ·)IILq :::; CoA(t)C,Cv't/logtrO (11r.pllw;+l + 1I'ljJllw;)

and

Ilut(t,·)IILq + IIV'xu(t,·)IILq:::; COA(t)c,t C0(1/2-,/(1-W» (1IlPllw;H + 1I'ljJllw;)

(3 1

in the cases .A(t) = (eo+t) 1=13 and .A(t) = exp(t 1 - W ), respectively, fort E [to+1, (0) and «(3,I',W, 8) E fl1.4 U fll.5 U fl 2 .5 U {fl 3 .8 : 8 < -I} U fl 3 .9 .

Proof of Theorem 8.1. We shall prove here only the non-critical case, that is, «(3,I',w,8) E fll.5 U fl 2 .5 U fl39 · For the critical case «(3,I',w,8) E fl1.4 U {fl 3 .8 :

8 < -I} we refer to Remark 8.1.

1. One lemma for equations with a periodic coefficient - Floquet's theory

Consider the auxiliary ordinary differential equation with a periodic coefficient

(8.3)

Let the matrix-valued function X = X(t, to) be the solution of the Cauchy problem

~X=(O dt 1 X(to, to) = (~ ~). (8.4)

Thus, X gives the fundamental solution to the equation (8.3). The following lemma, which concerns the instability of the solutions for Hill's equation with periodic coefficients [3], [15], is very important for our considerations.

Lemma 8.1: If b = bet) is a non-constant, positive, smooth function on ~ which is I-periodic, then there exists a .A = .Ao > 0 such that X (to + 1, to) has the eigenvalues p,o and P,OI satisfying 1P,01 > 1.

For the proof of the lemma we refer to Chapter 1 [3], for instance. It is no loss of generality to regard .A(t) = t1 (l = 1~;3) for (3 E [0,1), the

general mass term met) = A(t)-'(log A(t))-8 and aCt) = 109.JtCt) because we are

interested only in the asymptotic behavior for solutions of (8.1) for t -> 00.

Lemma 8.2: There exist positive constants C and T such that

)..(t)2 - )..(t - d)2 :::: Cdaji )..(t?

for any t 2: T, 0:::: d:::: ~vta(W-l and E > O.

Proof. By Taylor's formula, there exists a constant dE (0, d) such that

)..(t)2 _ )..(t _ d)2 = 2d)..'(t _ d))..(t _ d) = 2d a (t) )..(t)2 tA'(t - d))..(t - d) vt )..(t)2log )..(t)

Noting that for t 2: T it holds that vta(t)";-l :::: t, we have

2t)..'(t - d).,(t - d) = { )..(t)2log ).,(t)

with a positive constant C.

2(t d)21-1 - <C t 21 1 log t - ((3 < 1),

((3 = 1,w < 0),

2. Some properties of the fundamental matrix and its eigenvalues

Let tf, be the solution of the equation

o

(8.5)

where )..0 > 0 is taken from Lemma 8.1. Then, tf, is decreasing in I~I, such that

tf, --+ 00 as 1';1 --+ 0

since (3 < 'Y or (3 = 'Y and w < 6. This behavior of tf, is very important for our further approach. Using Lemma 8.2 we can prove the next result.

Lemma 8.3: There exists a positive constant C such that

1)"(t,;)2(~)~(td - )..(tf, - d)2(';)~(t<_d) 1 :::: Ca(tf,)" )..(tE.)2(';)~(t<)

for any small 1.;1, 0:::: d:::: ~~a(tf,)c-l and E > O.

Proof. We shall prove only the case (3 = 1 because the other case is evident. By using Lemma 8.2 we have

1)"(tf,)2(';)~(t<) - )..(tf, - d)2(~)~(t<_d) 1

< ( )..(tf,)2 - )..(tf, - d)2) (';)~(t<) + )..(tf, - d? 1 (';)~(td - (~)~(t< -d) 1

< Cdaj;! )"(tc.)2(';)~(td + I)"(tf, - d)2 (m(tf,)2 - m(tf, - d)2) I.

For the second term we have

where

1.A(tf, - d)2 (m(tf,)2 - m(tf, - d)2) I -2d.A(tf, - d)2m'(tf, - d)m(tf, - d)

2d.A(tf, - d)2 .A(tf, - d) m(tf, _ d)2 (1 + w _ ) A(tf, - d) log A(tf, - d)

a(tf,) 2 2 -) 2d If: .A(tf,) m(tf,) K(tf" d ,

ytf,

K(tf" d):= Vii (.A(tf, - d)) 2 (m(tf, - d)) 2 .A(tf, - d) . a(tf,) .A(tf,) m(tf,) A(tf, - d)

Taking into consideration the definition of .A and m gives

K(tf, d) _ t 1- a (.A(tf, - d)A(tf,))2 ( logA(tf,) )28 .A(tf, - d) , - f, .A(tf,)A(tf, - d) log A(tf, - d) A(tf, - d)

145

< Ce-a __ f, _ f, f, ( t )2(I-a)( ta+(l-o:)logt )28 1

- f, tf, - d (tf, - d)a + (1 - 0:) log(tf, - d) (tf, - d)l-a

~C.

Summarizing, we obtain

1.A(tf,)2(~I~(t<) - .A(tf, - d)2(~I~(t<_d) I ~ Cdaj;j .A(tf,)2(';I~(td ~ ~Ca(td'.A(tf,)2(';I~(td·

The statement of the lemma is proved.

According to the definition of tf, and .Aa, we get

1.A6 - .A(tf, - d)2(';I~(t<_d)1 ~ C.A6 a(td:

for 0 ~ d ~ ~Viia(tf,)c-l and small 1';1. Let us set

X(t + 1, t ) := (xu X12) f, f, X21 X22 '

o

(8.6)

then according to Lemma 8.1 the eigenvalues of this matrix are /-La and /-La 1 . Therefore

+ -1 Xu X22 = /-La + /-La

leads to

and implies

We suppose (8.7)

so that

IX22 -110 1 1 2 ~1110 -110 1 1. Let us consider for every given integer n 2 0 the equation

d2 2 2 2 dt2 W + )..(tE; - n + t) (f')rn(tf.-n+t)b(tE; + t) w = O. (8.8)

It can be written as a first-order system which has the fundamental matrix Xn = Xn(t, td solving the Cauchy problems

d ( 0 dt X = 1 X(t1' h) = I. (8.9)

Due to (8.6) we get the estimate

for

max IIXn(t,O) - X(tE; + t, tE;)11 ::; C1a(td' tE[O,lJ

(8.10)

(8.11)

where the constant C1 is independent of n and ~. Then the right-hand side of (8.10) tends to 0 if I~I tends to O. Moreover, it holds that

max IIXn (t,8)11::; C. (8.12) t,sE [0, 1J

Indeed, to prove estimate (8.12) we note that for t E [0, 1], ~ E IRn with sufficiently small I~I, and for n satisfying 0 ::; n ::; ~~a(td'-l we have 0 ::; tE; - n + t and, due to Lemma 8.3,

)..(tE; - n + t)2(~)~(tcn+t) ::; C)"(tE;)2(~)~(tf.) = C)..6·

Thus the coefficient matrix of system (8.9) is uniformly bounded for all t E [0,1], ~ E IRn with sufficiently smalll~l, and 1 ::; n ::; ~~a(tE;)c-l. Therefore, due to Lemma 8.3, we have for these t, ~, and n the inequality

1)"(tE; - n + t)2(~)~(tf._n+t) - )..(tE; - n - 1 + t)2(~)~(tcn_l+t) I a(tE;) 2 2

::; C ~ )..(tE; - n + t) (~)rn(tf.-n+t)

if 0 < n ::; ~~a(tE;)c-1. Consequently,

(8.13)

for 0 < n ::; ~~a(tE;)c-l. Using Lemma 8.1, (8.10) and noting the fact that

(8.14)


in the cases 01.5U02.5U03.9 the matrix Xn(l, 0), with the property det Xn(l, 0) = 1, has eigenvalues Vn and v;;l which satisfy

(8.15)

for any given positive number,.. with sufficiently small 1.;1 = 1';1("'). Hence we have

Ivnl ~ ~(lfLol + 1) > 1. (8.16)

Denoting

we get

Ixu(n) -vnl ~ Ixu - fLol- (Ixu(n) - xul + IfLo - vnl) ~ ~lfLO - fL011· (8.17)

Analogously we prove

IX22(n) _v;;ll ~ ~lfLO - fL011· The estimate (8.13) implies

a( tt;,) IXij(n + 1) - xij(n)1 ~ C -../fi.. (8.18)

Due to estimates (8.18) the eigenvalues Vn and v;;l are distinct for every n. Then, according to (8.18), we get

a(tr;) IVn+1 - vnl ~ C ..;ti,.

3. Lower bound for the energy of an unstable solution of an auxiliary ordinary differential equation

The calculations of the previous steps serve as a preparation to prove the next lemma.

Lemma 8.4: Let no be the largest integer satisfying (8.11). Then there exist positive numbers Co and C 1 such that the solution w = w( t,';) of

Wtt + A(t)2b(t)2(';)~(t)w = 0

with initial data

w(tr; - no - 1,';) = 1,

satisfies Iw(tr;,,';)1 + IWt(tr;,';)1 ~ Coexp(C1~a(tr;)c-1)

for all small 1.;1.

Proof. The function w = w(tr; - no + t,';) satisfies equation (8.8). Hence

(8.19)

( ftw(tr;,.;)) = X (1 O)X (1 0)··· X (1 O)X (1 0) (ftw(tr; - no - 1,.;)) w(tr;,';) 0, 1, no-I, no, W(tr; - nO -1,';) .

148

The matrix

F. Hirosawa and M. Reissig

X12(n) v n - x ll(n)

1

is a diagonalizer for Xn(l, 0), that is,

Xn(1,O)Bn = Bn (Va v~l). Since detXn (l,O) = 1 and since the trace of Xn(l,O) is Vn + v;;l we get

Vn - V;;l det Bn = 1 .

v;; - x22(n)

According to (8.12), (8.16) and (8.17) this leads to

IV;;l - x22(n)1 ~ C, I det Bnl ~ C > 0

for all n with 0 < n ~ ~~a(t.;-)"-l, respectively. Further, taking into account (8.12) we have IXij(n)1 ~ C and, consequently,

(8.20)

for all n with 0 < n ~ ~~a(t.;-y-l, where C is independent of n. The estimates (8.18) and (8.20) lead to

-1 -1 I a(t.;-) IIBn_lBn - III = IIBn_l(Bn - Bn-dl ~ C ~ .

Denoting Gn := B;:!..l Bn - I, we have

Xo(l,O)Xl(l,O)· ··Xno-l(l,O)Xno(l,O)

B ( Vo o 0

( Vno X 0

B ( Vo o 0

( Vno X 0

o ) -1 (VI VOl Bo Bl 0

o ) B- 1 -1 no Vno 0

) (I + Gd ( VI -1 0 Vo 0

) B;;ol. -1 Vno

~l ) B"II B2 ... B;:ol_l Bno vI

(8.21 )

From (8.20) we see that in order to prove Lemma 8.4 it is sufficient to show that the (I, l)-th element Yl1 of the matrix

(~ v~l) (I + G l ) (~ V~l) (I + G2)··· (I + Gno ) (Voo V~ol) can be estimated to below with some positive numbers Co and C l by

Coexp(Cl~a(t.;-)"-l) .

It is evident by (8.21) and Lemma 3.3 of [31] that

no no

IYll - II Vnl :::; Ca(t~r; II Ivnl· n=O n=O

Thus, it follows from (8.15) that

no IYlll > (1 - Ca(t~t) II Ivnl ~ (1 - Ca(t~r;) IfLO - Kino

n=O > (1 - Ca(t~)E) IfLO - KlhltE"a(tel,-l_l

with no = max{n E N+; n :::; ~vtea(t~)E-l}. Recalling (8.14) the lemma is proved. o

4. Construction of unstable solutions

The considerations of this step are based on a family of solutions u = u(t, x,~) of (8.1),

where ~ E ]Rn is regarded as a parameter. The functions of this family will violate the estimate (8.2),

Ilut(t, ·)II£q + IIV xu(t, ·)II£q :::; COA(t)C1a(W'O (11<pllw;+l + 11'Ij;lIw,f )

for sufficiently smalll~l. To construct this family we note that there exists a positive number N such that

{ lim tNl-l+~(llogt)l-E = 00

a(t)l-E t-+oo

lim exp (Na(t)Vi) = t-+oo v't 1· t(a-!)(l-E)-! Nt'" 1m 2 2e = 00

t-+oo

({3 < 1),

({3 = 1).

(8.22) Further, we fix a cut-off function X E CQ"(]Rn) such that X(x) = 1 when Ixl :::; 1, while X(x) = 0 if Ixl ~ 2. Then with the number no from Lemma 8.4 and with N of (8.22) we choose at t = to, to := t~ - no - 1, the initial data <p and 'Ij; as

<p(x) = eix.ex(x<~)~t~)), 'Ij;(x) = eiX·~x(x<~)~t~)) vn:~2;~O(~o) (8.23)

belonging to CQ"(]Rn). Using the theory of the Cauchy problem for strictly hyperbolic equations, especially the existence of a cone of dependence, there exists a unique solution u = u(t, x,~) to (8.1), (8.23), where u = u(t, .,~) has compact support for every given t and ~, t E [to, (0), ~ E ]Rn. By (8.22) there exists a positive number T such that

4v'ta(t)E-l :::; CA(t)N

for any given C and all t ~ T.

(8.24)

5. Cone of dependence and lower bound for the energy If we take into account the cone of dependence property of the solution, then u = u(t,x,~) is representable in B1(0) c IR~, in the ball of radius 1 centered at the origin, at time t = tf, as

(8.25)

with the function w from Lemma 8.4. To understand (8.25) let us calculate the lower base at t = to of the truncated cone starting from Bl (0) at t = tf, with the slope A(tf,) maxtE[O.lj bet) = b1.>..(tf,) and the height tf, - to = no + 1. This lower base is contained in the ball

B dep := {x E IR~ : Ixl :S 2h.>..(tf,)(no + I)}.

On the one hand we have due to (8.24) the inequalities

2h.>..(tf,)(no + 1) :S 4bl'>"(tf,)~a(td'-1 :S C'>"(tf,)N+l = C.>..b"+l(O:~~rl).

On the other hand, if x E B dep , then the module of the argument x(~)r:,t~) of the function X is less than or equal to 1 for sufficiently small I~I. Hence, we get

the identity X (x(Or:,t~») = 1 on B dep '

The function eix>f,w(t,~) solves the equation and takes at t = to the data

(8.23) if x E IRn, Ixl :S (O:i~r2). The uniqueness of solutions to (8.1), (8.23) implies the representations (8.25). In particular we obtain from this representation

Ilut(tf" ',0 II Lq(B, (0» + IIV xu(tf" "~) IILq(B 1 (0» = (1~llw(tf,' 01+ IWt(tf" ~)I) J dx. B, (0)

Now let us suppose that the inequality (8.2) holds. Then we have

Ilut(t, ',~)IILq(Bl(O» + IIVxu(t, ',~)IILq(Bl(O» < Cllut(t, ',~)IILq + IIV xu(t,', ~)IILq

< Co ( 1 + I vn:~2;~O(~o) 1)'>"(tf,)Cl a (td - Eo IleiX·f,x(x(~)r:,t~»)llw;. Using this inequality and recalling (8.12) and (8.17) we have

Iw(tf"OI + IWt(tf,,~)1 :S CoA(tf,)C2 +Cl a (t<)-EO

for all ~ E IRn with sufficiently small I~I, where the exponent C2 arises from the corresponding Sobolev norm of the data and the definition (8.5) for tf,.

But from Lemma 8.4 we already know that the inequality

Iw(tf" 01 + IWt(tr" ~)I :::: Co exp (C1 ~a(tr,)E-l) = Co'>"(tf,)Cla(td-2+E

holds with some positive numbers Co, C1 . According to the conditions of the theorem we have 0 :S c < 2, that is, 0 :S co < 2 - c < 2 for any c > O. Comparing the last two inequalities leads to a contradiction for sufficiently small I~I. The theorem is proved. 0

Remark 8.1: Let us make some comments on the critical cases 01.4 U {03,S : J < -I}. In these cases we have limt--->CXJ A(t)m(t) = C > O. If we suppose that b = bet) from (8.1) is I-periodic, non-constant, smooth and strictly positive, then due to [3] there exists an infinite number of intervals of instability, that is, we have a sequence {An}, n E N, tending to 00 such that the eigenvalues J-ln of the corresponding fundamental matrices satisfy IJ-ln I > 1. Defining as in (8.5) the function tf; it is sufficient to choose a natural number no and the positive constant Ana > C from the sequence as the right-hand side. Then the function tf; ----> 00 when I~I ----> O. But this allows us to repeat the approach to proving Theorem 8.1 and to conclude the same statement.

Remark 8.2: We can expect some intermediate behavior appearing in some parts of the areas 0l.3, 0;{.4, {03,S : J < -I} and 0 3,10. Indeed, by Lemma 4.2 (iv), (v) (see also Table 3), the growth order of the solution is not more rapid than

IJ that for classical Floquet cases. For instance, when A(t) = (1 + t) /3-1 and met) = (1 +t)- 12/3, that is the case of Example 1.1 with l = (1~j3)' the upper bound of the

')'~2{'+1

Lq-norm of the solution is O(exp(A(tp- 2f3+ 1 )) rv O(exp(t~)) rv O(exp(tf<o)) in 01.3 with a fixed 1'),0 = 1'),0(/3,ry) < 1. On the other hand, by Corollary 8.1, the Lq-norm of the solution cannot be controlled by O( exp( tf<)) for any I'), < 1 from below in 01.4 U 01.5. Consequently, the behavior in 0l.3, which we called modified Floquet behavior is better than the classical Floquet behavior in 01.4 U 01.5.

Concluding remark: Now let us give answers to the questions formulated in the Introduction. For this reason we recall the Cauchy problem for our starting equation

Utt - A(t)2b(t)2 D U + A(t)2b(t)2 28 U = 0

(e + A(t))2/( log(e + A(t))) (8.26)

under the assumptions for b, A and m from Section 1.3.

Case 1: Let us suppose (3 E (~, 1) in (1.7) and J = 0, ry :2: 0 in (8.26). Then the change-over from Klein-Gordon behavior to Floquet behavior can be described in the following way:

• 0 ::::: ry < 2(3 - 1, in 01.1, Klein-Gordon behavior; • ry = 2(3 - 1, in 01.2, critical Klein-Gordon behavior; • 2(3 - 1 < ry < (3, in 0l.3, modified Floquet behavior; • (3 ::::: ry, in 01.4 U 01.5, classical Floquet behavior.

Klein-Gordon behavior -+ critical Klein-Gordon behavior -+ modified Floquet behavior -+ classical Floquet behavior

Case 2: Let us suppose (3 = 1 in (1.7) and J = 0, ry > 1 in (8.26). Then the changeover from wave behavior to Floquet behavior can be described in the following way:

• wE (-1,0] in (1.7), in ~h1 U O2 .2 , wave behavior; • w = -1 in (1. 7), in O2 .3 , critical wave behavior; • w < -1 in (1.7), in O2.5 , classical Floquet behavior.

wave behavior --+ critical wave behavior --+ classical Floquet behavior

Case 3: Let us suppose (3 = 1, w = -~ in (1.7) and"( = 1 in (8.26). Then the change-over from Klein-Gordon behavior to wave behavior can be described in the following way:

• 8 < -1, in O:u U 0 3 .2 , Klein-Gordon behavior; • 8 = -1, in 0 3 .3 , critical Klein-Gordon behavior; • -1 < 8 < 0, in 0 3 .4 U 0 3 .8 U 0 3 .10 , modified Floquet behavior; • 0 = 8, in 0 3 .7 , critical wave behavior; • 0 < 8, in 0 3 .6 , wave behavior.

Klein-Gordon behavior --+ critical Klein-Gordon behavior --+ modified Floquet behavior --+ critical wave behavior --+ wave behavior

9. Appendix

Lemma A.I: Let ';0,6 E lR. If 1';01 2: 1';11, then the following inequality holds: t

/ .\(s) ( 1';01 _ 161 ) ds > _1_A(t) ( 1';01 _ 161 ) (';0)m(8) (6)rn(s) - 1 + "( (';O)rn(t) (6)m(t)

o

for any t > O.

Proof· We note that _«)Iel is monotone increasing with respect to 1';1. Let ';0,6 E <, =(s)

lR \ {O} satisfy 1';01 2: 161. By partial integration we have

t

/ .\(s)( 1';01 _ 161 )dS=A(t)( 1';01 _ 161 ) (';O)m(s) (6)rn(s) (';O)rn(t) (6)m(t)

o t

/ ( 8) A(s) .\() (m(s)21';01 m(s)2 161 ) d - 0 "( + log (eo + A(s» eo + A(s) s (';o):n(s) - (';d:n(s) s.

Here we remark that the inequality

1';01 _ m(s)21';01 > 161 (';O)m(s) (';o):n(s) - (6)m(s)

m(s)2161 (6):n(s)

holds for any s ::; t, that is, the function f : 1.;1 --+ f(I.;I), where

f(I.;I) := 1';1 (';)m(8)

m(s)21';1 (.;):n(s) ,


is monotone increasing with respect to I~I. Indeed, we have

8f(IW ol~1

(~);;'~s) (O~(s) -1~12(~);,,(s) - m(s)2(~);,,(s) + 3m(s)21~12)

3m(s)41~12(~);;,~s) ~ O.

Therefore, we obtain

t

/ >.(s) ( I~ol _ 161 ) ds (~O)m(s) (6)m(s)

o t

>A(t)( I~ol _ 161 )-'Y/>'(s)( I~ol _ 161 )dS. - (~O)m(t) (6)m(t) (~O)m(s) (6)m(s)

o

Thus the lemma is proved.

Lemma A.2: Let t be a positive constant and

E t ,C" C 2 = {77 E lR \ {O} ; 77 satisfies (6.3)}

153

o

with positive constants C1 and C 2 • If 770, 771 E E t ,C"C2 satisfy 1770 1 ~ 17711, then the following inequality holds:

t

~ />.(s) ( 17701 _ 17711 ) ds > _l_A(t) (~_ hl). A(t)f3 g(s, t, (77oh) g(s, t, (771h) - 1 + 'Y (77oh (771h

o

Proof. The proof is similar to that for Lemma A.I. We have only to show that the function 1: 1771 ----+ 1(1771), where

1(1771) := 1771 _ m(s)21771 = 1771 (1 _ m(s)2 ) , g(s, t, (77)Il g(s, t, (77h)3 g(s, t, (77)Il g(s, t, (77hJ2

is monotone increasing with respect to 1771. Indeed, it follows from (6.3) that

1 - (n;«)~ )2 ~ 0, hence 1(1771) is described by a product of two non-negative 9 s, , 1] 1

monotone increasing functions with respect to 1771. Thus, the lemma is proved. 0

Acknowledgements

The authors began to write this paper during the visit of the first author from November 2000 to February 2001 at the Technische Universitiit Bergakademie Freiberg. The first author is grateful to the Fakultiit fiir Mathematik und Informatik for the hospitality and to DAAD for financial support. The paper was finished during the work of the second author as a Foreign Professor at the University of Tsukuba in February, March 2002. He is grateful to the staff of the Institute of Mathematics, especially the research groups of Prof. Kajitani, Prof. Taira and Prof. Wakabayashi for their hospitality. Finally, the authors thank Karen Yagdjian for his valuable remarks.


References

[1] P. Brenner, On Lp - Lq estimates for the wave-equation, Math. Zeit. 145 (1975), no. 3, 251-254.

[2] F. Colombini, D. DelSanto and M. Reissig, On the optimal regularity of coefficients in hyperbolic Cauchy problems, Preprint del Dipartimento di Scienze matematiche dell'Universita di Trieste, n. 511 (2002).

[3] M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh and London, 1973.

[4] A. Galstian and M. Reissig, Lp - Lq decay estimates for a Klein-Gordon type model equation, Eds. H. Begehr et al.,Proceedings of the second ISAAC congress, vol. 2, 1355-1369, Kluwer (2000).

[5] F. Hirosawa, Energy decay for degenerate hyperbolic equations of Klein-Gordon type with dissipative term, Funkcial. Ekvac. 43 (2000), no. 1, 163-191.

[6] F. Hirosawa, On the Cauchy problem for second order strictly hyperbolic equations with non-regular coefficients, to appear.

[7] F. Hirosawa, Precise energy decay estimates for the dissipative hyperbolic equations in an exterior domain, Mathematical Research Note 2000-008, Institute of Mathematics, University of Tsukuba.

[8] L. Hormander, Translation invariant operators in LP space, Acta Math. 104 (1960), 93-140.

[9] L. Hormander, Remarks on the Klein-Gordon equation, Journees Equations aux Derivees Partielles, Saint Jean de Monts, 1987.

[10] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan 41 (1995), no. 4, 617-653.

[11] S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1,43-101.

[12] A. Kubo and M. Reissig, Construction of parametrix for hyperbolic equations with fast oscillations in non-Lipschitz coefficients, Mathematical Research Note 2002-003, Institute of Mathematics, University of Tsukuba.

[13] O. Liess, Decay estimates for the solutions of the system of crystal optics, Asymptotic Analysis 4 (1991), no. 1, 61-95.

[14] W. Littman, Fourier transformations of surface carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766-770.

[15] W. Magnus and St. Winkler, Hill's Equation, Interscience Publishers, New YorkLondon-Sydney, 1966.

[16] A. Matsumura, Energy decay of solutions of dissipative wave equations, Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), no. 7, 232-236.

[17] K. Mochizuki and T. Motai, On energy decay-nondecay problems for wave equations with nonlinear dissipative term in]RN, J. Math. Soc. Japan 41 (1995), no. 3, 405-421.

[18] H. Pecher, Lp-Abschatzungen und klassische Losungen fur nichtlineare Wellengleichungen. I, Math. Zeitschrift 150 (1976), no. 2, 159-183.

[19] R. Racke, Lectures on nonlinear evolution equations, Aspects of Mathematics, Vieweg, Braunschweig/Wiesbaden, 1992.


[20] M. Reissig, Klein-Gordon type decay rates for wave equations with a time-dependent dissipation, Adv. Math. Sci. Appl. 11 (2001), no. 2, 859-891.

[21] M. Reissig and K. Yagdjian, About the influence of oscillations on Strichartz-type decay estimates, Rend. Sem. Mat. Univ. Polito Torino 58 (2000), no. 3,117-130.

[22] M. Reissig and K. Yagdjian, One application of Floquet's theory to Lp - Lq estimates for hyperbolic equations with very fast oscillations, Math. Meth. Appl. Sci. 22 (1999), no. 11, 937-951.

[23] M. Reissig and K. Yagdjian, Klein-Gordon type decay rates for wave equations with time-dependent coefficients, Banach Center Publications, 52 (2000), 189-212.

[24] M. Reissig and K. Yagdjian, Lp - Lq estimates for the solutions of hyperbolic equations of second order with time-dependent coefficients - Oscillations via growth -, Preprint 98-5, Fakultiit fiir Mathematik und Informatik, TU Bergakademie Freiberg 1998, ISSN 1433-9307.

[25] M. Reissig and K. Yagdjian, Weakly hyperbolic equations with fast oscillating coefficients, Osaka J. Math. 36 (1999), no. 2, 437-464.

[26] M. Stoth, Globale klassische Losungen der quasilinearen Elastizitiitsgleichungen fur kubisch elastische Medien im ]R2, SFB 256 Preprint 157, Universitiit Bonn, 1991.

[27] R. Strichartz, A priori estimates for the wave-equation and some applications, J. Funct. Anal. 5 (1970), 218-235.

[28] H. Uesaka, The total energy decay of solutions for the wave equation with a dissipative term, J. Math. Kyoto Univ. 20 (1980), no. 1, 57-65.

[29] W. V. Wahl, LP -decay rates for homogeneous wave-equations, Math. Zeit. 120 (1971), 93-106.

[30] K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics, micro-local approach, Math. Topics, Akademie-Verlag, Berlin, 1997.

[31] K. Yagdjian, Parametric resonance and nonexistence of global solution to nonlinear wave equations, J. Math. Anal. Appl. 260 (2001), no. 1, 251-268.

[32] S. Zheng and W. Shen, Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems, Sci. Sinica, Ser. A 30 (1987), no. 11, 1133-1149.

Fumihiko Hirosawa Institute of Mathematics University of Tsukuba Ibaraki 305-8571, Japan e-mail: [email protected]

Michael Reissig Fakultiit fur Mathematik und Informatik TU Bergakademie Freiberg D-09596 Freiberg, Germany e-mail: [email protected]

Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations

Michael Dreher

Abstract. The purpose of this paper is to investigate weakly hyperbolic equations with degeneracies in the space and time variables. These degeneracies as well as the sharp Levi conditions of Coo type are formulated by means of certain weight functions. For Cauchy problems to such quasi-linear weakly hyperbolic equations, the following subjects are studied: local existence of solutions in Sobolev spaces and Coo, a blow-up criterion, domains of dependence, and Coo regularity. The main tools are the transformation of the higher-order equation to a first-order system, a calculus for pseudodifferential operators with non-smooth symbols, and a generalization of Gronwall's lemma to differential inequalities with a singular coefficient.

1. Introduction

Let us consider the differential operator of order m

j+lod:<=;m,j<m

where we adopted the usual notation D t = -iot , Dx = -i"'Vx . This operator P is called hyperbolic in the direction t if the roots Tj = Tj (x, t,~) of the equation

j+lal=m,j<m

are real for all real x, t,~, The operator P is said to be strictly hyperbolic in the direction t, if the roots Tj are real and distinct for ~ E IRn \ {a}. If P is hyperbolic, but not (necessarily) strictly hyperbolic, it is called weakly hyperbolic.

Hyperbolicity is a necessary condition for Coo well-posedness of the Cauchy problem (see [19], [22]). Well-posed ness (with respect to chosen topological spaces for the data, right-hand side and the solution) of a Cauchy problem means, as usual, the existence, uniqueness and continuous dependence (in the topologies of the given spaces) of the solution. However, hyperbolicity does not guarantee the

2000 Mathematics Subject Classification. 35L70, 35L80. Key words and phrases. Sobolev solutions, loss of regularity, blow-up criterion, domains of dependence.

158 M. Dreher

well-posedness in, e.g., Coo or Sobolev spaces. A sufficient condition for the wellposedness in Coo and in Sobolev spaces is the strict hyperbolicity, see [26], [20] and [14].

Therefore it is a natural goal to find classes of weakly hyperbolic Cauchy problems which are Coo well posed.

In the weakly hyperbolic case, new phenomena occur which may prevent the Cauchy problem from being well posed. These phenomena are the following:

Oscillations in the coefficients with respect to time

• Colombini, Jannelli, and Spagnolo [6], [7], constructed a smooth function a(t) 2: ° and smooth data uo(x), Ul(X) with the property that the Cauchy problem

Utt - a(t)uxx = 0, u(x,O) = uo(x), Ut(x,O) = Ul(X)

has no solution U in the distribution space V'(1R x [0,1]). This coefficient a(t) is positive for t > 0, oscillating for t --+ ° + ° and vanishing for t :::; 0.

• Let b(t) be a positive, periodic, smooth and non-constant function. Tarama [32] proved that the Cauchy problem

Utt - exp( -2CD!)b(Cl )2uxx = 0,

U(X, 0) = uo(x), Ut(X, 0) = Ul(X)

is Coo well posed if and only if 0: 2: 1/2.

The influence of lower-order terms

• Ivrii and Petkov [16] showed that necessary conditions for the Coo wellposedness of

Vtt - t 2l vxx + tkvx = 0, l, kENo,

Utt - x 2nu xx + xmux = 0, n, m E No,

(1.1)

(1.2)

are k 2: l - 1 and m 2: n. The sufficiency of these conditions was proved by Oleinik [25].

• If one wants to study well-posedness in Sobolev spaces, one has to pay attention to another phenomenon, which occurs in the border case k = l - 1 of the Coo well-posedness: the loss of Sobolev regularity. Qi [27] showed by an explicit representation of the solution to the Cauchy problem

Utt - t 2uxx = bux , (1.3)

u(x,O) = cp(x), Ut(x,O) = 0, b = 4m + 1, mEN

that u(·, t) E Hs-m if cp E HS. Or, let us look at it from another point: choose an arbitrary data function cp(x) with high Sobolev smoothness s » 1. Then a number b exists such that there is no classical solution of (1.3). The solution only exists in distribution spaces.

Quasi-linear Weakly Hperbolic Equations

The loss of regularity also occurs for equations of the form

Utt - t 2l u xx - btl-lux = 0,

u(x,O) = cp(x), Ut(x,O) = 'ljJ(x),

159

(1.4)

as shown by Taniguchi and Tozaki [31]. Equations of the type (1.3) and (1.4) are interesting because singularities of their solutions may propagate in a non-standard way, see [12], [13], and [31].

There are different ways to exclude the phenomenon of oscillations and to restrict the influence of the lower order terms. We pick the equation

Utt - a(x, t)uxx + b(x, t)ux + d(x, t)Ut + e(x, t)u = f(x, t)

as a model problem. First we consider the oscillations.

• If the degeneracy occurs for t = ° only, we may assume ([25])

0::::; Ca(x, t) ± Ota(x, t), t 2: 0, C> 0.

• We can suppose that the coefficient a(x, t) has the structure

a(x, t) = ao(x, t)a(x)2 )..(t)2

with some smooth ao(x, t) 2: 0: > 0, and )..(0) = 0, N(t) > ° (t > 0). The degeneration happens at the zeroes of the product a A. The functions a and A characterize the spatial degeneracy and time degeneracy, respectively. Assumptions of this type were made, e.g., in Nersesyan [24], Yagdjian [35], and [10], [11], [13]. We will follow this idea and generalize it to quasi-linear higher order equations in higher dimensions.

Second, we consider the lower order terms. Conditions which restrict the influence of these terms are called Levi conditions. Our aim is to find conditions which do not exclude the interesting equations (1.3) and (1.4). The following Levi conditions have been used widely in the past:

• If the degeneracy occurs for t = ° only, then we may take the condition

Btb(x, t)2 ::::; Aa(x, t) + Ota(x, t), t 2: ° (1.5)

from [25]; A and B are some positive constants. This Levi condition is sharp in the case of finite degeneracy: if one fixes a(x, t) = x 2nt 21 and b(x, t) = xmtk, (1.5) implies m 2: n, k 2: 1-1. These are exactly the necessary and sufficient conditions from Ivrii, Petkov and Oleinik. However, this condition is not sharp in the case of time degeneracy of infinite order. There exists an explicit representation of the solutions to

2 1 1 1 Utt - e-' t 4 Uxx + be-' t4 Ux = 0, t 2: 0, b = const, (1.6)

see Aleksandrian [1], which implies that the Cauchy problem for this equation is Coo well posed. Yet, the coefficients from (1.6) do not satisfy (1.5). Similarly to (1.3) and (1.4), the solutions to (1.6) lose regularity, too; and their singularities can propagate in an astonishing way, see [1].

160 M. Dreher

• If one wants to include more general degenerations, one may assume the rather general and crude conditions

b(x, t)2 :s: Ca(x, t),

at(x, t) :s: Ca(x, t) or at(x, t) 2 -Ca(x, t),

or, similarly,

Bb(x, t)2 :s: Aa(x, t) + at(x, t), A, B > 0,

compare D'Ancona [8], Manfrin [21]. However, these conditions are not sharp; they exclude (1.3), (1.4) and (1.6) .

• It can be presumed a(x, t) = ao(x, t)0"(x)2 A(t)2 with ao(x, t) 2 a > 0 and Ib(x, t)1 :s: CIO"(x)IX(t). Coefficients a(x, t) and b(x, t) satisfying such a Levi condition include the interesting cases (1.3), (1.4) and (1.6). We will follow this way and generalize these conditions to higher-order equations.

Let us list the main results of this paper. We are concerned with the hyperbolic Cauchy problem

D';'u+ j+lal=rn,j<m

= f(x, t, {D~(Ck,i3(X, t)Dfu)}Ik+li3l:S:m-l), m 2 2,

u(x,O) = 'Po(x), . .. , D,;,-lu(x, 0) = 'Prn-l (x)

for (x, t) E M x [0, T]; where M is either lRn or a smooth closed n-dimensional manifold. The functions A = A(t) and 0" = O"(x) describe the degeneration of the principal part of the differential operator, and the functions Ck,i3 = Ck,i3(X, t) are weight functions for the lower order terms and characterize the Levi conditions.

Example. The weight function A = A(t) has to satisfy a certain condition (see Condition 4.1). Examples of such A are

A(t)=tl, lEN, l2m-l,

A(t) = A'(t) with

A(t) = exp( -Itl- r ), r> 0,

A(t) = exp( - exp(exp(exp(ltl- r )))), r > O.

There are no restrictions on the choice of 0", any smooth real-valued function 0" = O"(x) is admissible.

The weight functions Ck,i3 are connected with A and 0" via the relations (4.11) and (3.22)-(3.25), and special examples are

{ A( t )rn-k A( t )k+li3l-rn 0"( x) 1131

Ck,i3(X, t) = 1

where A(t) = J~ A(T) dT.

: 1131 > 0,

: 1131 = 0,

Quasi-linear Weakly Hperbolic Equations 161

The following results are proved for such Cauchy problems in this paper.

Local existence in Sobolev spaces: For given data in Sobolev spaces, a solution is found which suffers from the loss of Sobolev regularity, as motivated by Qi's example.

Blow-up criterion: We will prove that a blow-up of the solution in high order Sobolev spaces is only possible if the C; Zygmund norm of certain weighted derivatives (up to the order m - 1) of the solution blows up. This is a generalization of a similar criterion from the strictly hyperbolic case, see Taylor [33].

Local existence in Coo: That blow-up criterion leads to the local existence of solutions in Coo immediately.

Domains of dependence: A special feature of strictly hyperbolic equations is the finite propagation speed. In other words, the value of the solution at a given point at a given time only depends on the values of the initial data and right-hand side from a certain bounded domain, the so-called domain of dependence.

We will define and study domains of dependence for quasi-linear weakly hyperbolic operators, and use them to prove the local in space and time existence and uniqueness of Sobolev solutions, and their Coo regularity provided that the data are from Coo. Our concept of domains of dependence extends the concept of Alinhac and Metivier [2] from the strictly hyperbolic to the weakly hyperbolic case. Geometrically speaking, these domains can be described by the condition that the principal part of the operator be hyperbolic at each point of the boundary of the domain in the normal direction of the boundary. Since the coefficients of the principal part depend on the solution, the domain of dependence for the solution will be dependent on the solution itself.

Next we give some remarks concerning the used methods and tools.

A crucial step of the investigation of hyperbolic Cauchy problems is an a priori estimate of the solution in Sobolev spaces, which is usually proved by means of pseudo differential operators. However, since the coefficients of the hyperbolic operator depend on the solution and its derivatives itself, and because the solution will be from some Sobolev space, the coefficients of this hyperbolic operator will not have Coo smoothness. Thus, the theory of pseudodifferential operators with symbols of infinite smoothness seems not to be applicable; hence we present a theory of pseudodifferential operators with symbols of finite smoothness (HS or C 1 or merely CO) in Section 2. We cite results of Taylor [33] concerning mapping properties, commutator estimates, adjoints and compositions.

Our methods for proving the local existence of a solution to (1. 7) are a unification of ideas taken from Taylor [33] who studied quasi-linear strictly hyperbolic equations; and Kajitani and Yagdjian [17] who studied quasi-linear weakly hyperbolic equations with time degeneracy.

162 M. Dreher

In Section 3, we study weakly hyperbolic Cauchy problems with pure spatial degeneracy, i.e., (1.7) without weight function A = A(t). Our approach in this case is as follows. We construct some vector-valued function U' which contains weighted derivatives of u up to the order m - 1 and solves a pseudodifferential hyperbolic system of first order

UtU' = K*(x, t, U*, D)(CTU*) + F*(x, t, U*) (1.8)

where K* is a strictly hyperbolic matrix pseudo differential operator of order 1, and F* contains the right-hand side and some other terms. We insert some smoothing operators JE; into (1.8) such that its right-hand side becomes an operator of order 0, and the existence of an approximate solution U; follows immediately from functional analytic arguments. Next we have to prove independent of E estimates of U; and its life-span. These estimates will allow us to show an interesting blow-up criterion:

A blow-up of U* in the H S norm is impossible as long as the Zygmund space norm IIU* lie; remains bounded.

We are able to extend the results won by Dionne [9] and Taylor [33] from the strictly hyperbolic case to the weakly hyperbolic case.

The general weakly hyperbolic Cauchy problem with spatial and time degeneracy is treated in Section 4. Here we face new difficulties which are typical for the time degeneracy:

One such obstacle is a singular coefficient in the energy inequality. Consider, as an example, the weakly hyperbolic equation

Utt - A(t)2uxx = f(x, t), A(O) = 0, A'(t) > 0 (t > 0).

If we choose the energy in the usual way, E(t) = Ilutll~ + IIA(t)uxll~, then we obtain, after some calculations,

I 2 A'(t) E (t) ::::; Ilf(-, t)IIL2 + A(t) E(t).

The lemma of Gronwall is not applicable, since the coefficient N(t)/A(t) becomes unbounded for t ---; O. But one can use Nersesyan's lemma (see Lemma 6.2) if the initial data vanish and Ilf(', t)llu has a zero of sufficiently high order at t = O.

Another obstacle is the loss of regularity. The example of Qi [27] shows that the solution can lose Sobolev smoothness in comparison with the initial data. The number of lost derivatives depends (in the linear case) on the LCXl-norm of the coefficients of some lower order terms. This makes the investigation of nonlinear Cauchy problems delicate, since the usual fixed point arguments cannot be applied directly. The crucial tool for solving this difficulty is the reduction (Section 4.3) of the Cauchy problem (1.7) to another Cauchy problem which enjoys the so-called strictly hyperbolic type property: let L be a weakly hyperbolic operator of order 2 (for simplicity). We say that a Cauchy problem

Lu(x, t) = f(x, t), u(x,O) = Ut(x, 0) = 0

has the strictly hyperbolic type property if there is a topological space B and a weight function w(x, t) such that fEB implies w(x, t)V'xu E Band Ut E B. Then the local existence in B of a solution to a quasi-linear version of the above Cauchy problem can be proved by standard arguments. In the strictly hyperbolic case, we choose w == 1 and B = C ([0, T], HS). In the weakly hyperbolic case, w is chosen according to the degeneracy, and B consists of functions which decay sufficiently fast for t --+ 0, see Section 4.1. For other applications of such adapted Banach spaces to weakly hyperbolic equations, see [12], [13], and Reissig, Yagdjian [30].

Concerning the investigation of domains of dependence in Section 5, our technique is as follows: we exhaust the domain of dependence with hypersurfaces, and change the variables such that these hypersurfaces become planes of constant time. Outside some small domain, we then change the operator slightly, and transform the equation into a Cauchy problem on a torus which can be treated with the methods of Section 3.

Finally, we introduce some notation.

By M we denote either JRn or a closed smooth n-dimensional manifold.

The Banach space of functions whose derivatives up to the order k are bounded and continuous is denoted by C~(M), kENo. Similarly, we introduce the Holder spaces Ct,(M), s E JR+, and write Lip1(M) for the space of Lipschitz continuous functions on M.

Let CZ(M) denote the Holder spaces for s tic N, and the Zygmund spaces for s E N+. The Zygmund spaces CZ, s E N+, consist of all functions u with the property that u E C;-l and (in local coordinates)

" ID"'u(x) - 2(D"'u)((x + y)j2) + D"'u(y)1 sup ~ I I < (Xl. xoJy X - Y IO'I=s-l

The spaces C~ are continuously embedded in C~, for k E N+.

Let 6 be the Laplace-Beltrami operator on M and set (D) = (1- 6)1/2. In case of M = JRn, (D) can be written as a pseudodifferential operator with symbol (~) = (1 + 1~12)1/2 (a thorough representation of the theory of pseudodifferential operators can be found in Hormander [15]). Then we define the Sobolev spaces HS(l\1) = (D)-S L2(M) for s E JR, where L2(M) is the usual Lebesgue space of square integrable functions on M.

Assuming local coordinates x = (Xl, ... , X n ) on M, we will employ the multiindex notation:

Acknowledgment. I would like to thank Prof. Reissig for many useful discussions and the referee for his careful reading and helpful criticism.

164 M. Dreher

2. Pseudo differential Operators with Finite Smoothness

2.1. Definition and Mapping Properties

In order to describe the smoothness of functions and pseudodifferential symbols, we introduce some scales (XS)s of function spaces:

XS = HS(M),

XS = C:(M),

Xs = Cb(M),

n "2 < s < 00,

0< s < 00,

0:::; s < 00.

Definition 2.1.1 (Space of symbols of finite smoothness). The space XS Sro consists of all functions p(x,~) : M x IRn -7 C with '

IIDfp(·,~)llxs :::; Ca(~)rn-Ial a ~ o. In other words, for all N E No it holds that

7rr;J,xs (p) = sup {IIDfp(-,~) Ilxs (~) -rn+lal : ~ E IRn , 101 :::; N} < 00.

Definition 2.1.2 (Classical symbols of finite smoothness). We say that p(x,~) E xs Sd if there is an asymptotic expansion p(x,~) rv 2:>0 X(~)Pj (x, ~), where J_

Pj (x,~) are positive homogeneous of degree m - j in ~ and p - 2:7=-;;1 XPj E

xsS'!:O-N. The function X E COO(IR~) vanishes in a neighborhood of 0 and equals 1 for I~I ~ C > o. Definition 2.1.3 (Operators of finite smoothness). Let M = IRn. The operator spaces OP XS S1:o, OPXs Sd,respectively, consist of all operators p(x, D) mapping Co (M) into the space of distributions 1)' (M) whose symbols p(x,O belong to X s S1:o, X s Sd' respectively, and satisfy

(p(x, D)u)(x) = (27r)-n r eiX~p(x, ~)il(Od~ u E Cg"(Rn). JlR n

If M is a Coo manifold, then the operator p(x, D) is defined as follows. Let (n, "') be a local chart of M, "': M J n -7 U c IRn. Define the pull-back ",*: Co(U) -7 Co(n) by (",*u)(x) = u(",(x)), and the push-forward "'*: 1)'(n) -7 1)'(U) by (",*F, u) = (F, ",*u), (F E 1)'(n),u E Co(U)). Then an operator P: Co(M) -7 1)'(M) belongs to OPXsSro, OPxsSd if, for every local chart (n,,,,), "'* oPo

",*: Co(U) -71)'(U) belongs to OPxsS1:o, OPXsSd , respectively.

The following two mapping properties are cited from [33] and [34], Chapter 13.9.

Proposition 2.1.4. Let p(x, D) E OPC;Sro. Then p(x, D) can be extended to an operator continuously mapping c;+rn int~ C; (-s < r :::; s) and Hr+rn into Hr (-s < r < s), respectively.

In the case of operators with coefficients from Sobolev spaces, we have fewer problems with the borderline case r = s:

Proposition 2.1.5. If p(x, D) E OP H S Sro, then p(x, D) can be extended to an operator which maps HT+m continuously into HT for -s < r ~ s.

2.2. Special Smoothing Operators

Definition 2.2.1. Denote the Laplace-Beltrami operator of M by 6.. For 0 < E: ~ 1, we define the smoothing operator Je; = (1 - d,:::.)-1/2.

The proofs of the following lemmas are straightforward.

Lemma 2.2.2. The operator Je; is invertible and commutes with (D).

Lemma 2.2.3. Let XS be either HS(M) with s E JR, or C:(M) with s > O. Then there is a constant 0 > 0 such that for every 0 < E ~ 1,

IIJdIIXs+l ~ OE:- 1 Ilfllxs , IIf - Je;fllxs-t ~ OE:t Ilfllxs, 0 ~ t ~ 1.

2.3. Commutator Estimates

We quote some estimates from Coifman, Meyer [5], Kato, Ponce [18] and Taylor [33]:

Proposition 2.3.1. The following inequalities hold:

II[P,f]llv->v ~ CllfllLipl, P E opst,o, f E Lipl,

II[P, flllv->Hl ~ C IlfllLipl, P E OPSP,o, f E Lip\

II[P,J]IIH-,->V ~ CllfllLipl P E OPSP,o, f E Lipl,

IIP(fu) - f Pu ll L 2 ~ 0 IIfllLipl IlullHs-l + C IlfllHs IlullLoo, (2.1)

P E OPSf,o, s > 0, f E Lipl n H S , u E DXJ n H s - 1 .

Lemma 2.3.2. Let Je; be the smoothing operator from Definition 2.2.1. Then the assertions of the previous proposition hold for P = Je; with a constant C independent of E:, 0 < E: ~ 1.

Exploiting the above estimates, we come to the central result of this section: commutator estimates for operators with non-smooth, classical symbols.

Proposition 2.3.3. Let a(x,D) E opotS'j, b(x,D) E opots~ with a,/3 E {O, I}. Then it holds (with some N)

II [a(x, D), b(x, D)]IIHQ +f3-'---->V

166 M. Dreher

where aj, bj are the homogeneous components of the expansions of a, b with remainders ra,N, rb.N, respectively. If 0: = f3 = 0, then we additionally have

II[a(x, D), b(x, D)]llo~Hl

The key idea of the proof is the following. Since the symbol a( x,~) is classical, it allows the expansion (in local coordinates)

N-l 00 h(l,n)

a(x,~) = L L L ajlm(X)Yzm(~)(~)Q-j + ra,N(x, ~), j=o 1=0 m=l

where Y = Yzm(~)' 0 ::; l < 00, 1 ::; m ::; h(l, n), are the spherical harmonics, i.e., the eigenfunctions of the Laplace-Beltrami operator on the unit sphere sn-l, which form an orthogonal basis of L2(sn-l). The sequence {ajlm}l,m is rapidly decreasing in the sense that

h(l,n)

"" IIajlrnllC1 ::; Ck1T~-61(aj)(1 + l)-k, k;:::: 0, ~ b 1 b m=l

for alll ;:::: 0, and N sufficiently large. Moreover, ra,N E C~StoN. Plugging this expansion and a similar one for b(x,~) into the commutator [a, b], and employing the above commutator estimates from Proposition 2.3.1, one can derive the desired estimates. For details, see [33].

Now we want to generalize (2.1), replacing f E Lipl n HS by A(x, D) E

OPC~S'j n OPHsoS'd with 80 > nj2, 0: E No. Here we run into a problem, since an operator P from OPSi 0 does generally not map Cb(M) into C~(M). For this reason we introduce the space C'tK of all functions u satisfying (D)C<Yzm(D)u E ., 0

C~ for alll, m such that sUP1,rn(1+l)-Ko II (D)C<Yzrn(D)ullc~ < 00. The constant Ko

is fixed in such a manner that IIYzrn(D)ullco ::; C(l+l)Ko Ilulico for all u E Co(M). b b

The profit of this definition is that the mapping B : C't K ----> C~ is continuous for ., 0

all B E OPS'j. The embedding C~+8 C C~Ko is continuous for any positive 6, see [33], p.126. We have the (set-theoretical) inclusions C'tK C Ct: c Ce;: .

• , 0

Proposition 2.3.4. Let P E OPSf,o, A(x,D) E OPC~S'j n OPHsoS'j with 80 > nj2, 0< 8 ::; 80, 0: E No and K ;:::: Ko. Then it holds that

IIIP, A(x,D)] ulI L, ,; C (%;' K~~J,:(aj) +K~~~(rN)) lIullw,o-,

+ CK (%;' K';;;".(aj) + K~~;'(rN)) lIullc'K

with some constant N and the terms aj, r N from the asymptotic expansion of the classical operator A.

A proof can be found in [33]. Now we list properties of the spaces C~K'

Lemma 2.3.5. For every a E No, a positive constant C exists such that

Ilullea + II (D)"ulleo ::; C Ilullea .' u E C~Ko' b b ~ ,KO

(2.2)

Let 0" E Cb , u E C~Ko and Kl > Ko be sufficiently large. Then O"U E C~K1 and a constant C = C(O", a) (independently of u) exists with

II00uilea ::; C Ilullea (2.3) ~,K1 ~,Ko

2.4. Adjoint Operators and Compositions

Pseudodifferential operators with symbols of finite smoothness form an algebra in the sense of the following propositions, whose proofs can be found in [33].

Proposition 2.4.1. Let K(x, D) E opct S;z be a matrix pseudodifferential operator. Then the adjoint operator K*(x, D) satisfies

symb(K* (x, D) - R) = K(x, I:,)T, IIRUll o ::; C7rJ\, e" (K) IIUll o . , b

with some operator R and some N > O.

Proposition 2.4.2. Let A(x, D) E OPC~S~z' B(x, D) E opcts~z-j (j 0 or j = 1) be pseudodifferential matrix operators. Then

A(x, D)B(x, D) = C(x, D) + R,

C(x, 1:,) = A(x, I:,)B(x, 1:,) E C~ S;z,

IIRUll o ::; C7r~,c~(A)7r~~bt (B) IIUll o .

3. Weakly Hyperbolic Cauchy Problems with Spatial Degeneracy

3.1. The Linear Case

We are concerned with the linear Cauchy problem

D~tu+ aj,,,(x, t)D':D~ (O"(x)"'u) = f(x, t), j+I",I=rn,j<rn

u(x, to) = IPo(x), ... , D,;,-lU(X, to) = IPrn-l (x),

under the following condition of hyperbolicity:

Condition 3.1. We assume that the roots Tj (x, t, 1:,) of the equation

j+I",I=rn,j<rn

(3.1)

are real and distinct, h (x, t, 1:,) -Ti(X, t, f,) I ;::: cll:,l, c> 0, for i -=1= j, and all (x, t, 1:,).

168 M. Dreher

Remark 3.1.1. The space variable x lives on some manifold M, where either M = lR.n or M is a smooth closed n-dimensional manifold, and (3.1) is to be understood in local coordinates. Of special importance is the case of M being a torus, M =

(lR./27r)n: during the investigation of domains of dependence (in Section 5), we will transfer a hyperbolic equation which is defined in some bounded domain of lR.n x [0, T) into a hyperbolic equation defined on (lR./27r)n X [0, TJ, and bring into play the results to be proved now.

Marking the regularity of the data with subscript "d", we suppose that

'Pj E HSd+m-1-j (M), Sd ~ 0, (3.2)

f E C([to, TJ, HSd(M)). (3.3)

The weight function (J is presumed to be real-valued and smooth,

(J E Ct:(M, lR.).

In case of M being a closed manifold, we assume n

se> "2 + 1,

(3.4)

(3.5)

where the subscript "e" means "coefficient" . And if M = lR.n , we assume that there are constants Cj,a such that

n Se > "2 + 1. (3.6)

The reason for this distinction is that functions of HSc (lR.n) have to decay at infinity, making Condition 3.1 impossible to hold. For unity of notation, we may define Cj,a = ° in the first case, and Aj,a = aj,a - Cj,a for general M.

Our approach is as follows. We insert a smoothing operator (see Subsection 2.2) into (3.1), such that we obtain an ordinary differential equation for a function with values in a Banach space. Then this equation will be transformed into a first-order pseudodifferential system. An a priori estimate and an existence result will be proved for this regularized system, see Proposition 3.1.3 (a), and an a priori estimate for the corresponding non-regularized system will be shown in Proposition 3.1.3 (b).

The question of existence of a solution to (3.1) will be answered in Section 3.2, after we have investigated a quasi-linear version of (3.1).

3.1.1. CONSTRUCTION OF A FIRST-ORDER SYSTEM For ° < E ::; 1, we consider a regularized version of (3.1):

(3.7) j+lal=m,j<m

u",(x, to) = 'Po(x), ... ,D';-lUE(X, to) = 'Pm-l (x).

The operator J E maps Hr into Hr+l for any r E lR. with norm (')(c 1). This allows us to regard (3.7) as a linear Banach space ODE which is globally solvable, and

we acquire a unique solution

UE E C=([ta, T], Hmin(sc,sd)(M)).

If we succeed in finding estimates of U E which do not depend on E, then there is hope that the limit limE->a U E in the corresponding topologies exists and is a solution to (3.1). We recall that the Sp a seminorms of the pseudo differential symbol of J E

can be estimated uniformly in E, 0 :::; E :::; 1. We define the vector of unknowns

UE = (UE,I, ... ,UE,=)T, UE,j = ((D)JEr- j (a=-jD~-luE)' (3.8)

and get the system

DtUE,j = (D)JE (aUE,Hd + (D)JE [( (D)JE)=-j-I, a] ((D)JE F+I-=UE,HI,

DtUE,= =j - JE L aj,o:PO:(D)(aUE.Hd j+lal==,j<=

- JE L aj.o:pO:(D) [((D)JE)lo:l-I, a] ((D)Je)l-laIUE,HI j+lo:l=rn,j<=

with Pj = DXj (D)-I, pa = Il7=1 p;j. This gives

GtUE = KE(aUe) + BEUE + F, Ue(ta) = <1>E,a,

k j = - L aj,npo:, 1001==-j

o 1 0 o 0 0

(2) o be 0 0 o 0 bP) 0

o 0 0 b~rn)

bE.I bE.2 be,3 0

bE.k = -JE L ak_I,apn(D) [((D)Je)m-k,a] ((D)Je)k-rn, lal=rn+l-k

b~j) = (D)Je [((D)JE)=-j,a] ((D)Je)j-m,

F = (0,0, ... ,0, iJ)1' E C ([ta, T], HSd),

<1>,.a = (( (D)J,yn-I(am-I<pa), ... , <Pm-If E HSd.

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

We remark that Ka(D) is a strictly hyperbolic operator, and does not depend on E. Next we construct a symmetrizer for K a, using ideas from Leray [20J. We

170 M. Dreher

introduce the notation Pj = ~j (0 -1, pa = IT]=l p;j, and denote the eigenvalues of Ko(x,t,p) by iTj(X,t,p). Obviously,

K (1 m-1)T. (1 m-1)T o , Tj, ... , Tj = Z Tj , Tj, ... , Tj .

Let So = V(TI (x, t,p), .. . ,Tm(X, t,p)) be the Vandermonde matrix of the numbers ( T1, ... , T m), which satisfies

KoSo = iSo diag(T1, ... , Tm) = iSoD.

We put 8k(X, t,p) = "L.'F=1 Tj(X, t,p)k, and see that the matrix

80 81 82 8m-l 81 82 83 8 m

T S = SoSo = 82 83 84 8rn+1

8m-l 8 m 8rn+l 82rn-2 is symmetric and positive definite. Vieta's theorem reveals that the 8k are some polynomials in aj,ape>. The symmetrizer is defined as R = det(S)S-l, and is obviously a symmetric positive definite matrix. It remains to check that RKo is symmetric: The matrix KoS is symmetric since

T· T . TT T KoS = KoSoSo = zSoDSo = (zSoDSo) = (KoS) .

Setting C = det(S), we see that R is a symmetrizer for Ko:

RKo = cS- 1 Ko = cS- 1(KoS)S-1 = cS- 1(KoS)(S-I)T

= (cS- 1(KoS)(S-1)T)T = (RKof·

The components rij of R are some polynomials of the aj,apa, that is,

rij(x,t,p) = L Cijl( . IT .. aj,a(x,t)) ( . IT pa). IEB'j C],a)ED,]l (],a)EDijl

(3.14)

with Cijl E C and some finite index sets Bij and Dijl . Since the Tdx, t,p) depend on Pj = ~j (~) -1, we have R(x, t,~) E C~ S~l' The property of R being a symmetrizer implies (see [20])

C- 1 11V11~2 :::; (RV, V)V(M) :::; C 11V11~2' V E £2. (3.15)

The product structure of the kij and rij gives us the estimates

IIRllv-+L2 :::; C(max Ilaj,alleo), ),Q b

(3.16)

(3.17)

where the term C(maxj.a Ilaj,alleg) denotes a universal constant which depends

on maxj,a liaj,a Ileg in a nonlinear way.

Next we consider (Ko(D))* R + RKo(D). Proposition 2.4.1 gives us an expression of (Ko(D))*, and Proposition 2.4.2 tells us how to compose (Ko(D))*

and R, as well as Rand Ko(D). This way, we see that the principal symbol of (Ko(D))* R + RKo(D) falls out, and we obtain

II (Ko(D))* R + RKo(D) 110->0 ::; C(max Ilaj.alleo) max(llaj,alle1 + 1). (3.18) ],0. b ],Ci b

Finally, mapping properties of the matrix operator BE are studied. The as

sumption (Y E C'b implies b~j) E OPS~I; hence Ilb~j)vIIHs ::; C Ilvll Hs , uniformly in

E. Similarly, IlbE •k vll L2 ::; Cmaxj,a Ilaj,allLoo Ilvllo, which yields

If s > 0, then we can make use of formula (3.1.59) from [33],

and the estimates

Ilpa(D) [( (D)JE)m-k, 0'] ((D)J,,Jk-mvIIHs ::; C IlvllHs ,

IIPC'(D) [( (D)JE)m-k, 0'] ((D)JE)k-mvIILoo ::; C Ilvlleo , ~,Ko

which give us the uniform in E estimates

Let us summarize the results:

(3.19)

Proposition 3.1.2. The regularized linear Cauchy problem (3.1) can be transformed into the equivalent system (3.9) with UE, K E , BE, F, <PE,o from (3.8), (3.10), (3.11), (3.12) and (3.13), respectively.

The matrix operator Ko (D) is a strictly hyperbolic pseudodifferential operator with finite smoothness, ~o(D) E OPC~S~I' In case M = ]Rn, a matrix pseudodifferential operator K exists, whose symbol does not depend on x, such that Ko(D) - k E OPHscS21' In case of a compact manifold M, we have Ko(D) E OPHscS~I'

Furthermore, a symmetrizer R assigned to Ko(D) exists, which is a zeroorder pseudodifferential operator with symbol of finite smoothness, REO pel S~l' and induces a norm in £2 which is equivalent to the usual norm, see (3.15).

The operators Ko(D), R, (Ko(D))* R + RKo(D) and BE have the mapping properties given in (3.16)-(3.20), respectively.

3.1.2. A PRIORI ESTIMATES Now we have all tools to show an a priori estimate of strictly hyperbolic type:

Proposition 3.1.3. (a) The linear system (3.9) has a unique global solution UE E C 1 ([to, T], Hmin(sC,sd) (M)) which satisfies the following estimates for 0 ::; s ::;

172 M. Dreher

at (R(D)SUe, (D)SUe)

~ C(max Ilataj,alleo) IlUell~. + 2J(R(D)SUe, (D)SUe)J(R(D)S F, (D)s F) ),0. b

+ C(max lIaj,alleo) max(llaj,allel + 1) IlUell~s ),O!. b ],0. b

+ C(max lIaj,alleo) max(IIAj,aIIH' + 1) IlUellHs IlUelle, . j,o. b ],Ot. ~,KO

(b) Consider the system of type (3.9) which we obtain from (3.7) in the way of Subsection 3.1.1, replacing everywhere Je by the identity operator. Let U E C([to, T], Hmin(sC,Sd)(M)) n C1([to,T],Hmin(sc,sd)-1(M)) be a solution of such a system and 0 ~ s ~ min(se, Sd) - 1. Then

at (R(D)SU, (D)SU)

~ C(max lIataj,alleo) 11U1I~s + 2J(R(D)sU, (D)sU)J(R(D)s F, (D)s F) ],O!. b

+ C(max Ilaj,alleo) max(llaj,allel + 1) 11U1I~s j,o. b ),0. b

+ C(max Ilaj,alleo) max(IIAj,aIIHs + 1) IIUIIH8 11U11e' . J,o. b ),0. ti,Ko

Proof of (aJ We can write

at (R(D)SUe, (D)SUe)

= (Rt(D)SUe, (D)SUe) + (R(D)S(KeaUe + BeUe + F), (D)SUe)

+ (R(D)SUe, (D)S(KeaUe + BeUe + F)).

It is easy to estimate the first term on the right:

Since (R-,.) is a scalar product of L2, the Cauchy-Schwarz inequality yields

I (R(D)S F, (D)SUe) + (R(D)SUe, (D)S F) I ~ 2J(R(D)SUe, (D)sUe)J""-(R--""(D--"--)S F-,-.,.-(D----,)-s F--,-).

From the formulas (3.20) and (3.17) we see that

I (R(D)S BeUe, (D)8Ue) + (R(D)SUe, (D)S BeUe) I ~ C(max Ilaj,alleo) IlUell~. + Cmax(IIAj,aIIHs + 1) IIUeileo IIUEIIHs,

J,o. b j,Q tt,Ko

It remains to consider the terms

The scalar product h can be written in the form

h =111 + h2 + h3 + h4 + h5 + h6 + 117 + hs + h9

= (RJe [(D)S, Ko](D)a-Ue, (D)SUe) + (RJEKO [(D)S+1 ,a-] Ue, (D)S Ue)

+ (RJe [Ko, a-] (D)s+IUe, (D)SUe) + (RJea [Ko, (D)] (D)SUe, (D)SUe)

+ (R Pc, a-] (D)Ko(D)SUe, (D)SUe) + ([R, a] (D)JeKo(D)SUe, (D)SUE)

+ (a- [R, Jc] (D)Ko(D)SUe, (D)SUe) + (a-JcR [(D), Ko] (D)SUE, (D)SUc) + (a-JERKo(D) (D)SUE, (D)SUE).

173

We estimate now Ill, ... ,hs. From (3.17), Proposition 2.3.4 and Lemma 2.3.5 it can be deduced that

Ihll ::; C(max Ilaj,a Ileo) max(llaj,alle! + 1) IIUcll~< J,a b J,a b

+ C(max Ilaj,alleo) max(IIAj,aIlH< + 1) IIa-Ucll e ! IlUcllH8 J,a b J,a ~,K,

::; C(max Ilaj,alleo) max(llaj,alle! + 1) IIUcll~s ],0. b ),n b

+ C(max Ilaj,alleo) max(IIAj,aIIHs + 1) IlUclle! IlUelIH< . J,O: b ),Ct ~,KO

From (3.16), (3.17), and [(D)S+I, a-] E OPSl,O we can conclude that

Ih21 ::; C(max Ilaj,alleo) IlUell~s . J.a b

From (3.17) and Proposition 2.3.3 (0: = f3 = 0) it follows that

lId::; C(max Ilaj,alleo) max(lIaj,alle! + 1) 1Ia-lle! IlUcll~s . J,a b J,a b b

By (3.17) and Proposition 2.3.3 (0: = 0, f3 = 1) we have

11141 + Ihsl ::; C(max Ilaj,allen) max(llaj.alle! + 1) IlUcll~8 . J,CX b ],CX b

We use (3.17), Lemma 2.3.2, Proposition 2.3.3 (0: = f3 = 0), (3.16), and get

1115 1 + 11161 + 11171

::; C(max Ilaj,alleo) max(llaj,allel + 1) II(D)Ko(D)SUEIIH-l IlUellHs ),CX b ),a b

::; C(max Ilaj,alleo) max(llaj,alle! + 1) IlUell~s . ],a b J.a b

Summing up shows

Ih - h91 ::; C(max Ilaj,alleo) max(llaj,alb + 1) IlUcll~s l,Ct b ],a b

+ C(max Ilaj,alleo) max(IIAj,aIIHs + 1) IlUellHs IlUcllel . ),a b ),Q ~,KO

174 M. Dreher

The scalar product h can be decomposed into

(R(D)SUc' (D)S JcKo(D)aUc) = 121 + 122 + h3 + 124

= (R(D)SUc, J c [(D)S, Ko] (D)aUc) + (R(D)SUc, JcKo(D) [(D)S, a] Uc)

+ (R(D)SUc' [Jc, Ko(D)] a(D)SUc) + (a Jc(Ko(D))* R(D)SUc, (D)SUc) ,

where we used the self-adjoint ness of J c and the fact that a is real-valued. Similarly as above we obtain

Ih - h41 :::; C(max Ilaj,alleo) max(llaj,alle1 + 1) IlUcll~s ),Q b ),0: b

+ C(max Ilaj,alleo) max(IIAj,aIIHs + 1) IlUcllHs IIUcilel . ),Q b ],0: ~,KO

Finally, (3.18) yields

Ihg + h41 = I (aJc (RKo (D) + (Ko(D))* R)(D)SUc, (D)SUc) I

~ C(max Ilaj,alleo) max(llaj,alle1 + 1) IIUcll~s . J,O: b J,O'. b

Summing up we obtain the estimate of (a).

Proof of (b). We verify this estimate in a similar way as the previous one replacing the operators J c by the identity operator.

The Proposition 3.1.3 is proved. 0

Remark 3.1.4. The restriction 3 :::; mine 3 e , 3d) - 1 in the part (b) (instead of 3 :::;

min(Se, Sd) in the part (a)) can be explained as follows: The attempt to estimate at (R(D)sou, (D)SOU) (30 = min(Se,Sd)) leads to a term ((D)SoKaU, (D)soU) which does in general not exist, if U(" t) E HSo.

Or, seen from a different perspective: it is well known [9] that the assumptions (3.2), (3.3), (3.5), (3.6) lead to a solution U E C ([0, T], HSo) in the strictly hyperbolic case a == 1. Then the energy Eso(t) = (R(t)(D)soU(t), (D)soU(t)) is a continuous function of t, but in general not C 1 . Hence one cannot expect the estimate from the part (b) to hold for S = So.

Remark 3.1.5. In case of 3 = 0, we can slightly improve the estimates of Proposition 3.1.3: we may replace (3.20) by (3.19); and in the estimates of 111 and hI, we substitute Proposition 2.3.4 with Proposition 2.3.3, leading to

at (RU, U) :::; C(max Ilotaj,alleo) 11U11~ + 2V(RU, U)V(RF, F) J,a b

+ C(max Ilaj,alleo) max(llaj,alle1 + 1) IIUII~ ),0. b ),Q b

for 3d 2: 1. The advantage is that no Sobolev norm of the coefficients aj,a appears, and we can weaken the assumptions (3.5), (3.6) to aj,a E C~ ([to, T] x M). A similar estimate holds for operators with lower order terms D';: D~ (a1a1u), j + lal :::; m -1, and will be used to study domains of dependence.

3.2. The Quasi-linear Case

Now we consider the quasi-linear Cauchy problem with spatial degeneracy

D';u + I: aj,a(x, t, {D~(c~,!3(x, t)D~u)} )D~Di (0"(x)la 1u ) j+lal=m,j<m

= f(x, t, {D~(c~,!3(x, t)D~u)}Ik+I!3I::;m-l)' m 2: 2,

u(x, to) = 'Po(x), . .. ,D,;-lU(X, to) = 'Pm-l (x),

175

(3.21)

where the real-valued function 0" E Cb(M) describes the degeneracy, which occurs at the zeroes of 0", and the weight functions cZ,!3 characterize the Levi conditions as follows:

c~,!3 E Cl([to, T], H s c+I!3I(M)),

II (8tc~,!3(-, t))v(-) IIHs+li11 ::; C IIC~,i3(" t)v(·) IIH8+1i31 '

Ilc~,!3(" t)v(·) IIHsH31 ::; C Ilc~+l,!3(-' t)v(·) IIHs+Ii31 '

c~.6(x, t) = O"(x)I!3I, k + 1,61 = m - 1,

for Se > n/2+ 1, Se 2: S 2: O.

Example. We give some examples of cZ,!3' 0::; k + 1,61 ::; m - 1:

• cZ,!3(x, t) = O"(x)I!3I,

(3.22)

(3.23)

(3.24)

(3.25 )

• c~.!3 E C l ([to,T]'Hs c+I!3I(M)) with the property that (8tc~.!3)/c~.!3 and

c~.!3/C~+l.!3 belong to Cl([to, T], H s c+I!3I(M)) + Cl([to, T], C~c+I!3I(M)).

However, coefficients cZ,!3 with non-continuous quotients (8tc~,!3)/c~,!3 and c~.!3/ c~+ 1.(3 are also possible: let M = 1R'./27r be the unit circle, and set O"(x) =

exp(-(sin(x))-2), which has zeroes of infinite order for x E {0,7r}. Then the following weight functions cZ,6 E Cb([O, T] x M) are admissible:

{O"(X)I!31

cZ,!3(x, t) = (t + k + l)O"(x)I!31

(2t + k + 2)0"(x)I!31

: k + 1,61 = m - 1,

: x E [0,7r), k + 1,61 ::; m - 2,

: x E [7r,27r), k + 1,61 ::; m - 2.

We suppose that the coefficients and the right-hand side are defined in a suitable neighborhood Kc of the initial data,

(3.26)

for some G > O. Next, we introduce some set Xc c H m -1 (M x [to, T]) containing those functions v which can be reasonably inserted into aj,a and f:

v E Xc {=} (x, {D~(c~,!3(x, t)D~v(x, t))}) E Kc Vex, t) EM x [to, T].

Our regularity assumptions on the right-hand side f and coefficients aj,a are:

176 M. Dreher

• The mapping

n;~l Ck([to,T],Hs+rn-l-k(M)) nXc -7 C([to,T],HS(M)),

v(x, t) ,..... f(x, t, {D~(c~,(3(t, x)D~v(t, x))}),

is bounded and continuous for every s with n/2 + 1 < s ::::: Se.

• There are constants Cj,a such that the mappings

n;=ol Ck([to, T], H s+rn - 1- k) n Xc -7 nt=oCk([to, T], H s- k),

v(x, t) ,..... aj,a(x, t, {D~ (c~,{3(t, x)D~v(t, x))}) - Cj,a,

are bounded and continuous for every s with n/2 + 1 < s ::::: Se.

(3.27)

(3.28)

Remark 3.2.1. In case of a bounded manifold M, these conditions are satisfied if f E C([to, T], CSc(Kc)) and aj,a E C1([to, T], CSc(Kc )). If M = ]Rn, appropriate decays of f(x, t, {Vk,{3}) and aj,a(x, t, {Vk,{3}) - Cj,a for Ixl -700 are required.

The initial data are supposed to satisfy

'Pj E Hsc+rn-1-j(M), j::::: m - 1. (3.29)

Finally, we assume the hyperbolicity of the Cauchy problem (3.21):

Condition 3.2. The roots Tj(X, t, v,~) of

j+iai=rn,j<rn

are real and distinct, ITj(X,t,v,~) - Ti(X,t,v,~)1 :2: cl.;l, c > 0, #- j, for all (t,x,v,~) E [to,T] x Kc x ]Rn.

The main result of this subsection is the following theorem:

Theorem 3.2.2. Under the above assumptions, there is a To, to < To ::::: T, such that the Cauchy problem (3.21) has a uniquely determined solution u with

(D)k(uk D;n-l-ku ) E C([to, To], HSc(M)) n C1([to, To], Hsc-1(M))

for 0 ::::: k ::::: m - 1. This solution persists as long as the vector of weighted derivatives (x,{D~(cg,{3(x,t)Dfu(x,t))}) stays in Kc for all x and the norms II(D)k(uk D;n-k-lu)llc; remain finite, 0::::: k ::::: m - 1.

The following well-posedness result in Coo is an immediate consequence:

Theorem 3.2.3. Consider the Cauchy problem (3.21) assuming (3.4) and 'Pj E HOO(M). Furthermore, we assume that (3.22)-(3.25), (3.27), (3.28) hold for all s :2: 0, Se > n/2+1 and that Condition 3.2 is satisfied. Then the Cauchy problem (3.21) has a unique solution

u E Crn([to, To], COO(M)).

If the equation is linear, then we have global existence:

Remark 3.2.4. Proposition 3.1.3 and Theorem 3.2.2 give us a unique solution u to the linear Cauchy problem (3.1) which cannot blow up, i.e.,

for 0 :::: k ::: m - 1 and So = min(sc, Sd). Of course, we can include lower-order terms D~(cL3(X, t)D~u(x, t)) as long

as the equation remains linear.

The proof of Theorem 3.2.2 comprises the Propositions 3.2.5-3.2.8. At first, we construct a regularized hyperbolic first-order system for a vector of weighted derivatives up to the order m - 1. Employing the ideas from Subsection 3.1 we prove the existence and estimates of the solution U; to this perturbed system. Then we show that the life-span of the U; does not tend to zero as c approaches zero. For c ---> 0, the UE converge to a solution U of the asserted regularity. Last, the blow-up criterion will be proved.

3.2.1. CONSTRUCTION OF A FIRST-ORDER SYSTEM We start with a regularized version of (3.21),

D';'uE + JE L aj,a(x, t, {D~ (c~.f3J!f31 D~UE)} )D'; J~ Di (0'Ia1uc ) j+lal=rn,j<m

= f(x, t, {D~(c~,f3J!f3ID~UE)})' (3.30)

uE(x, to) = 4?o(x), ... , D;n-luE(x, to) = 4?m-l (x).

This is a quasi-linear ODE for a function U E with values in the Banach space Hsc(M); hence it has a solution

U E E em ([to , TE ], HSc(M)), To > to.

We define UE.k = ((D)JEJm-k(O'm-k D~-lUE) as in (3.8), and

UE.I,k(X, t) = ((D)JE)I(O'(x)1 D~UE(X, t)), k + l ::: m - 2,

UE,k.f3(X, t) = (D)If3I(cZ.f3(x, t)J!f3ID~UE(X, t)), k + 1;31 ::: m - 2.

Obviously,

OtUE,k,f3 = (D)If3I((Otc~,f3)J!f3ID~uc) + i(D)If3I(c~.f3J!f3ID~+lUE)' Assuming k + 1;31 ::: m - 3, the HS norm of the right-hand side is bounded by

II(OtCZ (3)J!f3ID~UEII + Ilc~ f3J!f3ID~+lUEII ' Hs+If3I' Hs+If31

(3.31 )

::: e Ilc~ f3J!f3ID~UEII + e Ilc~+l f3J!f3ID~+lUEII ' Hs+I(J1 ' Hs+If31

= e IIUE.k,f3II H s + e IIUE,k+l,f3II H s •

178 M. Dreher

And for k + 1,61 = m - 2, we obtain

II (Ote~ (3)Jl!3ID~UEII + Ile~ (3J~(3ID~+lUEII . ~ Hs+lfll' Hs+lfll

< C Ilea JI 6 1Dku II + C Il a.l6IJI(3IDk+luE II - k,(3 E t E Hs+lfll E t H,'+If31

~ C IIUE,k,(3II Hs + C L 11«(D)JE)I(O'ID~+lUE)IIHs , I:S: 1(31

see (3.22)-(3.25). We introduce the vector

U; = ({UE,k,(3}, {UE,I,d, u'[f and obtain

a U* (0 0 ) (U*) (GE(X, t, U;)) t E = 0 K (x t U* D) 0' E + F (x t U*)

c " c' E: " E

with IIGEllHs ~ C 11U:IIHs, cf. (3.31), (3.32). This system can be written as

(3.32)

OtU; = K;(x, t, U;, D)(O'U;) + FE*(X, t, U;), U;(to) = <1>;. (3.33)

The matrix R*(x,t,U*,D) = diag(E,R(x,t,U*,D)) is a symmetrizer for K*, where E is the identity matrix and R is the (independent of c) symmetrizer from Subsection 3.1.

3.2.2. ESTIMATES AND COMMON EXISTENCE INTERVAL According to Theorem 3.2.2, the solution U: E C l ([to, TE ], HSc) persists as long as it stays in KG and as long as 11U: II HSc < 00. Applying Proposition 3.1.3 and Cl,b C Ci,Ka we get

at (R*(D)SU;, (D)SU;)

~ C(max IIOtaj,allca) 11U;II~s ),0: b

+ 2J(R*(D)sU;, (D)sU;) J(R* (D)s F;, (D)s F;)

+ C(max Ilaj,allcD) max(llaj,allcl + 1) 11U;II~s ],0: b ],0. b

+ C(max Ilaj,allcD) max(IIAj,aIIHs + 1) IIU;IIHs IIU;ll c ",8 ],Q b ],Q b

for Sc 2: S > n/2 + 1 + <5, 0 < <5 < 1. The Moser-type estimates

Ilaj,allcl ~ C(IIU;llcD)(IIU;llc" + 1), b b b

IIAj,aIIHs ~ C(IIU;IIL=)(IIU;IIHs + 1),

and the embedding inequality IIU:llcu ~ C 11U:IIHs can be applied on the right. b

Let us consider the term C(maxj,a Ilotaj,allcg) which denotes some constant that

depends in a nonlinear way on maxj,a Ilotaj,a II cg' The computations which lead to this term show that it has the form

Appealing to Lemma 2.3.5 and (3.16), we have

Ilataj,e>llc~ ::::; C(l + IlatU;llc~)::::; C(l + IIK;O'U;llc~ + IIFE*llc~) ::::; C(l + 11U;ll c 1,8).

b

Taking into account all these inequalities we obtain

at (R*(D)SU;, (D)SU;) ::::; C(IIU;IIL=)(IIU;llc~a + 1) 11U;II~s

+ 2J(R* (D)sU;, (D)sU;)J(R* (D)s F;, (D)s FE*)'

179

(3.34)

In the next step we show indirectly that there is a common existence interval of the solutions U:. We define tE E (to,TeJ by the inequality 11U:(t)IIHs ::::; 2 sUPO<c'<EO 1I;,IIHs + 1, to ::::; t ::::; te; and by the condition that the components of the vector U: (t) be in the interior of the domain of definition of the coefficients aj,e> and the right-hand side j, for to ::::; t ::::; tEo

To obtain a contradiction, let us assume that for every small 'I > 0 an E = E( 'I) exists with to < te ::::; to + T Now we study estimates of 11U: II H' and 11U: - <1>; II L='

The norms 1IVIlu and J(R*V, V) are equivalent as long as R* is defined (i.e., for t ::::; te). From (3.34) and IlFe*IIHs ::::; C(IIU:IIL=)(IIU:IIHs + 1) we see that

at (R*(D)SU;, (D)SU;) ::::; Q((R*(D)SU;, (D)SU;)),

where Q is a smooth nonlinear increasing function, independent of E. Let To be a number with the property that the nonnegative solutions of

aty(t) ::::; Q(y(t)), (3.35)

y(to) = (R*(x, to, <1>*, D)(D)s;, (D)s;) , 0::::; E ::::; EO,

satisfy y(t) ::::; 2 sUPo<e'<eo 11*IIHs + 1 for to ::::; t ::; To. To estimate 11U: - <1>; IIL=' we write U: = <1>; + ~* and get, by Proposi

tion 3.1.3,

at (R*~*, V/) ::::; C(IIU;IIL=)((R*Ve:*, Vc*) + (R* F;, FE*))'

(R*VE*' ~*) (to) = O.

The norms 11U:II L= are uniformly bounded for t::; te, due to the definition of te' From Gronwall's lemma it can be concluded that

(R*~*, Vc*) (t) ::::; g(tf, to::::; t ::; min (To , te),

g(to) = 0, g continuous and increasing. We obtain 11~*(t)ll~ ::::; Cg(t)2; and an interpolation argument gives us a continuous function gl, such that

(3.36)

This demonstrates that te cannot come arbitrarily close to to, which is a contradiction. Hence there is a common existence interval.

We have proved:

180 M. Dreher

Proposition 3.2.5. There is a constant To > to with the property that the systems (3.33) have unique solutions U: E C 1 ([to, To], HS) for 0 < c ::; co and Se 2: S > n/2 + 1. It holds for all 0 < c ::; co and all to ::; t ::; To that

IIU;(t)IIHs ::; c, IIU;(t) - <1>* liD'" ::; gl(t)

with some continuous function gl (t), gl (to) = O.

3.2.3. CONVERGENCE AND REGULARITY OF THE LIMIT Our previous a priori estimates allow us to show

IIU;C t) - U;'(" t)llt ::; C(To - to)(c + c'), to::; t ::; To.

By the uniform bound 11U:IIHsc ::; C and interpolation, the following result is easily established:

Proposition 3.2.6. The above constructed sequence (U:) c C 1 ([to, To], HSc) converges in C ([to, To], H S ) and C([to, To], Ct,li) for any sand 8 with n/2 + 1 + 8 < S < Se. The limit U* belongs to C 1 ([to, To], Hs- 1) and is a solution of (3.33) with c = O.

It remains to study the regularity of U*. Here we make use of standard techniques, which can be found e.g., in [28]. The uniform estimate of U: in HSc gives U* E LOO([to, To], HSc) and U* E Lip1([to, To], H sc- 1). We want to show that the HSc norm of U* is not only bounded, but also continuous:

Proposition 3.2.7. The above constructed solution U* to (3.33) with c = 0 belongs to C ([to, To], HSc) n C 1 ([to, To], H sc- 1).

Proof. We fix to ::; t1 < To and will show the continuity of IIU*(" t)IIHsc at the point t 1 . To this end, we consider the forward Cauchy problem (recycling the variable U; which we do not need anymore)

OtU; = K;(x, t, U;, D)(aU;) + FE*(X, t, U;), U;(td = U*(td,

for some small c > O. The only difference to (3.33) is the initial condition. Defining the equivalent norm for the Hilbert space HSc

(3.37)

we deduce from (3.34) that

Ot 11U;(t)II~sc,t ::; C'(IIU;llc~'o) 11U;(t)ll~sc,t + C 11F:(t)ll~sc' Gronwall's lemma gives, with some C' = C'(IIU:ll c1,O),

b

11U;(t)ll~sc,t ::; 11U*(tdll~sc,tl eC'(t-t,j + clt eC'(t-T) 11F:(T)II~sc dT.

t,

The weak compactness of bounded subsets in Hilbert spaces implies (c --+ 0)

IIU*(t)ll~sc,t ::; IIU*(tdll~sc,tl eC'(t-t,) + Cit - t11·

The function IIVllHsc t defines an equivalent norm in the Hilbert space HSc if t is fixed. For proving ~ontinuity in t we need a norm which does not depend on t. Using the Lipschitz continuity of R*, we find

IIU*(t)ll~sc.i1 :::; IIU*(tl)ll~sc,t1 eC'(t-til + CI/lt - tIl,

resulting in

lim sup 11U*(t)ll~sc t :::; IIU*(tI)ll~sc t :::; liminf IIU*(t)ll~sc t , t-+t1 +0 ' 1 , 1 t-+t1 +0 ' 1

which gives the HSc-continuity of U* at tl from the right. Inverting the time direction the reader can show the continuity from the left. Since tl can be chosen arbitrarily, the proof is complete. 0

In the last step a criterion for the blow-up is given. The idea of the proof is taken from [33], Proposition 5.1.F.

Proposition 3.2.8. Let U* E C([to,T),HSc) n C l ([to,T),HSc- l ) be a solution of (3.33) (c = 0) with

sup IIU*(t)llcl < 00, tE[to,T) •

inf dist((x,{Uk,/3(x,t)}),oKG) 2: J > O. tE[to,T)

Then a constant Tl > T exists with U* E C ([to, T I ], HSc) n C l ([to, T I ], Hsc- l ).

Proof. We consider the non-regularized version of (3.33), apply JE; to it, and estimate the terms on the right as in the proof of Proposition 3.1.3. This gives

at (R*(D)ScJE;U*, (D)ScJE;U*) (3.38)

:::; C(IIU*IILoo)(l + IIU*llcl + IIU*llcl ) 11U*II~sc + 11F*11~8C . b ~,KO

We suppose that the HSc norm is arranged in such a way that 11V*llcl :::; 11V*IIHsc holds for every function V* E HSc. Then the inequality •

IIV*II 1 :::; C IIV*II 1 (1 + In (11V*IIHsc)) C~,KO c. IIV* Ilc,;

can be shown, see [33], (B.2.12). Consequently,

IIU*llcl :::; C IIU*llcl (1 + In+ IIU*IIHsc) + C. ~,Ko *

According to Lemma 2.3.5, we have IIU*llcl :::; C IIU*llcl , and from IIF*II~sc :::; b ~,KO

C(IIU*IILoo)(e + 11U*II~sc) it follows that

at (R* (D)Sc JE;U*, (D)Sc JE;U*)

:::; C(IIU*IILoo)(l + IIU*llc,;)(l + In+ IIU*II~sJ(e + 11U*II~sJ. Using the equivalent norm 11·IIHsc,t from (3.37) and IIU*llc,; :::; C, we get

at IIJE;U*(t)IIHsc,t :::; Co(l + In+ IIU*(t)II~8c,t)(e + 11U*(t)IIHsc,t).

182 M. Dreher

We integrate, let E tend to 0 and see that

IIU*(t)II~'c,t ~ IIU*(to)ll~sc,to

+ Co it (1 + In+ 11U*(T)II~sC'T)(e + IIU*(T)II~sC'T) dT. to

Introducing N(t) = e + IIU*(t)ll~sc.t we deduce that

N(t) ~ N(to) + 2Co it In(N(T))N(T) dT = Q(t). to

The continuity of N(t) yields Q(t) E C 1 [to, T). Obviously,

Q'(t) = 2Co In(N(t))N(t) ~ 2Co In(Q(t))Q(t);

hence Q(t) ~ C for all to ~ t < T. Taking into account all these inequalities we find 11U*(t)IIHsc ::; c' for to ~ t < T.

Next we have to extend U* continuously to some longer time interval. Therefore, we review the regularized Cauchy problems (3.33) with data U;(T - "1) = U*(T - "1) for some small "1 > O. The functions U;(t) persist as long as their HSc norms remain bounded and as long as each component of these vectors stays in the domain of the functions aj,a and f. An estimate of the life-span of U; is provided by (3.35) and (3.36), which are autonomous (differential) inequalities (independent of E), hence the length of the existence interval only depends on 5 and C'. It follows that for small "1 the point T is contained in the common existence interval of the U;, and consequently, of the limit U. 0

This completes the proof of Theorem 3.2.2.

Remark 3.2.9. Crucial for the proof of Proposition 3.2.8 was the fact that the right-hand side of (3.38) grows at most linearly in IIU* lie; and IIU* Ilei,KQ'

Remark 3.2.10. Using similar arguments as above, one can easily show that the solution of a quasi-linear weakly hyperbolic Cauchy problem continuously depends on the data, weight functions, coefficients and right-hand side.

4. Weakly Hyperbolic Cauchy Problems with Spatial and Time Degeneracy

Now we are ready to study the general equation (1. 7) which incorporates both types of degeneracy: spatial degeneracy and time degeneracy.

Our approach is divided into three steps:

• First, we study a linear Cauchy problem with vanishing initial data, whose right-hand side has a zero of sufficiently high order at t = O. We establish an estimate of strictly hyperbolic type in Theorem 4.1.1.

• Secondly, we consider a quasi-linear Cauchy problem. Its initial data also vanish, and the right-hand side has a zero of high order at t = 0. The estimates of strictly hyperbolic type for the linear problem and the usual iteration procedure imply local existence, see Theorem 4.2.6.

• Thirdly, we transform the general equation (1.7) into a special equation which can be treated with the methods of the second step, see Theorem 4.3.1.

The idea of transforming a weakly hyperbolic problem with general righthand side into another weakly hyperbolic problem with special right-hand side has been widely used, for example in Kajitani, Yagdjian [17], Oleinik [25] and Reissig [29].

4.1. A Special Linear Case

We analyze the Cauchy problem

D';'u + L aj.a(x, t)A(t)lalD';D~ (CT(x)la 1u ) = f(x, t), (4.1) j+I",I=m.j<m

u(x,O) = ... = D~'-IU(X, 0) = 0

under the decay assumption

(4.2)

Later, the number p will be presumed sufficiently large. Additionally, we suppose Condition 3.1, (3.3), (3.4), and (3.5) or (3.6). For the function A = A(t) we assume:

Condition 4.1. Fix A(t) = J~ A(T) dT. The function A = A(t) satisfies one of the following conditions:

• A(t) = tl, lEN, l ~ m - 1, • A E C 2 ([0, T]), A(O) = 0, A/(t) > 0 (t > 0), X'(t) ~ 0 (t ~ 0),

A(t) C A'(t) A(t)::; ,\ A(t) ,

X(t) C' A(t) A(t)::; ,\ A(t)'

m C,\ < --.

m-l

Examples for the second case are A(t) = exp(-Itl- T ), r > 0. The central result is the following theorem:

Theorem4.1.1. If the constantp is sufficiently large, then the Cauchy problem (4.1) has a solution u with the property that

U E C ([0, T], HSc) n C1 ([0, T], Hsc- l ) ,

U = {(D)m-i«ACT)m-iD~-lu) : i = 1, ... , m}.

There are constants C I , C 2 , C3 (independent of u) such that the estimate

t C,(t-T) (A(t))C2 IIU(t)IIHsc ::; C3 Jo e A(T) Ilf(T)II H8 c dT

holds for p > C2 + 1.

184 M. Dreher

Proof. We obtain the system

>..' OtUj = (m - j)>:Uj + )..i(D)(o"Uj+l)

+ >.i(D) [(D)m- j-l, 0] (D)j+l- mUj+1, 1 S; j < m,

OtUm = -i L aj,a)..laID'.;Di (ola lu) + if j+lal=m,j<m

= -iA L aj,apa(D)(oUj+d

This leads to

j+lal=m,j<m

- iA L aj,a pa (D) [(D) la l-l, a] (D) l-Ial Uj+l + if. j+lal=m,j<m

A' OtU = )"Ko(D) (aU) + )"BU + H >:U + F, U(O) = 0 (4.3)

with Ko(D), B, F as in (3.10), (3.11), (3.12) and H = diag(m -1, m - 2, ... ,1,0). We regularize this system and eliminate the time degeneracy:

>..' Ot Ui5E; = JE ().. + J)Ko(D)(aUsc ) + ABUsE + H A + JUs E + F, (4.4)

USc(O) = 0,

where 6 > 0 is small. This is a weakly hyperbolic system with pure spatial degeneracy; therefore we can take advantage of the methods of Subsection 3.1. We may choose the same symmetrizer R, since the function A + 6 has no influence on the operator Ko(D). This operator does not feel the time degeneracy.

We choose s = 0 or n/2 + 1 < s S; sc, and define the norm

Then Proposition 3.1.3 gives us the estimate

By Gronwall's lemma we see that

Hs(U/ic(t)) S; it eC2 ,8(t-T)C1 IIF(r)IIHS dr

S; exp (Ct) C it )..'(r) .. (r)Pdr S; CSA(t)P+I,

C2,/i = sup {C1 + C2 A~~;~S} = 0(6-1). tE[O,Tj

This allows us to apply Nersesyan's lemma (see Lemma 6.2) to (4.5) if we assume p> C2 + 1. The result is

Hs(U(jc(t)) ~ it eC, (t-T) (~~~) C 2 C1 IIF(T)IIHs dT

~ c)"'(t) C2 it eC, (t-T) ... (T)P-C2)...'(T)dT

C C,t < e )...(t)p+1. - p- C2 + 1

We emphasize that this estimate is independent of 5 and c. It is known that U(jE belongs to the spaces C ([0, T], HSc) and C 1 ([0, T], H sc- 1). Employing the methods from Subsection 3.2 one can show that there is a limit U(j = limc---+o U(jE which belongs to the same spaces and solves

and that the following a priori estimate holds for n/2 + 1 < S ~ Sc and s = 0:

(4.7)

In the next step we send 5 to 0 and study the convergence properties of the sequence (U(j). The difference U(j - U(jf solves the equation

)...' = ()... + 5')Ko(D)(0'(U(j - U(jf)) + )...B(U8 - U8 f) + H )... + 5' (U8 - U8f)

+ (5 - 5') ( Ko(D) (O'U(j) - H ()... + 5;()... + 5') U8) .

From (4.7) with s = 0 it can be concluded that

HO((U8 - U8 f )(t)) ~ CI5 - 5'1)...(t)P.

It is standard to verify that the sequence (U8) converges to a limit U which is a solution of (4.3) and satisfies

On the other hand, we have U(O) = 0 and IIU(t)IIHsc ~ C)...(t)p+1. This gives the continuity for t = 0 and the theorem is proved. D

186 M. Dreher

4.2. A Special Quasi-linear Case

Now we assume that the initial data vanish and that the right-hand side decays sufficiently fast for t ----> O. More precisely, we consider the Cauchy problem

D';'u+ j+lal=1n,j<m

= f(x, t, {D~Ck,,6(X, t)Dfu}lk+I,6I:Sm-I),

u(x,O) = ... = D,;,-IU(X, 0) = 0

with the following asymptotic behavior for f:

Ilf(·, t, O)IIHsc S CfoA(t)P A'(t),

L 1i00f Ct,{gk,,6})11 SCf k+I,6I:Sm-1 gk,,6 £00

( 4.8)

(4.9)

(4.10)

for all t E [0, TJ and all {gk,,6} E ]Rna from a suitable chosen compact set near zero. Furthermore, we suppose (3.4), (3.27), (3.28), (replace c~,,6 by Ck,,6, to by 0 and KG by some compact set near zero) and Condition 3.2. Concerning the weight functions Ck,,6, we assume the Levi conditions

{A(t)m-k A(t)k+I,6I-1nc~ (x, t)

Ck,,6(X, t) = 0 ,,6 ck,,6(x, t)

: 1;31 > 0,

: 1;31 = 0, (4.11)

where A(t) = I~ A(T) dT; and (3.22)-(3.25). Our intention is to show the local existence of a solution to (4.8), see Theorem

4.2.6. For this purpose we transform the equation into an equivalent system of first order and study a linearized version of this system. In other words, we introduce U = (UI , ... , U1n )T, V = (VI, ... , V1n )T, V* = (V{,6' VT)T,

Uj = (D)1n- j ((Aa)1n- j Dtlu),

Vk,,6(X, t) = D~(Ck,,6(X, t)Dfv(x, t))

and get the system

Vi = (D)m- j ((Aa)m- j Di-Iv),

(4.12)

OtU = AKo(x, t, V*, D)(D)aU + AB(x, t, V*, D)U + F(x, t, V*) + H~ U,

U(O) = O.

We obtain a mapping U = (V), defined via V f---4 V* f---4 U, and can construct a sequence {Vk} by V k = (Vk- I ), V O = O. This sequence will be shown to converge for large p and small times, after some preparatory estimates.

4.2.1. AUXILIARY ESTIMATES We proceed with estimating V* in terms of V, and K o, B, F in terms of V*.

Lemma 4.2.1. Let T < 1 and IIV(t)IIHs :S CVp>'(t)p+l. Then there is a constant C4 , independent of V and p, such that

(4.13)

Proof. The assertion is obvious for 1;31 = 0, k = m - 1. Now let 1;31 = 0, k :S m - 2. Then (3.23) and (3.24) imply

at Ilck,pD~vIIHS :S II(atCk,p)D~vIIHs + Ilck,pD~+lVIIHs :S Cc Ilck,pD~vIIHs + Cc Ilck+l,pD;+lVII Hs .

By Gronwall's lemma and induction, we get (4.13) for 1;31 = 0 and k :S m - 2. Now 1;31 > O. We choose k = m - 1 - 1;31 as base of induction, and deduce that

II(D)IP1Ck,p(-, t)D~vIIHS = ~~~~ 11>'(t)IPI(D)IPI(O'li3I D ;v)IIHS

yet) , :S C), >.(t) IIV(t)IIHs :S C)'Cvp>'(t)P>' (t).

Let k+ 1;31 :S m - 2. Making use of Y ~ >.2 A (see Condition 4.1) and the induction hypothesis, we obtain

Ilc~+l,pD;+lVIIHs+Ii31 :S CCvp>.(t)P+k+l-m >.'(t)A(t)rn-k- 1i3I -l

:S CCvp>.(t)P+k-l- m (>., (t))2 A(t)rn-k- 1P1 .

By (3.23) and (3.24) we then conclude that

at Ilc~,pD;vIIHS+Ii31 :S CCVp >.(t)p+k-l-rn(>.'(t))2 A(t)m-k- 1P1 + C IIC~"i3D~VIIHs+Ii31 .

Bringing Gronwall's lemma into play, we then find

Il co, Dkvll < cc it >'(T)p+k-l-m(>.'(T))2 A(T)rn-k-IPI dT k.,i3 t Hs+Ii31 _ Vp

° :S CCvp>.(t)p+k-rn A(t)m-k- 1P1 >.' (t),

which concludes the proof. D

Remark 4.2.2. The conclusion of this lemma can be sharpened in the following way. If 0 :S s :S Se, then the estimates

IlVk,pllHs :S II(D)IPI(Ck,pD~v)11 :S >'(t)P>"(t)C4 sup 11~(~)I~~s Hs rE[O,t] T P

hold for k + 1;31 = m - 1. And for k + 1;31 :S m - 2 we have

II(D)IPI(c Dk+lV)11 II Vi II < >.(t)P>"(t)C k+l,p t HS k,p Hs _ 4 sup '( )p \I( ) .

rE[O,t] /\ T /\ T

188 M. Dreher

This lemma gives us estimates for U* and V* if bounds of U and V are known. An estimate of U in the terms of the right-hand side is given by Theorem 4.1.l. The next lemma will be practical to find an estimate of F in terms of V* .

Lemma 4.2.3. Let K c ]Rna be compact and M be an n-dimensional smooth closed manifold. Let f E C N (M x K) and Vi E HN (M) with (x, VI (x), ... , vno (x)) E

M x K for x E M. If N is sufficiently large and 0 ::::; m < no, then a constant Nl < N (independent of N) exists with

Ilf(" VI (-), ... , Vno (-)) II~N

::::; 'PN(llvllleNl , ... , IIVno IleN1 ) b b

X (1IvIIIHN + ... + IlvrnllHN + Ilvrn+Iil HN-" + ... + Ilvno II HN-l)

no II of II a + L OV("Vl(')""'vno (-)) L II0xVjllo j=rn+l J £00 lal=N

+ L Ilf(a,o, ... ,O)(-,vl(·), ... ,vnJ))llo· lal::;N

Here we used the equivalent norm Ilwll~N = L:1al::;N 110~wllo'

(4.14)

This lemma generalizes Remark A.l in [11] and describes precisely the dependence of Ilf("vl,,,.,vn)II~N on the highest orders of some Vj (see the terms 110~vj 110)' The proof is omitted. We will use this lemma to determine the loss of Sobolev regularity (it depends on Ilfvj 11£00)' or, in other words, to determine the space in which the solution exists.

From now on we assume Se = N E N and set SI = N 1 .

Lemma 4.2.4. Assuming II {Vk,!3} IIeS1 ::::; 1, there are constants C5 and T*, such b

that for 0 ::::; t ::::; T*,

(4.15)

Proof. Lemma 4.2.3 and (4.14) allow us to estimate

Ilf(x, t, {Vk,P} )IIHsc ::::; cSc Ilf(x, t, {Vk,!3} )11~sc

::::; Csc'Psc(II{Vk,!3}lle~d II{Vk,!3}IIHsc-1 + cSc L II ~:: (x, t, {Vk,!3})11 lal::;sc 0

+csc L Iloi II L Ilo~Vk,!3llo' k+I!3I::;m-l k,!3 £00 1<>I=sc

Repeated application of Remark 4.2.2 yields (for 1(31 > 0)

II (D) 1i3lcm_l_Ii3I,i3D;n-l-li3lv II IlVk,i3II H s c- 1 ::; G).,(t)p ).,'(t) sup ).,( ) )..I( ) Hsc- 1

TE[O,t] T P T

::; G).,(t)P).,'(t) sup ).,(T)-P II(D)'i31- 1 (>.a)'i31- 1 D;n-l-'i3lvll . TE[O,t] Hsc

Considering the time derivative of the last norm and exploiting Nersesyan's lemma, we find (restricting the time interval)

II{Vi }II < >.(t)p+l s IIV(T)IIHsc k,i3 Hsc-1 - up '( )p+l .

TE[O,t] /\ T

Again by Remark 4.2.2,

~ 118a Vi II < G>.(t)P>"(t) sup IIV(T)IIHsc. ~ x k,i3 0. - >'(T)p+l

1<>1=8c TE[O,t]

Lastly, from Hadamard's formula and (4.9) it can be deduced that

Summing up, we can estimate for small t:

o

4.2.2. ITERATION AND CONVERGENCE Now we have all tools to find a bound for the mapping V f---+ V* f---+ U.

Lemma 4.2.5. We assume that p is sufficiently large and that T* is sufficiently small. If IIV(t)IIHsc ::; >.(t)P+l for 0 ::; t ::; T*, then IIU(t)IIHsc ::; ).,(t)p+l for o ::; t ::; T* .

190 M. Dreher

Proof. Due to Theorem 4.1.1 and (4.15) we have

IIU(t)IIHsc ::; c31t eC, (t-s) (~~!D C2 Ilf(s)IIHsc ds

Ctlt('\(t))C2 () f()( IIV(T)IIHsc ) ::; C3 e 1 -( ) C5 ,\ s P,\ S C f sup \() +1 +Cfo ds ° ,\ s TE[O.s] /\ T P

,\(t)P+ 1 < C C (C + C )eC,t_-'--':c-__ - 3 5 f fO P _ C2 + 1

::; A(t)p+1

if eC, t ::; 2 and p is large. o

Now we restrict the constant T* in such a manner, that the assumption IIV(t)IIHsc ::; ,\(t)p+1 for all t E [0, T*] implies (x, (Vk,{3(X, t))) E KG for all (x, t) E

M x [0, T*]. All these results enable us to define a sequence (Vi) C C ([0, T*], HSc)n C 1 ([0, T*], H s c- 1 ) by VO(t) == 0 and

Vi(O) = 0, l2: I,

at Vi = AKo(x, t, V*,i-1, D) (D)(jVI + '\B(x, t, V*,1-1, D)Vi

A' + F(x,t, V*,1-1) + H):.Vi, l2: 1.

Due to Lemma 4.2.5 and Remark 4.2.2 the functions Vi fulfill

Using the above technique once more and choosing p larger if necessary, we are able to show the estimate

l!Vi+l(t) - Vi(t)11 2 1 sup "'------l---'-'-"L"-- < - sup

tE[O,T*] ,\(t)p+ - 2 tE[O,T*]

I!VI(t) - V i - 1 (t)llu A(t)p+1

This confirms that the sequence (Vi) converges in C ([0, T*], HO). By interpolation we see that (Vi) converges in C ([0, T*], H s c- 1 ), too; and we can prove in a standard way that the limit U is a solution of (4.8). Exploiting the arguments which gave Proposition 3.2.7, we get U E C ([0, T*], HSc). Thus, we have proved:

Theorem 4.2.6 (Existence). Let the conditions mentioned at the beginning of this section be fulfilled. Let sc, pEN be sufficiently large and T* > 0 be sufficiently small. Then the Cauchy problem (4.8) has a solution u with U E C ([0, T*], HSc) n C 1 ([0, T*], H sc- 1 ).

4.3. Reduction of a General Quasi-linear Equation to a Quasi-linear Equation with Special Right-Hand Side

In this subsection we reflect upon a general quasi-linear weakly hyperbolic Cauchy problem and find a solution using the technique of the previous subsection. Namely,

we will transform the Cauchy problem

D';'u+ j + I a I ='Tn ,j < 'Tn

= f(x, t, {D~Ck,(3(X, t)D~u}),

u(x,O) = 'Po(x), . .. ,D,;,-lU(X, 0) = 'Pm-1 (x)

into another Cauchy problem

D';'v+ j+lal=rn,j<m

= f p' (x, t, {D~Ck,(3(X, t)D~v}),

v(x,O) = ... = D,;,-lV(X, 0) = 0,

191

(4.16)

( 4.17)

whose right-hand side f p ' fulfills the requirements (4.9) and (4.10) with p = p(p'). It will be shown that these two Cauchy problems are equivalent in the sense that functions U1, U2, ... , up' exist with

u = U1 + U2 + ... + up' + v.

The functions U1, U2, ... , up' are solutions of ODEs in t with parameter x. If pis large enough, then Theorem 4.2.6 guarantees the existence of a solution to (4.17). This idea has been used in [17] and [29].

The example of Qi [27] shows that a loss of Sobolev regularity (in comparison to the data) must be expected. This corresponds to the fact that the smoothness of the Uj decreases by m, as j increases by 1.

We list the assumptions. The functions Ck,{3 are assumed to satisfy (4.11) together with (3.22)-(3.25).

The functions aj.a and the right-hand side f are defined in the set KG; see (3.26). Finally, we suppose (3.4), (3.27), (3.28), (3.29), Condition 3.2 and Condition 4.1.

Theorem 4.3.1 (Existence). If Se E N is large enough, then some number T* E

(0, T] and some I > 0 (independent of se) exist with the property that there is a solution u of (4.16) with

Proof. We define

U E C ([0, T*], HSc-,) n C1 ([0, T]*, H s.-,-l) , U = {(D)rn-i(()..a)m-iD~-lu) : i = 1, ... , m}.

{I :1(3I=Oorl>i,

Cl,i,{3 = 0 : 1(31 > 0 and l = i

192 M. Dreher

and consider the system of ODEs in t with parameter x,

Dr;'Ul (x, t) = f(x, t, {D~Ck,(3(X, t)D~Ul (x, t)Cl,I,(3}),

Ul (x, 0) = 'Po(x), ... ,Dr;'-lu1 (x, 0) = 'Pm-l (x),

Dr;'Ul(X, t) = gl(X, t, Ul(X, t), ... , Dr;'-IUl (X, t)) l

= f (x, t, {D~Ck,(3(X, t)D~ L cl,i,(3Ui(X, t) }) i=1

l-1 - f (x, t, {D~Ck,(3(X, t)D~ L cl-l,i,(3Ui(X, t)} )

i=1 l-1

L aj,a,l(X, t) L A(t)laID~ Di (0"(x)la1ui(x, t)) j+lal=m,j<m i=1

l-2 + aj,a,l-1 (x, t) L A(t)la l D~ Di (0"(x)la1ui(x, t)),

j+lal=m,j<m i=1

Ul(X,O) = ... = Dr;'-IUl (X, 0) = 0, l = 2, ... ,po

The following abbreviations have been used here:

l

aj,a,l(X, t) = aj,a (x, t, {D~Ck,(3(X, t)D~ L Ui(X, t)cl,i,(3 }), i=1

o

L=O. i=1

These equations can be solved step by step, and they possess solutions Ul E C m ([0, Ttl, H sc- rn(l-I)) , Be - ml > n/2 + 1. For a proof, see Theorem 3.2.2 with 0" == O.

The functions Ul have a special asymptotical behavior for t ---7 O. If A satisfies the second set of assumptions of Condition 4.1, we fix some positive number c with 2c < m - (m - l)C)... Otherwise, c = O. Inductively, we prove

m-l l-1 '"' iiDtulC t)ii ::; C6 (t 1/2 A(t)C) ~ Hsc-rn(l-l) j=O

with Be - ml > n/2 + 1. This is obvious for l = 1. To prove a corresponding estimate for Ul+l, we consider Ilgl+l(t)IIH s c-=I. Hadamard's formula yields

1+1 l

f( x, t, {D~Ck,(3D~ LCl+1,i,(3Ui}) - f( x, t, {D~Ck,(3D~ LCl,i,(3Ui}) i=1 i=1

= L d1lk (x, t)ck,oD~ul+l + L d2lk(3(X, t)D~Ck,(3D~Ul. k 1(3I>o,k

The Hsc-ml_norms of the functions d llk , d2lk(3 are bounded, by the induction assumption and Ul+l E cm ([0, Tl+1], Hsc- ml ). The function A(t)m-2c A(t)l-m is

monotonically increasing due to the choice of s, see Condition 4.1. This implies

Ilck,,B(·,t)IIHs :::; CsA(t)2c, S E JR+, 1,61 > O.

The other contributions to gl+1 can be discussed similarly. We conclude that Ul+l is a solution of

k<m

(4.18)

Utilizing a standard technique one shows (restricting the time interval) m-l t

2..: IIDiul+l(·,t)IIHsc_=1 :::; Cllllhl+l(.,T)IIHsc-=1 dT:::; (t 1/ 2A(t)"'Y )=0

This is the desired estimate. Summing up the differential equations for Ul, ... , Ul we deduce that

I

D"('(UI + ... + Ul) = f(x, t, {D~Ck,j3D~ 2..:S1,i,j3Ui}) i=1

[-1

'" a I'" AlalDOI Dj(aIOllu·) ~ ],CY., ~ x t 'l ,

j+IOII=m,j<m i=1

Dt (Ul + ... + Ul)(X, 0) = tpj(X), j = 0, ... , m ~ 1.

We define I

aj,a,I(X, t, {D~Ck,,BD~v}) = aj,a (X, t, {D~Ck,,BD~ (2..: Ui + v) }), i=1

I I

= f (X, t, { D~Ck,,BD~ (2..: Ui + v) }) ~ f (X, t, {D~Ck,,BD~ 2..: S[,i,,BUi}) i=1 i=1

I [

2..: aj,OI (X, t, {D~Ck.,BD~ (2..: ui + v)} )AlaID~Di (a lOiI LUi) j+lal=m,j<m i=1 i=1

I I-I + 2..: aj,a (x, t, { D~ ck,,BD~ 2..: S[,i,,BUi} ) Alai D~ Di (a lOiI LUi)'

j+IOII=m,j<m i=1 i=1

If a function v has homogeneous initial data and satisfies

D"('v+ 2..: aj,a,I(X, t, {D~Ck,,BD~v} )AlaID~D{ (aIOllv) = fleX, t, {D~Ck,j3D~v}) j+lal=m,j<m

(which is (4.17) with pi = I), then the function U = L~=1 Ui + v solves (4.16).

194 M. Dreher

It remains to verify the conditions (4.9) and (4.10) for this function fz: We fix 5 > ° and restrict all the intervals [O,l1J in such a way that

liD! t Ui(., t) - <pj(-) II'F'-~' ,,8, 0" t ,,1l, 0" j " m- 1.

The condition (4.10) is obviously satisfied for t ::; 11 and the constant C j only depends on 5, but not on l. In a similar way as in the proof of (4.18) one shows

which proves (4.9). Theorem 4.2.6 gives the existence of a solution v to (4.17) which satisfies

(D)k(()..a)kv) E C ([O,T*],Hsc- m1 ), ifp and Se - ml are large enough. 0

Remark 4.3.2. Since the time degeneracy occurs only for t = 0, the blow-up criterion of Proposition 3.2.8 is still valid; we only have to take into account the number of lost derivatives.

5. Domains of Dependence

We will construct so-called domains of dependence. It turns out that our definition generalizes the definition of [2J from the strictly hyperbolic case to the weakly hyperbolic case. These domains can be exhausted with hypersurfaces, and the Cauchy problem is weakly hyperbolic in the normal direction at each point of each hypersurface, see Definition 5.1.2. One example of such domains is a cone, whose slope does not exceed some critical value. This concept will be applied to prove some results of uniqueness, finite propagation speed and regularity:

Global uniqueness for linear equations: A solution to a linear Cauchy problem is unique in any domain of dependence, see Theorem 5.2.1.

Local uniqueness for quasi-linear equations: For every ball in the initial plane one can find a cone (with suitably small slope) over this ball with the property that a solution is unique in this cone, see Theorem 5.3.2.

Local existence for quasi-linear equations: For every rectangle in the initial plane one can find a rectangular parallelepipedon over the rectangle with the property that a Sobolev solution of the quasi-linear equation exists in this parallelepipedon, cf. Theorem 5.3.1. This solution exists in the whole domain of dependence if the equation is linear, cf. Corollary 5.3.4.

Coo regularity: We consider a quasi-linear Cauchy problem, whose coefficients, right-hand side, weight functions and initial data are Coo. Let us be given a Sobolev solution in some domain of dependence. Then this solution is Coo in this domain. cf. Theorem 5.4.2.

5.1. Definition of Domains of Dependence We come to the definition of a domain of dependence, see also [2].

We consider the Cauchy problem

D';'u + L aj,a(x, t)D~Di (0'(x)la 1u ) = f(x, t), (5.1) j+lal--::::m,j<m

u(x,O) = 'Po(x), ... ,D,;,-lU(X, 0) = 'Pm-l (x)

for (x, t) E no x [0, T] c IR~ x IR t ; no is an open and bounded domain with smooth boundary.

We suppose Condition 3.1 and

0' E cb'(no),

{ C; (no x [0, T]) a E

J,Di Cr(no x [0, T]) : j + 10'1 = m,

: j + 10'1 < m.

The principal part of the operator on the left is

j+IDiI=m,j<m

To this operator we assign the strictly hyperbolic operator

j+lal=m,j<m

(5.2)

(5.3)

The domain of dependence n over a bounded domain no c IRn has to satisfy the following conditions. At first,

n = n' n {(x, t) : t 2:: O}, n' E IRn +1 , n' open,

no = n' n {(x, t) : t = O}.

(5.4)

(5.5)

Next, the projections 7r : (x, t) f--+ x of the level sets nto = n' n {(x, t) : t = to} shall become "smaller" for increasing to,

7rn t1 E 7rnto 0:,,:: to < tJ :":: T.

The set n can be exhausted with hypersurfaces ST'

n = U S1' = U {(x, t) : g(x, t) = r}.

We suppose 9 E cb'(no x [0, T]) and

og > 0 . n at III H.

(5.6)

(5.7)

(5.8)

Furthermore, we assume that each hypersurface S1' intersects with the initial domain no and

( no n "'\::!"" 8") " (no n o,\::!", s,) 0" co < r; < r'. (5.9)

196 M. Dreher

Finally, we need a connection between the slope of the normal vector to Sr at the point (x, t) and the largest characteristic root of Pm,u at that point. Let Qm be the principal part of a hyperbolic in direction T operator. Then the largest characteristic root Amax(X, t; Qm) of Qm at the point (x, t) is defined by

The slope of the normal vector and the largest characteristic roots satisfy

( ) I \lxg(x, t) I Amax x, t; Pm,u () < 1.

gt x, t (5.10)

This condition can be interpreted in the way that the polynomial P m,u is weakly hyperbolic at the point (x, t) in the normal direction of Sr.

For technical reasons we assume the following condition:

The domain 00 has the J-extension property defined below. (5.11)

Definition 5.1.1. We say that a domain no has the J-extension property if for every small c > 0 there is an operator £0 : Cb(nO) ---4 Cb(IRn) such that

• (£ou)(x) = u(x) for x E 00, dist(x, ano) > c, • there is a Coo mapping IJ! : lRn -7 no n {x: dist(x, ano) > c} such that

I(£cu)(x) - u(lJ!(x))1 < c for all x E lRn.

Example. Every star-shaped domain has the J-extension property.

Definition 5.1.2. A set 0 is called a domain of dependence over no for the operator Pm,u if the conditions (5.4)-(5.11) are satisfied.

Example (Characteristic cone). The characteristic cone K(B) for the ball B B(x*, d) in the initial plane is defined by

with

K(B) = {(x, t) : Ix - x* I < d - A'rnax,ut, O<t<_d_} - A~ax,a

A'rnax,u = Ilallnoo(B) sup {Amax(X, t; Pm,l): (x, t) E B x [0, Tn. In this section we will also examine the quasi-linear Cauchy problem

D';'u + L aj,a(x, t, {D~a(x)Ij3ID~u} )D~Di (a(x)la 1u ) j+lal:Sm,j<m

= f(x, t, {D~a(x)Ij3ID~u}Ik+Ij3I:Sm_I)'

u(x,O) = 'Po(x), ... , D,;,-IU(X, 0) = 'Pm-I(X).

This equation will be written as pi~~u = f(u).

(5.12)

If u solves this Cauchy problem, then one can define a domain of dependence n(u), which itself depends on u, since the coefficients of the principal part depend

on u. For this Cauchy problem we will assume almost the same conditions as in Section 3, see (3.27), (3.28) and Remark 3.2.1:

aj,a E e 1([0, T], eSc(no x ]Rna)),

!.pj E H s c+m - 1-j(no),

f E e([o, T], esc (no x ]Rna)).

And we modify Condition 3.2 in an obvious way:

Condition 5.1. The roots Tj(X,t,v,~) of

j+lal=m,j<m

n Se> 2' + 1, (5.13)

(5.14)

(5.15)

are real and distinct, h(x,t,v,~) - Ti(x,t,v,~)1 ~ cl~l, c > 0, i =f. j, for all (x,t,v,~) E no x [O,T] x ]Rna X ]Rn.

5.2. Uniqueness for Linear Equations

There is another way to define domains of dependence. A set n c ]R+ x]Rn is called domain of dependence over no c ]Rn for some hyperbolic operator if the vanishing of data in no and the vanishing of the right-hand side in n imply the vanishing of the solution to the Cauchy problem in n. The following theorem states that the two definitions are in concordance.

Theorem 5.2.1 (Uniqueness). We suppose (5.2), (5.3) and Condition 3.1. Let n c no x [0, T] be a domain of dependence over no for the operator Pm,a. Assume !.po == ... == !.pm-1 == 0 in no and f == 0 in n. Then u == 0 in n for every solution u of (5.1) with

Proof. Let us sketch the proof. The solution is defined in n, the coefficients aj,a are defined in no x [0, T]. In a first step we extend u and aj,a to the domain no x (-00,0]; the function u vanishes there. This gives a new Cauchy problem with solution u and vanishing initial values. In a second step we transform the variables. The domain n is mapped to some domain n = {(y, r) : y E no, g(y, 0) :::; r < r*}, and v, the image of u, has compact support with respect to the spatial variable y if the time variable is frozen. We extend the transformed coefficients aj,a and the weight (j to ]Rn x [0, T], apply an energy estimate, and obtain v == O. This will prove the theorem.

The coefficients aj,a satisfy Condition 3.1, (5.3) and are defined in no x [0, T]. For 0 < t :::; c we set aj,a(x, -t) = 2aj,a(x, 0) -aj,a(x, t). Then we change aj,a(x, t) appropriately for -c :::; t :::; -c/2, set aj,a(x, t) = aj,a(x, -c) for -00 < t < -c, and conclude that aj,a E el(no x (-00, T]) for j + lal = m and aj,a E e~(no x (-00, T]) for j + 1001 < m. Condition 3.1 is true in this domain if c is small.

The same method can be used to extend the derivative gt(x, t) ofthe function g(x,t) to no x (-oo,T]. The result is 9 E eOO(Oo x (-oo,T]), gt E e;:(no x

198 M. Dreher

(-00, TJ) gt(x, t) = gt(x, -E), t ::; -E. We see that the function g takes arbitrarily small values and conclude that

n u Z = 0 U (00 x (-00,0]) = -oo<r<r'"

We extend the solution u by u(x, t) = 0, (x, t) E Z, the letter Z stands for "zero". Then u solves (5.1) in Z with homogeneous data for t = -1, f == ° in Z, and

D;n-I"'ID~alalu E C(O U Z), lal::; m. We apply a Holmgren type transform to change the variables,

y = x, r = g(x, t), \7x = '\ly + ('\lxg)ar , at = (atg)ar, v(y, r) = u(x, t).

The dual variables fulfill

~ = 'f7 + (\7xg)e = 'f7 + c(y, r)e, T = gte = co(y, r)e. The domain 0 is mapped to the set

n = {(y,r): y E OO,g(y,O)::; r < r*},

and Z is the image of Z:

Z = {(y,r): y E 0 0 ,-00 < r::; g(y,O)}.

It is easy to verify that D~-laID~a(y)I"'lv(y,r) E C(O U Z) for lal ::; m. The function v is a solution of

D~v+ L aj,,,,(y,r)D~Dt(a(y)lalv)=O, j+lal:5m,j<m

v(y,O) = '" = D~-1V(Y, 0) = 0, (y, r) E 00 x [0, r*),

aj,a(y, r) are the transformed coefficients. We have to check whether this Cauchy problem is weakly hyperbolic and satisfies the Levi conditions. We remember the definition of (strict) hyperbolicity, see [23]:

Definition 5.2.2. A differential operator Q(z, Dz ) = E1al=m a",(z)D'; is called hyperbolic at the point Zo in the direction N =I- ° if

• Q(zo, N) =I- 0, • Q(zo,TN + (") = ° has only real roots T for every (" =I- 0.

A differential operator Q(z, Dz ) is called strictly hyperbolic at (zo, N) if it is hyperbolic at (zo, N) and if Q(zo, TN + (") = ° has m real and distinct roots T for every (" 1- N, (" =I- 0.

By definition, the operator Pm,a(X, t, D x, D t ) is hyperbolic in the direction N = (1,0, ... ,0) E ]RHn and the operator Pm ,1 (x, t, D x , D t ) is strictly hyperbolic in this direction N. The symbol of the principal part Pm,a of the transformed operator is

To be able to apply the results of Section 3, we have to verify that the operator F m,a is hyperbolic in the direction IV and that the operator Fm .1 is strictly hyperbolic in the direction IV. Here IV = (1,0, ... ,0) E lR~ x lR~ is the normal direction of the hypersurfaces r = const.

It can be seen that

- m ( C co) Pm,a(y,r,O,l)=Pm.a(x,t,c,co)=l cl Pm,a x,t,~,~ .

From (5.10) and the definition of Amax(X, t; Pm,a) we then conclude that

Fm,a(Y, r, IV) = Fm,a(Y, r, 0,1) #- 0.

In the next step we show that the equation Fm.a(y, r, TJ, (2) = ° has m real roots (21, ... , (2m for every TJ #- 0. It holds that

Pm,a(Y, r, TJ, (2) = Pm,a(X, t, TJ + C(2, Co(2) = c~ Pm,l (x, t, ~ (TJ + C(2), (2) .

If O'(x) = 0, then the only roots are (21 = ... = (2m = 0. If O'(x) #- 0, then

- m ( CO) Pm,a(y,r,TJ,(2)=0' Pm,l x,t,TJ+C(2,-;;(2 .

The polynomial Pm ,I (x, t, TJ, Co (2/0') is strictly hyperbolic in the direction N. From (5.10) we gather

I O'(x)c(x, t) I 1 co(x, t) < Amax(X, t; Pm.d'

and Proposition 6.1 reveals the strict hyperbolicity of the polynomial Pm .1 (x, t, TJ+ C(2, (co/O')(2) in the direction N +O'c/co. We get that Fm,a(Y, r, TJ, (2) = ° has m real roots, which are distinct if O'(Y) #- 0.

Exploiting Proposition 6.1 once more, and making use of

Pm,l (y, r, TJ, (2) = Pm,l (x, t, TJ + O'C(2, Co (2 ),

it is easy to verify that Pm,l is strictly hyperbolic.

We come back to the function v. Our aim is to show that v == ° in n, by the aid of Remark 3.1.5 and Gronwall's lemma. But first the coefficients aj,Cl' and the weight 0' have to be extended to the whole lR~ x [0, T], lR~, respectively, since they are only defined on 0 0 x (- 00, T] and 0 0 ,

We fix some arbitrary ° < ro < r* and will show v(y, r) = ° in no x [0, ro]. The function v(y,r) vanishes for y close to aoo; hence we are allowed to change the coefficients aj,Cl' and the weight 0' there. We extend these functions by

where £E is the extension operator of Definition 5.1.1. Condition 3.1 is satisfied everywhere in lR~ x [0, ro] if E is small enough.

200 M. Dreher

We get coefficients a),a E cl(JRn x [0, roJ) for j + 10:1 = m and a),a E C~(JRn x [0, roJ) for j + 10:1 < m. The function v can be extended by zero, and solves

Dr;'v + L aj,a(Y, r)D~ D~(a(y)lalv) = 0, (y, r) E JRn x [0, ro], j+a=m,j<m

v(y,O) = ... = D:;,-lv(y, 0) = 0, Y E JRn.

Remark 3.l.5 and Gronwall's lemma yield v(y,r) = ° for y E JRn, ° :::; r :::; rD. Since ro < r* can be chosen arbitrarily, we have v = ° in s1, hence u = ° in O. The theorem is proved. D

5.3. Existence and Uniqueness for Solutions of Quasi-linear Equations

Theorem 5.3.1 (Local existence). We suppose (5.13)-(5.15) and Condition 5.l. Let Qo = II~l [ai, bi ] be a rectangular parallelepiped (RP for short), Qo IS 0 0 . Then a constant ° < To :::; T and a solution u of (5.12) exist with

Di am-1-ju E C([O, To], Hsc+m-1-j (Qo)), j = 0, ... , m - l.

Proof. We take a cut-off function cp(x) which is supported in a neighborhood of Qo and is identical to 1 on Qo. Then we replace the functions a(x), f(x, t, {Vk,,a}), cPj(x) by cp(x)a(x), cp(x)f(x, t, {Vk,,a}) and cp(x)cpj(x).

Let Q be an RP with supp cp <s Q. Using suitable "reflections" (see the proof of Theorem 5.2.1), we can extend the coefficients aj,a from Q x [0, T] x JRna to the larger set Q' x [0, T] x JRna, Q' being an RP with twice the edge lengths of Q which can be regarded as a torus. We get a Cauchy problem on Q'. Theorem 3.2.2 shows that a solution u exists with the desired smoothness on the torus Q'. This function is a solution on Qo x [0, To], since cp == 1 on Qo. D

So far, nothing has been said about the uniqueness of this solution u. This question is taken up in the next theorem.

Theorem 5.3.2 (Local uniqueness). Let the conditions of Theorem 5.3.1 be satisfied. Let Bo <s 0 0 be a sufficiently small ball. Then a number To > ° and a cone 0 over Bo exist such that a uniquely determined solution u with

Diam-1-ju E C([O, To], Hsc+m-1-j(nt )), j = 0, ... , m - 1,

exists in O. The notation v E C([O,T],HS(Ot)) means that

• SUPtE[o,T]llv(-, t)IIHs(Sltl < 00,

• for all T', n~ with n~ x [O,T'] IS 0 it holds that v E C([O,T'],HS(O')).

Here HS(Ot) denotes the usual Sobolev space on the bounded domain Ot.

Proof. Theorem 5.3.1 shows that a small To and a solution u exist with

D{am-1-ju E C([O,To],Hsc+m-l-j(Bo)), j = 0, ... ,m -1,

P:n~~u = feu) in Bo x [0, To], D{u(-, t) = CPj(-) in Bo.

We define the cone 0 = K(Bo) over Bo as given in the Example following Definition 5.1.2 with

Pm,l (x, t, T,';) = Tm + j+lal=m,j<m

Then it follows that

D~am-l-ju E e([O, To], Hsc+m-1-j(K(Bo)t)), j = 0, ... , m - 1.

Let U' be another solution in 0, i.e.,

P;;:*}u* = feu') in Bo x [0, To], D~u* = rpj in B o.

The difference u - u* then solves

p(U) (u - u*) = m,O"

k+I,6I:":m-l

with gk,,6 E C(O), and Theorem 5.2.1 applied to the homogeneous Cauchy problem reveals u - u * == ° in O. D

Remark 5.3.3. The techniques of Section 5.4 enable us to show that solutions to quasi-linear Cauchy problems are unique not only locally, but also globally in domains of dependence. Since the proof is quite similar to that of Theorem 5.4.2, it is dropped.

If the equation is linear, we even have global existence:

Corollary 5.3.4 (Global existence). Let us consider the Cauchy problem (5.1). We suppose Condition 3.1 and

a E CCXl(O~), O~ ~ 0 0 ,

a E {C1([0,T]'HSC(00)) :j+lal=m, n J,a C([O, T], HSc(Oo)) : j + lal < m, Sc > "2 + 1,

rpj E Hsc+m-l-j(o~), f E C([O, T], HSc(O~)).

Let 0 be a domain of dependence for the operator P m,O" over the domain 0 0 . Then a unique solution u exists with

Proof. We pick a cut-off function rp(x) that is identical to 1 in a neighborhood of 0 0 and supported in 0 0, Then a cube Q with supp rp <s Q is chosen. We replace the functions a(x), fex, t), rpj(x) by rp(x)a(x), rp(x)f(x, t) and rp(x)rpj(x). The coefficients aj,ae-, t) are extended from 0 0 to Q \ 0 0 by the aid of the procedure from the end of the proof of Theorem 5.2.1. We acquire a linear Cauchy problem on a torus. Remark 3.2.4 gives us a solution defined in Q x [0, T] which has the desired smoothness. Theorem 5.2.1 shows that this solution is the only solution in O. D

202 M. Dreher

5.4. C= regularity First, we show a local regularity result in C=.

Lemma 5.4.1 (Local C=-regularity). Let u be a function defined in n(u) which is a domain of dependence over no for the operator PSn~~. Assume that u with

Df(J"Tn-l-ju E e([O,T],Hsc+Tn-l-j(nt )), j = 0, . .. ,m-1,

is a solution of (5.12). The coefficients aj,ex and the right-hand side are supposed to be C= with respect to all their arguments. Finally, we assume that

u E COO(n(u) n {(x, t) : t ~ to})

for some to ~ 0. Let B(xo, d) E nta be a ball. Then a number tl = tl (xo, to, d) > to exists with

Proof. We apply the procedure given in the proof of Theorem 5.3.1 to extend the Cauchy problem from B(xo, d) x [to, T] to the set Q' x [to, T], with Q' being a torus. We get a quasi-linear weakly hyperbolic Cauchy problem on a torus with C= coefficients and Coo data. Theorem 3.2.3 gives us a local Coo solution, and Theorem 5.3.2 shows that this solution is unique in some domain of dependence which contains some cylindrical set B(xo, d) x [to, tl] with small tl -to. This implies u E C=(B(xo, d) x [to, tIl). D

This lemma is the key tool to prove the following theorem:

Theorem 5.4.2 (Global Coo regularity). Let u be a function defined in n(u) which is a domain of dependence over no for the operator PSn~~. We suppose that u with

Df(J"Tn-l-ju E e([O, T], HSc+Tn-l-j (n~u))), j = 0, ... , m - 1,

is a solution of (5.12). Moreover, we assume aj,ex, f E Ch(nO x [0, T] x ]Rna),

<pj E Ch(nO). Then u E COO(n(u)).

Proof. If B(xo, d) E no is a ball, Lemma 5.4.1 gives us a numberh(xo,d) with

u E COO(B(xo,d) x [O,tl(xo,d)]).

This implies that a smooth function h = hex) > ° and a set M o exist with

M o = {(x, t) : x E no, ° ~ t < hex)}, u E C=(M).

The domain n(u) can be exhausted with hypersurfaces Sr. We transform the variables in the same way as in the proof of Theorem 5.2.1. This results in a quasi-linear weakly hyperbolic initial value problem for the function v,

P:::,~v = j(y,r,{D~(J"(y)Ij3ID~v}), y E no, g(y,O) ~ r < r*,

v(y, g(y, 0)) = 1/;o(y), ... , Dr;-lv(y, g(y, 0)) = 1/;Tn-l (y).

Denoting the images of Mo and n(u) under the transformation of variables by Mo and n(u), we know that v is Coo on the sets

Mo, M(ro) = n(u) n {(y, r): y E no, r ::; ro},

for some ro > 0. The proof will be complete if we verify two properties:

A: If v E COO(M(r», then v E C=(M(r'» for some r' > r. B: If v E COO(M(r» for all r with ro ::; r < rl, then v E CCXJ(M(rl»'

For A: The set 8M(ro) = n(u) n {(y, r): y E flo} can be covered by Mo and finitely many balls. For each ball, we apply Lemma 5.4.l.

For B: We cover the set 8M(rd by Mo and finitely many balls B(yo, d). For each such ball, we consider a Cauchy problem with initial data prescribed at B (Yo, d) x {r = rl - C:}, E > ° small. This Cauchy problem has the solution v, which is locally Coo. Proposition 3.2.8 and Remark 3.2.10 tell us that the life-span of this solution cannot approach zero for E --+ 0. This shows that v is Coo on B(yo, d) x {r ::; rd. 0

6. Appendix

Here we provide some auxiliary results.

Proposition 6.1 ([3], [4]). Let the homogeneous differential operator P(x, D) of order m be strictly hyperbolic at the point Xo in the direction N, INI = 1. By Amax we denote the largest absolute value of the characteristic roots, i. e.,

Amax = SUp{ITI : P(xo,TN + 0 = 0, lei = l,~ 1- N}.

Then P is strictly hyperbolic at Xo in any direction N +e with N 1- e, lei-I> Arnax.

The following lemma is a generalization of Gronwall's lemma to differential inequalities with a singular coefficient.

Lemma 6.2 ([24]). Let yet) E C([O, T]) n CI((O, T» be a solution of the differential inequality

y'(t) ::; K(t)y(t) + f(t), 0 < t < T, where the functions K(t) and f(t) belong to C(O, T). We assume for every t E (0, T) and every J E (0, t) that

i 5 K(T) dT = 00, jT K(T) dT < 00,

lim jt exp (it K ( T) dT) f (s) ds exists, 8---+0 8 s

lim y(r5) exp (it K(T) dT) = O. 6---+0 s

Then it holds that

yet) ::; it exp (it K(T) dT) f(s) ds.

204 M. Dreher

References

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[2] S. Alinhac and G. Metivier. Propagation de l'analyticite des solutions de systemes hyperboliques non-lineaires. Inv. math., 75:189-204, 1984.

[3] M.F. Atiyah, R. Bott, and L. Garding. Lacunas for hyperbolic differential operators with constant coefficients. I. Acta Math., 124:109-189, 1970.

[4] M.F. Atiyah, R. Bott, and L. Garding. Lacunas for hyperbolic differential operators with constant coefficients. II. Acta Math., 131:145-206, 1973.

[5] R. Coifman and Y. Meyer. Commutateurs d'integrales singulieres et operateurs multilineaires. Ann. Inst. Fourier (Grenoble), 28(3):177-202, 1978.

[6] F. Colombini, E. Jannelli, and S. Spagnolo. Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time. Ann. Scuola Norm. Sup. Pisa IV, 10:291-312, 1983.

[7] F. Colombini and S. Spagnolo. An example of a weakly hyperbolic Cauchy problem not well posed in C=. Acta Math., 148:243-253, 1982.

[8] P. D'Ancona. Well-posedness in C= for a weakly hyperbolic second order equation. Rend. Sem. Mat. Univ. Padova, 91:65-83, 1994.

[9] P.A. Dionne. Sur les problemes hyperboliques bien poses. J. Analyse Math., 10:1-90, 1962.

[10] M. Dreher and M. Reissig. About the C=-well-posedness of fully nonlinear weakly hyperbolic equations of second order with spatial degeneracy. Adv. Diff. Eq., 2(6):1029-1058, 1997.

[11] M. Dreher and M. Reissig. Local solutions of fully nonlinear weakly hyperbolic differential equations in Sobolev spaces. Hokk. Math. J., 27(2):337-381, 1998.

[12] M. Dreher and M. Reissig. Propagation of mild singularities for semilinear weakly hyperbolic equations. J. Analyse Math., 82:233-266, 2000.

[13] M. Dreher and I. Witt. Edge Sobolev spaces and weakly hyperbolic equations. Ann. mat. pura et appl., 180:451-482, 2002.

[14] L. Garding. Cauchy's Problem for Hyperbolic Equations. University Chicago, 1957.

[15] L. Hormander. The Analysis of Linear Partial Differential Operators. Springer, 1985.

[16] V. Ivrii and V. Petkov. Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well posed. Russian Math. Surveys, 29(5):1-70, 1974.

[17] K. Kajitani and K. Yagdjian. Quasi-linear hyperbolic operators with the characteristics of variable multiplicity. Tsukuba J. Math., 22(1):49-85, 1998.

[18] T. Kato and G. Ponce. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. M., 41:891-907, 1988.

[19] P.D. Lax. Asymptotic solutions of oscillatory initial value problems. Duke Math. J., 24(4):627-646, 1957.

[20] J. Leray. Hyperbolic Differential Equations. Inst. Adv. Study, Princeton, 1954.

[21] R. Manfrin. Analytic regularity for a class of semi-linear weakly hyperbolic equations of second order. NoDEA, 3(3):371-394, 1996.


[22] S. Mizohata. Some remarks on the Cauchy problem. J. Math. Kyoto Univ., 1(1):109-127, 1961.

[23] S. Mizohata. The Theory of Partial Differential Equations. Cambridge University Press, 1973.

[24] A. Nersesyan. On a Cauchy problem for degenerate hyperbolic equations of second order (in Russian). Dokl. Akad. Nauk SSSR, 166(6):1288-1291, 1966.

[25] O. Oleinik. On the Cauchy problem for weakly hyperbolic equations. Comm. Pure Appl. M., 23:569-586, 1970.

[26] I.G. Petrovskij. On the Cauchy problem for systems of linear partial differential equations. Bull. Univ. Mosk. Ser. Int. Mat. Mekh., 1(7):1-74, 1938.

[27] M.-Y. Qi. On the Cauchy problem for a class of hyperbolic equations with initial data on the parabolic degenerating line. Acta Math. Sinica, 8:521-529, 1958.

[28] R. Racke. Lectures on Nonlinear Evolution Equations. Initial Value Problems. Vieweg Verlag, Braunschweig et a!., 1992.

[29] M. Reissig. Weakly hyperbolic equations with time degeneracy in Sobolev spaces. Abstract Appl. Anal., 2(3,4):239-256, 1997.

[30] M. Reissig and K. Yagdjian. Weakly hyperbolic equations with fast oscillating coefficients. Osaka J. Math., 36(2):437-464, 1999.

[31] K. Taniguchi and Y. Tozaki. A hyperbolic equation with double characteristics which has a solution with branching singularities. Math. Japonica, 25(3):279-300, 1980.

[32] S. Tarama. On the second order hyperbolic equations degenerating in the infinite order. - example -. Math. Japonica, 42(3):523-533, 1995.

[33] M.E. Taylor. Pseudodifferential Operators and Nonlinear PDE. Birkhiiuser, Boston, 1991.

[34] M.E. Taylor. Partial Differential Equations III. Nonlinear Equations. Springer, 1996.

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Michael Dreher Faculty of Mathematics and Computer Science Freiberg University of Mining and Technology Agricola-StraBe 1 D-09596 Freiberg, Germany e-mail: [email protected]


An Approach to a Version of the S (M, g)pseudo-differential Calculus on Manifolds

F. Baldus

Abstract. For appropriate triples (M, g, M), where M is an (in general noncompact) manifold, g is a metric on T* M, and M is a weight function on T* M, we develop a pseudo-differential calculus on M which is based on the S(M, g)-calculus of L. Hormander [27] in local models. In order to do so, we generalize the concept of E. Schrohe [41] of so-called SG-compatible manifolds. In the final section we give an outlook onto topological properties of the algebras of pseudo-differential operators. We state the existence of "order reducing operators" and that the algebra of operators of order zero is a submultiplicative \{f*-algebra in the sense of B. Gramsch [18] in L (L 2 (M)).

1. Introduction

When pseudo-differential operators were introduced in 1965 (cf. J.J. Kohn and L. Nirenberg [29], A. Unterberger and J. Bokobza [51], [52], L. Hormander [24], [25]), there arose naturally a desire to prove invariance under changes of variables and thus, to define pseudo-differential operators on manifolds. Having developed a pseudo-differential calculus on IRn or on a manifold, properties of the algebra WO of pseudo-differential operators of order zero as a topological algebra came to be of interest. With the help of "order reducing operators", many questions of the full filtered algebra of pseudo-differential operators reduce to this algebra woo In a series of articles (cf. R. Beals [4], [5], B. Gramsch [19], H.O. Cordes [12], J. Ueberberg [48], E. Schrohe [43], [44]. B. Gramsch, J. Ueberberg, and K. Wagner [22], J.-M. Bony and J.-Y. Chemin [9], and [1], [2]), spectral invariance of WO in the Banach algebra B of bounded operators on L2 or, more generally, on LP- and Orlicz-spaces and on the corresponding (possibly weighted) Sobolev spaces was proved in many situations, i.e.,

2000 Mathematics Subject Classification. AMS 1991 Subject Classification: Primary 35S05; 58G15 Secondary 47A60; 47A10; 47D30; 46F05. Key words and phrases. pseudo-differential operators on manifolds; Weyl-Hormander calculus; SG-calculus; spectral invariance; submultiplicativity.

208 F. Baldus

holds for the groups of invertible elements. As a matter of fact, in many cases \[10

is a W- or w* -algebra in B in the sense of B. Gramsch [18], [19], i.e., a continuously embedded Frechet subalgebra of B with the same unit element, such that for the groups of invertible elements the above equation holds, and which is symmetric in the "*-case". In addition to implying an open group of invertible elements and continuous inversion, the properties of a \[I-algebra guarantee stability with respect to the holomorphic functional calculus of L. Waelbroeck [53). In addition to [19], a rather detailed discussion of the concept of W-algebras can be found in [31). In connection with non-commutative cohomology and Oka's principle, submultiplicativity of the algebra of operators of order zero is of importance (cf. B. Gramsch [20) and B. Gramsch, W. Kaballo [21)).

Originally, the calculus of pseudo-differential operators on manifolds was based on symbols which satisfy local estimates, i.e., estimates which are only uniform for x in a compact set (cf. [51], [52], [24], [25)). On ]Rn, symbols satisfying global estimates were considered (cf. L. Hormander [25], M.A. Shubin [45], C. Parenti [36], H. Kumano-go and K. Taniguchi [30], R. Beals and C. Fefferman [6], R. Beals [3], H.O. Cordes [11], [13], B. Helffer [23)). In connection with the Weyl calculus, L. Hormander [27) (cf. also [28, Section 18.5)) presented a vast extension of these classes: To a metric g on T*]Rn and a weight function M on T*]Rn

he associated the symbol class S (M, g) and developed the corresponding pseudodifferential calculus. For essential contributions to the S(M, g)-calculus we wish also to refer to A. Unterberger [49], [50], R. Beals [5], N. Dencker [15], J.-M. Bony and N. Lerner [10], J.-M. Bony and J.-Y. Chemin [9], J.-M. Bony [8).These data g and M also make sense on the cotangent bundle of a manifold but, for a generalization of the S(M,g)-calculus, we have to make assumptions concerning both the manifold, expressed by the existence of a suitable atlas, and the data M and g. In [27) L. Hormander investigated already the behavior of operators associated to metrics not too far from the standard p, b-metric under certain canonical transformations, introduced Fourier integral operators, and proved a generalization of Egorov's theorem.

In the present paper we generalize the concept of E. Schrohe (cf. [40), [42], [44)) of so-called SG-compatible manifolds which are appropriate in order to transfer the SG-calculus of M.A. Shubin [45], C. Parenti [36], and H.O. Cordes [11). In so doing, we develop an S(M, g)-based pseudo-differential calculus on manifolds M, at least for appropriate combinations of (M, g, M). We introduce conditions expressed by the existence of appropriate local models and compatible changes of coordinates (cf. Section 2). Associated to M and g, we define - up to negligible terms - pseudo-differential operators on M as operators CO'(M) ----> D'(M) possessing representations as pseudo-differential operators in local coordinates (in the corresponding classes). For the study of transformations under diffeomorphisms we apply the proof of Kuranishi (cf. [17, Appendix of Chapter I], [35, p. 155], [26, Proposition 2.1.3)). Concerning the negligible terms we assume a behavior based on characterizations with the help of mapping properties of iterated commutators with distinguished operators in a scale of Sobolev spaces corresponding to g as in

S(M, g)-calculus on Manifolds 209

[1, Theorem 3.6.6]. More precisely, we define the negligible class as the set of all operators such that these iterated commutators have order -00 in this scale of Sobolev spaces. Moreover, for an appropriate choice of cut-off functions, we have representations of the negligible terms as integral operators with rapidly decreasing kernels.

H. Widom [54] constructed a complete symbolic calculus associated to pseudodifferential operators on compact manifolds, defined with the help of a connection. Yu. Safarov [39] proceeded in this direction to define p, J-type classes of pseudodifferential operators without the restriction 1 - P ::; J which is known to be necessary in order to define pseudo-differential operators through their representations in local coordinates. For different approaches to pseudo-differential operators on non-compact manifolds we wish also to refer to H.O. Cordes and S.H. Doong [14], R Beals and P. Greiner [7] and M.A. Shubin [47]. In [37], [38], B. A. Plamenevskij and V.N. Senichkin develop an approach to a pseudo-differential calculus on stratified manifolds and study irreducible representations of the C* -algebra generated by pseudo-differential operators of order zero. The stratified manifolds are locally represented by "models" located in IRn and "admissible diffeomorphisms" between the local representations. The SG-calculus is closely related to the scattering calculus of RB. Melrose (cf. [33, Section 4] and [34, Section 6.3]). Moreover, Y.V. Egorovand B.-W. Schulze apply SG-calculus to define a pseudo-differential calculus on the infinite stretched cone IR+ x X over a closed compact Coo-manifold X (cf. [16, Section 8.2]). For a groupoid approach to spectral questions on non-compact Riemannian manifolds we refer to R Lauter and V. Nistor [32].

Under more restrictive assumptions, spectral invariance and submultiplicativity can be proved. Common tools for these two topics are commutator methods for the characterization of pseudo-differential operators. In order to derive spectral invariance and submultiplicativity, the existence of "order reducing operators" is of essential importance. We state these results without proof in Section 8.

Instead of explaining the organization of this paper at this point, we refer to brief introductions at the beginning of each section.

This paper is part of the author's Ph. D. thesis [2] written under the supervision of Professor B. Gramsch at the University of Mainz, Germany.

2. Admissible coordinates

There is a natural way to generalize g and M and thus, S(M, g) to cotangent bundles over manifolds. But, on the other hand, it is well known that the S;;>classes are invariant under (local) diffeomorphisms - and thus can be considered on the cotangent bundles of manifolds - if and only if 0 ::; 1 - P ::; J ::; p ::; 1, J < 1. Therefore, for an extension of the S (M, g)-calculus, we have to make assumptions concerning both the manifold, expressed by the existence of a suitable atlas, and the data M and g. In [41] E. Schrohe transfers a calculus of M.A. Shubin [45], C. Parenti [36], and H.O. Cordes [11], which is based on global estimates, to

210 F. Baldus

so-called SG-compatible manifolds (cf. also [40], [42], [44]). The local charts are assumed to satisfy conditions which allow us to prove invariance of the pseudodifferential calculus with respect to the changes of coordinates. Here we generalize these conditions in order to treat classes of operators associated to more general calculi on IRn. For instance, we can transfer the general Cordes' classes corresponding to metrics

<~>28 2 Ild~112 9( t:).- Ildxll + 2 0 S; Pl,P2 S; 1,1- P2 S; <5 S; P2,<5 < 1

x,~ .- <X>2Pl 2 <~>2P2'

and classes corresponding to metrics

(; 28 lid 112 Ild~ll; 9(x~) :=<x,.,,> x 2 + (; 2 ' , <X,.,,> P o S; P S; 1,1 - P S; <5 S; p, <5 < 1

to SG-compatible manifolds (cf. 7). In contrast to most of the already existing calculi, our approach allows us also to patch different local models together, provided that we have compatible changes of coordinates (cf. 7.1). Moreover, we can admit anisotropic versions of the above examples. In order to treat them, we make the following definition:

Definition 2.1. Let r := (')'oJaEA be a family of positive definite quadratic forms on IRm.

1. We call a decomposition IRm = El EBJ.. ... EBJ.. El r -admissible if

• Vj E {I, ... ,l}, 3Cj > 0, Vtl, t2 E E j \ {O}, Va E A, 11~~~ti· ~~(~I}) S;

Cj .

• 3C> 0, Vtl EEl'···' Vtl E Ez, Va E A, 2:~=1 ,a (tj) S; C,a(tl + ... + tl).

2. We call r quasi-decomposable if there exists a r-admissible decomposition. 3. We call an orthonormal (with respect to the Euclidean metric) basis B =

{tl' ... , tm } r -admissible if there exists a r -admissible decomposition IRm =

El EBJ.. ... EBJ.. El and for each j E {I, ... ,m} a constant Vj E {I, ... ,l} such that tj E Evj •

Remark 2.2.

1. Let r := (')'a)aEA, r := ("Ya)aEA be two families of positive definite quadratic forms on IRm such that with a suitable constant C > 0 for all a E A the estimate C-l'a(·) S; '"Ya(-) S; C,a(-) holds. Then r is quasi-decomposable if and only if r is quasi-decomposable, and IRm = El EBJ.. ... EBJ.. El is a r -admissible decomposition if and only if it is a r -admissible decomposition.

2. Let r := (')'a)aEA be a family of positive definite quadratic forms on IRm and let IRm = El EBJ.. ... EBJ.. El be a decomposition such that for all j E {I, ... , l} and all a E A, E j is contained in an eigenspace of ,a. Then this decomposition is r-admissible.


3. Let r := baJaEA be a quasi-decomposable family of positive definite quadratic forms on IRrn and let IRm = E1 E8.L ... Ef).L Ez be a r-admissible decomposition. For j E {I, ... , I} let 0 E Ej with 110112 = 1. For ex E A, we define

1a(t1 + ... + tz) := I:~=1 '"'fa (0) Iltjll~ (tj E Ej,j = 1, ... , I). Then, with a suitable constant C > 0, for all a E A we have C- 1 1a :s; '"'fa :s; C la.

Lemma 2.3. Let r := ba)aEA be a quasi-decomposable family of positive definite quadratic forms on IRm. Then there exists a unique coarsest r -admissible decomposition.

Next we introduce canonical weight functions which play an important role in the following:

Definition 2.4. Let U be an open subset of IRn and 9 a metric on U x IRn.

1. For x E U and'; E IRn let hg(x,';) := (SUPTE IR2n \{O} ~r::;~g:O 1/2 where

cr (T) cr(W,T)2 9(x,fJ := SUPWEIR2n \{O} g(x,o(W)'

2. Let B be a basis of IRn. For x E U and'; E IRn we define

Hg,B(X,';) := ~~ vg(X,O)(t, O)g(x,O(O, t) .

Now we are prepared to define admissible local models and metrics on these models which are suitable in order to generalize the corresponding calculus on IRn to manifolds. Apart from some technical conditions concerning the metric, e.g., a generalization of the assumption 1 - P :s; 6' (cf. 2.6), we require that the diffeomorphisms have differentials which are matrix-valued symbols in the calculus and that they extend to larger domains which are chosen in such a way that the existence of admissible cut-off functions and partitions of unity can be proved (cf. 4.15). The conditions concerning the manifold M with atlas A depend on two metrics 9 and 9 on T* M and M, respectively, which determine the pseudodifferential calculus to be defined in Section 5. In the SG-calculus we have 9 = Rl (g), but, as the examples in Section 7 show, in some cases it is reasonable to consider 9 =I- R1(g). Consequently, we describe admissible (in order to develop a pseudo-differential calculus) structures by the data (M, A, 9, g) (cf. 4.8 and 4.10). Moreover, in 2.6 we introduce conditions on weight functions and diffeomorphisms.

Definition 2.5. 1. Let U, U be open subsets of IRn with U ~ U, and let 9 be a smooth metric

on U. We call (U, U, 9) an admissible model if the following conditions are satisfied:

• 3rg > 0, \Ix E U, \ly E IRn, 9x(x - y) :s; r~ ===} y E U . • 3Cg > 0, \Ix, y E U, 9x(x-y) :s; Cg 1 ====? Cg19x(-) :s; 9 y (-) :s; Cg9x(-) . • The family of forms (9X)XEU is quasi-decomposable.

2. Let U1 be an open subset of IRn, let (U2 , U2 , {h) be an admissible model, and let K, : U1 ---> U2 be a CCXl-diffeomorphism. K, is called (U2, U2, 92)-compatible if the following statements hold:

212 F. Baldus

• (U1, U1, 9I) is an admissible model, where U1 := K- 1(U2) and 91 := K*92' i.e., 91,x(t) = 92,I'«x) (K'(X)(t)).

• [U1 3 x f---7 dK(X)] E S (1, 91; £ (]Rn)) and1

[U2 3 Y f---7 d (K-1) (y)] E S (1, 92; £ (]Rn)).

• 3C> 0, Vx E U1, Vy E U2, IIK(X)112:S; C <x>, IIK- 1 (y)112:S; C<y>. • For i = 1,2 there exists a (9i,x)xEu;-admissible decomposition ]Rn =

E~i) Ef).L .. , Ef).L Eti) such that we have E?) = dK(X)-1 (Ey») for all

x E U1 and j E {I, ... , I}.

Definition 2.6.

1. We call a split H6rmander metric on ]R2n of type (8) if there exists a constant C> ° such that VX,f"t,T E ]Rn,

g(x,O)(t,O):S; Cg(x,~)(t,O) , g(x,O)(O,T) 2:: C-1g(x,~)(0,T)

and there exists a basis of ]Rn which is orthogonal with respect to g~~:~)

and g~!:~) for all (x,f,) E ]R2n, where gg:~)(t) := g(x,~)(t,O) and g~!:~)(T) :=

g(x,~)(O,T). 2. Let (U, U, 9) be an admissible model and let 9 be a smooth metric on U x ]Rn.

We call (U, U, 9, g) an admissible quadruple of type (p), if the following conditions are satisfied: Concerning both 9 and g, we suppose that the following conditions hold:

• For all x E U we have 9x(-) :s; R1 (g)x(-), where Rl (g)x(t) := g(x,O)(t, 0). This definition generalizes naturally to metrics on the cotangent bundle over a manifold.

• 3C> 0, Vx E U, Vt,f, E ]Rn, 9x(t) g(x,~)(O,f,) :s; Cg(x,~)(t,O). • The family of forms (9x, g((1) C), g((2) c») is quasi-decomposable.

x,~ x,~ xEU,~ElRn

We continue with conditions concerning g: • There exists a Hormander metric gs of type (8) on ]R2n such that with

a suitable constant Cs > ° for all x E U and all f, E !R.n the estimate C;lgs,(x,~) (.) :s; g(x,~) (-) :s; Csgs,(x,~) (-) holds.

• 3Cg > 0, Vx,y E U,Vf, E !R.n , g(x,O)(x - y,O) :s; C;1 ===} C;lg(x,~)(-) :s; g(y,~)(.) :s; Cgg(x,~)(-) .

• VK > 0, 3CK > 0, "Ix E U, "If, E ]Rn, VA E £(]Rn)-l such that IIAIIC(lRn) :s; K and IIA-11IC(lRn) :s; K,

Ci/g(x,~)(-) ::; g(x,A~)(-) :s; CKg(x,~)(-) .

• 3C> 0, Vx E U, "It E ]Rn, ~~~2:S; Cg(x,O)(t,O).

1 We recall that, for a metric 9 on an open set U, a weight function M on U and a Banach space E, iiaTl ... aT a(X)ii

S(M, 9j E) is the space of all COO-functions such that for alll E No, sup I I ~ < 00, M(X) TIj=l 9X (Tj)

where the supremum is taken over all X E U and all Tl, ... , Tl E ]R2n \ {a}.

S(M,g)-calculus on Manifolds 213

Moreover, with a g-admissible basis B the following conditions are satisfied: • 3C > 0, 3N E N, '<Ix E U, '<Ie E ]R.n, <~>'5:. CHg,B(X,e)-N . • '<IT E ]R.2n, :3C > 0, :3N E N, '<Ix E U, '<Ie E ]R.n, c-1 Hg,B(X, e)N '5:.

gCx,~)(T) '5:. CHg,B(X,e)-N . Finally, with a g, O-admissible basis the following condition holds:

3C > 0, 3N E N, '<Ix E U \ U, '<Ie E ]R.n,

<x>'5:. C (~ag JOx(t) Hg,B(X, ON) -1

3. Let (U, U, 0, g) be an admissible quadruple of type (p) and let M : U x ]R.n ~ (0,00) be a continuous function. We call M a (U,U,g)-admissible weight function of type (p), if

• for each split Hormander metric gs as in 2.6 there exists a gs-admissible weight function M : ]R.2n ----+ (0,00) and a constant 0 > ° such that for all (x, e) E U x ]R.n we have 0-1 M(x, e) '5:. M(x,~) '5:. OM(x, e).

• '<IK > 0, :3CK > 0, '<Ix E U, '<Ie E ]R.n, '<IA E £(]R.n)-1 such that IIAIICClRn)

'5:. K and IIA- 1 1ICClRn) '5:. K, CKIM(x,~) '5:. M(x,Ae) '5:. CKM(x,e)· • '<Ix,y E U, '<Ie E ]R.n, 9Cx,O)(X - y,O) '5:. C;1 ===} CJ:,/M(x,e) '5:.

M(y,e) '5:. CMM(x,e) . 4. Let U1 be an open subset of]R.n, let (U2, U2, 02, g2) be an admissible quadruple

of type (p), and let K, : U1 ~ U2 be a (U2, U2, 02)-compatible diffeomorphism. K, is called (U2, U2, 02, g2)-compatible of type (p) if the following statements hold:

• (U1,U1,01,gl) is an admissible quadruple of type (p), where U1 .K,-1 (U2), 01 := K,*02, gl := X*g2' X : U1 X ]R.n ----+ U2 X ]R.n, (x, e) 1-+

(K,(x),t dK,(x)-l e ) .

• For i = 1,2 there exists a (gix,gCIC) <),gC2C) <))Cx<)EuxlRn-admissible , ~, x,~ 2, X,"" ,"" 1-

decomposition ]R.n = Eii) Ef).L ... Ef).L ElCi ) such that we have Ey) = dK,(X)-1 (EY)) for all x E U1 and j E {I, ... , I}.

If moreover M2 is a (U2 , U2 , g2)-admissible weight function of type (p) and Ml := X* M2 is a (U1 , U1 , gd-admissible weight function of type (p), then we call K, (U2, U2, 02, g2, M 2)-compatible of type (p).

Remark 2.7.

1. Let (U, U, g) be an admissible model. In the following we will suppose that Cg 2': 1 and rg '5:. llVVQ. If (U,U,O,g) is an admissible quadruple of type (p), we suppose that Cg 2': Cg, Cg is chosen such that for x, y E U and ~,'f/E]R.n,

9Cx,O(X - y,e - 'f/) '5:. C;1 ==? C;lgCx,O(-) '5:. gCy,1/)(') '5:. Cg9Cx,~)(-) ,

and rg '5:. min{C;I/2,rg}.

214 F. Baldus

2. Let (U, U, 9, g) be an admissible quadruple of type (p). We call an orthonormal (with respect to the Euclidean metric) basis B of IRn g-admissible, re-

spectively, g, 9-admissible, if B is (g((I) e), g((2) e)) -admissible, respec-x,~ x.~ (x,';)EUxlRn

. I ((1) (2) 9) d . 'bl tlve y, g( e), g( e), x -a mlSSl e. x.~ x,~ (x.OEUxlRn

3. Let U1 ~ IRn be an open set, let (U2, U2, 92, g2) be an admissible quadruple of type (p), and let K, : U1 --+ U2 be a (U2, U2, 92)-compatible diffeomorphism. For i = 1, 2 let IRn = Ei i) EB.l ... EB.l E?) be a Wi,x) XEUi -admissible decomposition. Then there exists an orthogonal map 7 : ]Rn --+ ]Rn such

that 7 (EP)) = Ey) (j = 1, ... , l). If K, is (U2 , U2 , 92, g2)-compatible of

type (p), then we choose the decompositions IRn = Ei i) EB.l ... EB.l E z( i)

(gi,x, g(I() e), g(2() e)) -admissible (i = 1,2). ',x,~ ',x,~ (X,OEUi xlRn

4. B2 can be chosen as a g2-admissible, respectively, g2, 92-admissible basis such that Bl := 7-1 (B2) is gl-admissible, respectively, gl, 91-admissible, too.

3. Transformation of pseudo-differential operators

Now we investigate the behavior of pseudo-differential operators under consideration when changing coordinates. In so doing, we make use of Kuranishi's method (cf. [17, Appendix of Chapter I], [35, p. 155], [26, Proposition 2.1.3]). More precisely, as usual, we split a general pseudo-differential operator into an operator with Schwartz kernel being supported in a neighborhood of the diagonal and a remainder term. As far as the examination of the first term is concerned, applying approximation arguments (cf. 3.12) we only have to apply Kuranishi's method to pseudo-differential operators associated to amplitudes in CD(IR~~.;,y) with Schwartz kernels being supported near the diagonal (cf. 3.11). Afterwards we must be able to "reduce amplitudes into symbols" within the classes under consideration (cf. 3.8). For this reduction of an amplitude a(x,~, y) to a symbol b(x, 0 we freeze the x-variable in a and consider a(x,',") as a "right symbol" which can be transformed into a "left symbol" (cf. [28, Theorem 18.5.10]; cf. also 3.10).

Definition 3.1. Let (U, U, 9, g) be an admissible quadruple of type (p). For z E U and P > 0 let

Uz,p,g := {x E ]Rn I 9z(x - z) :::; p2} .

For 0 < P :::; rg let Up,g := UzEU Uz,p,g (~ U). For abbreviation, we will write Uz,p and Up in case no confusion can arise. Moreover, if 9 = R 1 (g), then we let Uz.P,g := Uz,p,g, Up,g := Up,g.

Theorem 3.2. Let (U, U, g, g) be an admissible quadruple of type (p). Let 0 < PI < P2 < P3 < P4 :::; rg and P > 0 such that Pi + 3Cg P:::; PHI, i = 1,2,3. Then, for each z E Up2 , there exist functions cpz E CD (Uz,p) and 'ljJz E CD (Uz,2p) with the following properties:

SCM, g)-calculus on Manifolds 215

1. The families (CPZ)ZEU and (1/JZ)zEU are bounded in 5(1,9). P2 P2

2. For z E UP2 we have supp (cpz) <;;; UP3 (<;;; U) and supp (1/Jz) <;;; UP3 (<;;; U). 3. For x E IRn we dejine2 cp(x) := fu cpz(x) 19zI1/ 2 dz. Then we have O:s; cP:S; 1,

P2

supp(cp) <;;; Up3 , and for x E UP1 we have cp(x) = 1. 4. For z E UP2 and x E supp (cpz) we have 1/Jz(x) = 1, i.e., 1/Jz . cpz = cpz· 5. The functions UP2 '3 Z f---7 cpz E 5(1,9) and UP2 '3 Z f---7 1/Jz E 5(1,9) are

continuous.

Proof. Let cP E Co ([-I, 1]) such that 0 :s; CPo :s; 1 and CPo(s) = 1 for s E [0,1/4]. For z E U we define

CPZ,p(x) := CPo (9z(:2- Z)) , 1/Jz,p(x) := CPo (9z~p~ Z)) .

Moreover, let rex) := fu CPz,p(x) 19z1 1/2dz and finally, for x E U and z E Up2 ,

( ) ._ CPz,p(x) n/, () ./, () cpz x .- rex) ,'f/z X := 'f/z,p X .

o

Definition 3.3. Let U be an open subset of IRn and let 9 be a metric on U x IRn. Moreover, let M : U x IRn ---+ (0, (0). We define the space of amplitudes by

A(M, g) := 5 (V(M), V(g)) and Ilallk,A(M,g):= Ilallk,s(v(g),V(M)) (k E No)

where

D(g)(X1,C;,X2)(t1, T, t2)

V(M)(X1'~' X2)

on U x IRn xU.

.- g(X1 ,0 (tl, 0) + g(X2,0 (0, T) + g(X2,t;) (t2' 0) ,

M(X2'~)

Lemma 3.4. Let U1 <;;; IRn be an open set, let (U2,U2,92,g2) be an admissible quadruple of type (p), and let", : U1 ---+ U2 be a (U2, U2 , 92, g2) -compatible diffeo-

morphism of type (p). Let x: U1 X IRn ---+ U2 X ]Rn, (x,~) f---7 ("'(x),td"'(X)-1~), U j := ",-1 (U2 ), and g1 := X* g2. Then the following statements hold:

1. For x E U1 and y E IRn with g1,(x,O)(x - y, 0) :s; r~, we dejine

k(x,y) := r1 (d",)(y+(}(x-y))d(} (E £ (IRn)) . io Then: :3C > 0, :30 < r :s; r g" \Ix E U1, \ly E IRn with g1,(x,O) (x - y, 0) :s; r2, k(x,y) E £ (IRn)-1 and Ilk(x,y)IICCJRn):S; C, Ilk(x,y)-111£(JRn):S; C hold.

2We use the notation 19z1 := det(Gz) where 9z(t) = (Gzt,t) for t E ]Rn.

216 F. Baldus

2. There exists an open subset W 1•r of JR3n with (a) W1,r <:;;; U1 X JRn X U1 .

(b) (x,';, y) E W1,r ===> gl,(x,O)(x - y, 0) :S r2. (c) 30 < ro :s r, \:Ix E U1, \:Iy E JRn, gl,(x,O)(x - y,O) :S r5, \:I'; E JRn,

(x,';, y) E WI,r'

3. Now we restrict D(gl) to W 1,r' Then we have (a) [Wl,r:3 (x,~,y) J----+ k(x,y),tk(X,y)-l] E S(l,D(gl);£(JRn». (b) [Wl,r:3 (x,~,y) J----+ Idet(k(x,y»I±l] E S(l,V(gl».

Proof. First we fix x E U1 and let y E JRn with gl,(x,O)(x-y, 0) :S r;,. With suitable constants and with the definition Xg := y + e(x - y), for t E JRn the estimate

II (t, \ly) k(x, y) 11£(JRn) < 11 (1 - e) II (Otdfi:) (xg) 1I.C(JRn) de

< C1 11 vgl'(Xe,O)(t,O)dfJ :S C2Vg1,(X,0)(t,0)

holds. By virtue of the mean value theorem we obtain

Ilk(x,y) - k(x,x)II.c(lRn) II (\X-y,\ly)k(x,y)ly=xsdsll Jo £(JR")

< C2Vgl,(X,O)(x-y,0).

We set D := SUPXEUI Ildfi:(x)-lIL::(lRn ) and r := min{(2DC2)-1,rgJ. Then, for x E U1 and y E JRn with 91,(x,O) (x - y, 0) :S r2, we have

1 Ilk(x, y) - k(x, x)II£(lRn) :S 2I1k(x, x)-lll.c(lRn ) ,

where k(x,x) = dfi:(x) E £(JRn)-l. This yields k(x,y) E £ (JRn)-l and

Ilk(x, y)-l - k(x, x)-lll.c(lRn ) :S D,

hence Ilk(x, y)II.c(IR") :S C3 and iik(x, y)-lll.c(lRn) :S C3 ·

For x E U1 there exists 0 < Ex :S r/ (2y1C;) (:S rg, ) such that x' E IRn with 91,(x,O)(x - x',O):S E; implies x' E U1. Let ~r(X) be the set of all (x',y) E IR2n such that 91,(x,O)(x - x',O) < E; and 91.(x,O)(x - y,O) < (r/ (2VC;)( Then ~r(x) is open and for (x',y) E ~r(X) we have

V91,(X 1 ,O)(x' - y, 0) :S r :S rg , .

In particular, we have ,6,r(x) <:;;; U1 X U1. Moreover, we define ,6,r := UXEU, ~r(X) (<:;;; U1 X Ur) and W 1,r;= {(x,,;,y) E U1 X JRn X IRn I (x,y) E ,6,r}. Then W 1.r is an open subset of JR3n with W1 ,T <:;;; U1 X JRn X U1 .

The last assertion is obvious. 0

5(M, g)-calculus on Manifolds

Lemma 3.5. We apply the notation from 3.4. 1. \ik E No, ::ICk , V(x,~) E U1 X ffin , \iT1 , ... ,Tk E ]R2n,

k

g2,x(x,t;,) (X(k) (x, O(T1 , ... , T k )) ::::; Ck IT gl,(x,t;,) (Tj ) . j=l

2. For x E U1 , Y E]Rn with gl,(x,O)(x - y,O)::::; r2 (cf. 3.4), and ~ E]Rn let

K(x,~,y) := (K:(x),tk(X,y)-l~,K:(Y)).

Then:

with

k

D(g2)K(x,t,.y) (K(k)(x,~,y) (T1 , ... , T k )) ::::; Ck IT D (gd(x,t"y) (Tj) j=l

217

Proof. The proof is elementary. For details we refer to [2, Lemma 2.3.6]. 0

Corollary 3.6. We apply the notation from 3.4 and 3.5. Moreover, let M2 be (U2, U2, g2)-admissible of type (p). We define Ml := X* M 2.

1. Then we have K:*5(1, 92) '--+ 5(1,9d and X*S(M2,g2) '--+ 5(Ml,gd. 2. We restrict D(gl), D(Md to WLr and D(g2), D(M2) to the set {(K:(x),~,

K:(y)) I (x,~, y) E WI,1'}' Then we have K* S (D(M2), D(g2)) '--+ 5 (D(M1 ),

D(gd)·

Definition 3.7. If (U, U, 9, g) is an admissible quadruple of type (p), then we define (0 < Pl,P2 ::::; rg)

1:.9 •P' ,P2 := {(x,~,x') E UPl x]Rn x UPl I g(x.O)(x - x',O)::::; pD and I:. g '1'9 := I:. g ,1'g ,1' 9 (d. 3.1 for the notation).

Theorem 3.8. Let (U, U, 9, g) be an admissible quadruple of type (p) and let M be a (U,U,g)-admissible weight function of type (p). For a E A(M,g) with support in U1'g x]Rn x U1'g and x,x' E U, ~ E]Rn we define ax(x',~) := a(x,~,x'). Then,

for each x E U, we have ax E 5(M,gs) (cf. 2.6) and thus we can define ((x',~) E U x ffin)

R(a)(x, ~)

Rl(a)(x,~)

:= exp(i(Dx"Dt,))ax(x"~)lx'=x,

:= a(x,~, x) - R(a)(x,~) .

Then the following statements hold:

218 F. Baldus

1. For each a E A(M,g) with supp(a) ~ tlg,ry we have R(a) E S(M,g), RI(a) E S(Mhg,g), and supp(R(a)),supp(RI(a)) ~ Ury x JRn. More precisely, for each kENo there exist constants Ck > 0 and lk E N, depending on M only through the constant CM in 2.6 and the global constants of a fixed gs-admissible weight function M as in 2.6, such that

IIR(a)llk,S(M,g) ~ Ck IlaI11k,A(M,g) , IIRI(a)llk,S(Mhy,g) ~ Ck IlaI11k,A(M,g)·

2. If (ak)kEN ~ A(M,g) is a bounded sequence such that supp(ak) ~ tlg,ry for each kEN, and if furthermore (ak)kEN converges in coo(JR3n) to a, then (R( ak)) kEN converges in Coo (JR2n ) to R( a) and (RI (ak)) kEN converges in coo(JR2n) to RI(a).

Proof. 1. First we observe that [x f---+ ax] E Coo (JRn,coo(JR2n)). Moreover, since

supp(a) ~ tlg,ry, for all kENo and all tl, ... ,tk E JRn \ {O}, the function

JRn ---+ S (M(t l ' ... ,tk),gS)'

x f---+ (tl' \)x) ... (tk' \)x) ax = ((tl' \)x) ... (tk' \)x) a)x

is bounded, where M(tl, ... ,tk)(X',~) := M(x',~Hl~=IVgs,(xl,O(tj,O). Together with the continuity of exp (i (DXI, Df.)) on bounded subsets of S(Mo, gs) with respect to the COO-topology, where Mo is a gs-admissible weight function, by virtue of the mean value theorem we obtain that

[(x,~, x') f---+ exp(i (DXI, Df.) )ax(x', ~)] E c oo (JR3n)

and thus in particular R( a) E Coo (JR2n) with

OXj R(a)(x,~) = exp(i (DXI, Df.)) (oXja)x (x', ~)IX=XI

+OXj exp(i (DXI, Df.) )a(x', ~)IX=XI

(j = 1, ... ,n). Thus, we obtain R(a) E S(M,g) and the asserted estimates for R(a). For the corresponding statements concerning RI(a) we apply [28, Theorem 18.5.10]).

2. By assumption, for each x E JRn we have [(ak)x ---+ ax] E COO (JR2n). This yields exp(i (DXI, Df.) )(ak)x(x',~) ---+ exp(i (DXI, Df.) )ax(x',~) for all x, x', ~ E JRn. In particular, this yields R(ak)(x,~) ---+ R(a)(x,~) for all x,~ E JRn.

(R(ak))kEN is bounded in S (M, gs) and thus in particular in COO (JR2n). Since

coo(JR2n) is Montel, the standard indirect argument, taking into account the pointwise limits, yields the convergence of R(ak) to R(a) in COO (JR2n). For R I (a) the proof is analogous.

Definition 3.9. For a E C(f(JR3n), U E S(JRn), and x E JRn we define

a(x,D,x')u(x):= (2!)n 11 ei(X-Y,f.)a(x,~,y)u(y)dyd~. o

Remark 3.10. We assume the situation of the preceding definition. By virtue of [28, Theorem 18.5.10J we obtain

a(x, D, x')u(x) = (2~)n J J ei(x-y,f;)ax(y,f,)u(y)dyde,

(exp (i (Dx" D~)) ax(x', f,)lx'=x) (x, D)u(x) = R(a)(x, D)u(x) .

The following lemma makes use of Kuranishi's method.

Lemma 3.11. Let U1 ~ JRn be open, let (U2, U2, 92, g2) be an admissible quadruple oJ type (p), let M2 be a (U2,U2,g2)-admissible weight Junction oJ type (p), and let K, : U1 ----> U2 be (U2, U2, 92, g2, M 2)-compatible oj type (p). There exist constants o < rK,PK < rg2 such that Jor a E Co(JR3n) with supp(a) ~ D..g2 ,rK,PK and u E S(JRn) with supp(u) ~ U1 we have (x E Ud

K,*a(x,D,x') (K,-l)*U(X) b(x,D,x')u(x) , where

b(x, e" x')

k(x, x')

In particular, b E Co (JR3n).

a ( K, ( x ) , tk ( x, x') -1 e, , K, ( x') )

.- 11 (dK,) (x' + B(x - x'))dB .

I det(dK,(x'))I

I det(k(x, x'))1 '

Proof. For 0 < r sufficiently small, (U,U2,T)92,g2) is admissible of type (p) and K, is (U, U2 ,T) 92, g2)-compatible of type (p). There exists a constant p> 0 such that for x, x' E U1 and e, E JRn with (K,(x), f" K,(x')) E D. g2 ,r,p we have (x, e" x') E W 1 ,r'

with a suitable constant r' > 0, W 1 ,T' as in 3.4 with (U2, U2,T) 92, g2) instead of (U2' U2, 92, g2), and thus there exists k(x, X')-l, and k(x, x') and k(x, X,)-l are uniformly bounded with respect to those x, x'. Then, applying transformation formula twice, we have for x E U1 ,

(2nTK,*a(x, D, x') (K,-I)*U(X)

J J ei("(X)-K(X'),E.) a (K,(x), f" K,(x')) u(x')1 det(dK,(x'))ldx' de,

J J ei(x-x', t k(x,x')E.) a (K,(x), f" K,(x')) u(x') I det( dK,(x')) Idx' de,

if i(x-x'~) ( ( ) t ( ,)-1 (,) (') I det(dK,(x'))I , e . a K,X, kx,x e" K,x) ux Idet(tk(x,x'))ldxdf,.

o Theorem 3.12. We apply the notation oJ3.11. Let rp,7jJ E coo(JRn) such that

• supp ((x, e" y) f--> rp(x) . 7jJ(y)) ~ D..g2 ,TK,PK'

• [(x, £,) f--> rp(x)], [(x, e,) f--> 7jJ(x)J E S(l, g2). Then, Jor each a E S(M2,g2) and u E S(JRn) with supp(u) ~ U1 , we have (x E U 1 )

K,* [(rpa)(x,D) (7jJ. (K,-I)*(U))] (x) = b(x,D)u(x) ,

220

where

b(x, ~)

k(x,x')

F. Baldus

._ R (ip(II;(X)) a (II;(X), tk(x, X')-l~)

.- 11 (dll;) (x' + O(x - x'))dO .

'IjJ( ( ')) I det(dll;(x')) I ) II; x I det(k(x, x'))1 '

In particular, we have b E S (M1, gl) . Moreover, if U <;;; ]Rn such that supp (a) <;;; U x ]Rn, then we have supp(b) <;;; lI;-l(U) X ]Rn. Finally, the definition

c(x,~) := b(x,~) - ip(lI;(x))a (lI;(x),tdll;(X)-l~) 'IjJ(II;(X))

yields C E S (M1hgl1 gl).

Proof. By assumption we have supp (ip),supp ('IjJ) <;;; U2,rK and'IjJ E CO"(JRn). 3.2 yields the existence of functions rp1, rp2 E Coo (JRn) with supp (rpi) <;;; U2 (i = 1, 2) such that [(x,~) I-t rpi(X)) E S(1,g2), rp1(X) = 1 for all x E U2,rK and rp2(X) = 1 for all x E supp(rpd. Defining a(x,~) := rp1(X)· a(x,~) (x,'; E JRn) we have a E S(M2, g2,s). There exists a sequence (av)vEN <;;; CO"(JR2n) which is bounded in S(M2,g2,s) such that av ~ a in COO (JR2n). For lJ E N and x,~,x' E JRn we define av(x,~, x') := ip(x)av(x, ~)'I/1(x'). Noting the continuity of the mappings [S(JRn) 3 u I-t 1I;*(IP2 . u) E S(JRn)) and [S(JRn) 3 u I-t (11;-1 )* (11;* (rp2) . u) E S(JRn)] we have

11;* [(ipa) (x, D) ('1/1. (lI;-l)*(U))] (x)

11;* [rp2· ip. a(x,D) ('IjJ. (11;-1)* (lI;*rp2 . u))] (x)

S~) 11;* [rp2. ip av(x, D) ('1/1. (1I;-1)*(II;*rp2· u))] (x)

11;* [av(x, D, x') (II;-l)*U)] (x) .

By virtue of 3.10 and 3.11 we have

bv(x, D)u(x)

where

( ( ( )) _ ( ( ) t ( ,)-1) « ')) I det(dll;(x')) I ) bv = R <pIl;X a" II;X, kx,x ~ 'ljJII;X Idet(k(x,x'))1 .

We denote the argument in the bracket by Cv and without lJ simply with c. With the notation from the proof of 3.11 we have K-1(.6.g2 ,rK,PK) <;;; W1,rl, where K :

W1,rl ~ JR3n, (x,~,y) I-t (lI;(x),tk(x,y)-l,II;(Y)). Noting that K maps compact

subsets of W 1,r l onto compact sets, by virtue of the chain rule we have c" ~ c in coo(]R3n). Furthermore, by virtue of 3.4 and 3.6, the sequence (Cv)"EN is bounded in A(M1,gt). By virtue of 3.8 this yields bv = R(cv) ~ R(c) = b in coo (JR2n) and (R(cv))VEN is bounded in S(M1,gl,s), thus bv(x,D)u ~ b(x,D)u in S(JRn). The assertion concerning c is a consequence of 3.8. D

8 (M, g)-calculus on Manifolds 221

In the following proposition, we study the behavior of a pseudo-differential operator under compatible diffeomorphisms and show that the transformed operator is the sum of an operator with symbol in the pull-back of the original symbol class and a remainder part which is a bounded operator on L2 (l~n). The first summand comes from the part of the original operator with Schwartz kernel supported near the diagonal to which we can apply Kuranishi's method (cf. 3.12). The second summand is in fact much better and is even of order -00 in the original global symbolic calculus. But, so far we have no description of this smoothing class invariant under compatible diffeomorphisms which is the case for .c (L2 (l~n)). This proposition will be applied in the investigation of the behavior of local Sobolev spaces under changes of coordinates in the following section (cf. 4.7). In 5.12 we will apply similar arguments and hence give a detailed proof of this proposition and will omit the proof of 5.12.

Proposition 3.13. Let UI <;;; ~n be open, let (U2, U2, <;12, g2) be an admissible quadruple of type (p), and let K, : UI -+ U2 be (U2,U2,g2,g2)-compatible of type (p). We define M2 := H;;'~62 (cf. 2.4), where B2 is a g2-admissible basis of~n. Let P K, r K :'S r g2 be the constant from 3.12 and let C > 0 be chosen such that K,-I(U2.z,p) <;;; UI,I'O-'(z),Cp for Z E U2 and 0 < P < min{rg"rg2 }/C. Moreover, let 0 < PI < P2 < P3 < P4 := min{rK,rg,/C} and let 0 < P :'S PK/(2VC;;) be sufficiently small such that Pi + 3Cg2 P :'S PHI (i = 1,2,3). Let cp be the function from Theorem 3.2 applied with (U2, U2, R I(g2)) instead of (U, u, g). Furthermore, let'ljJ E CCXJ(~n) be a function with supp ('ljJ) CUI such that [~2n :3 (x,~) f---+ 'ljJ(x)] E

8(1,gd. Then, for S E~, a E 8(M!J.,g2), and u E L2(~n) we have

K,*[(cpa)(x, D)] (K,-1 )*('ljJ. u) = A 1u + A 2u

where Al E w(M1,gl.s), gl := X*g2' Ml := X* M2 (cf. 2.6), and A2 E .c (L2(~n)).

Proof. Let rpl,rP2 E CCXJ(~n) with supp(rpd C U2 (i = 1,2), rpl(X) = 1 for all x E U2.P3 , rp2(X) = 1 for all x E supp(rpd, and [(x,O f---+ rpi(X)] E 8(1,g2) (i = 1,2) (cf. 3.2). Let a := rpl . a. Then, for J E s(~n) and x E ~n we have

cp(x,lJ)a(x,lJ)J(x)

l2,P2 cpz (x, lJ)a(x, D)'ljJz(x, lJ)J(x) IRI (g2)z II/2dz

+ l2,P2 cpz(x,lJ)a(x,lJ)(l-'ljJz)(x,D)J(x) IR1(g2)zI1/2dz.

Here we note that for u E s(~n)

(K,* l2,P2 [cpz(x,lJ)a(x,lJ)'ljJz(x,lJ)] (K,-I)*('ljJ·U) IRI(92) z l l/2 dZ ) (x)

l2,P2 K,* [cpz(x,lJ)a(x,lJ)'ljJz(x,lJ)] (K,-I)*('ljJ. u)(x) IR1(g2lz!1/2dz.

222 F. Baldus

For Z E U2,P2 and x, y E Uz,P,92 we have g2,(x,O)(x - y, 0) :::; 4Cg2p2 and

supp ((x,y) f---+ 'Pz(x), 'lj;z(y)) s;;: U2,P3 X U2,P3 s;;: U2,TK X U2,TK'

hence supp ((x,~, y) f---+ 'Pz(x) . 'lj;z(y)) s;;: D..g2 ,TK,PK· Thus we find for z E U2,P2 (cf. 3.12)

",* ['Pz(x, D)a(x, D)'lj;z(x, D)] (",-1)*('lj;' u) = bz(x, D)('lj; . u)

with bz E 8(Ml' gl). More precisely, 3.6, 3.2, 3.8, and 3.12 yield that U2 ,P2 3 Z f---+

bz E 8(Ml' gd is continuous, bounded, and supp (bz ) s;;: ",-1 (U2,z,p) x ]Rn. For Z E ",-1(U2,P2) let bz := bK(z)' Then we have supp(bz ) s;;: ",-1(U2,K(z),p) x]Rn s;;: U1,z,cp X ]Rn and ",-1(U2,P2) s;;: U1,cP2' The definition

b(x,~):= j bz(x,~) IR1(g2)zI1/2dz = r bz(x,~) IR1(gl)zI1/2dz U2,P2 JK- 1 (U2,P2)

yields a symbol b E S(M1,gd with supp(b) s;;: ",-1 (U2 ,P3) x]Rn s;;: U1 X ]Rn, and we have

("'* i2P2 ['Pz(x, D)a(x, D)'lj;z(x, D)] (",-1 )*('lj;. u) IRI (g2) z I1/2 dZ )

= b(x, D)('lj;· u) .

Furthermore, 'Pz(x, D)a(x, D)(l - 'lj;z)(x, D) = cAx, D) where U2,P2 3 z f---+ Cz E

8(1,g2) is continuous, Defining c :=.£u cz(x,~) IR1(g2)zI1/2dz, we have c E 2,P2

8(1,g) and

i2,P2 'Pz(x, D)a(x, D)(l - 'lj;z)(x, D)f IR1(g2)zI1/2dz

<P2j 'Pz (x, D)a(x, D)(l - 'lj;z )(x, D)f IRI (g2)z 11/2dz = <P2 c(x, D)f . U2,P2

This yields the assertion for u E S(]Rn). For u E L2(]Rn) we make use of a sequence (UV)VEN C S(]Rn) with U v ----+ U in L2(]Rn) '----7 S'(]Rn). 0

4. Sobolev spaces

We will describe negligible terms in the calculus of pseudo-differential operators on manifolds (cf. 5.4, 5.9) with the help of mapping properties of iterated commutators with distinguished vector fields and multipliers between suitable global Sobolev spaces on the manifold (cf. 4.20). Here we first define and discuss the local models for these spaces. For an admissible triple (U, U, g) and a (U, U, g)-admissible weight function M we define the associated Sobolev space with the help of a Hormander metric g8 and an admissible weight function M as in 2.6. Nevertheless, 4.4 gives descriptions of the local Sobolev spaces which consider g8 and M only on U and thus, the definition of H (g, Mj U') is independent of the choice of g8 and M. In 4.7 we apply these descriptions together with 3.13 to show that the local Sobolev


spaces associated to the standard weights hg and Hg,B behave well under changes of coordinates. Aside from the just mentioned results, this theorem is based on the fact that the L2-spaces behave well under compatible diffeomorphism.

Then we define and discuss the Sobolev spaces on a generalized SG-compatibIe manifold M which are associated to powers of the weight Hg (cf. 4.16). More precisely, a distribution u E D/(M) is said to be in H (H~,g;M,A) (cf. 4.20) if for all admissible charts the localization of u with an admissible cut-off function is in the corresponding local Sobolev space on JRn. In 4.22 we show that these localizations are in H (H ~ , g; M, A) if and only if they are pull-backs of elements of the local Sobolev space. In the proof we make use of 4.7.

At this stage we avoid endowing the global Sobolev space with a topology. To have nevertheless meaningful extensions of operators CaCM) ---+ D/(M) to operators between suitable Sobolev spaces, we introduce the notion "bounded in charts" (cf. 4.25).

We start with a repetition of the definition of the global Sobolev spaces H(M, g). For an investigation of these weighted L2-Sobolev spaces adapted to the S(M, g)-calculus we refer to R. Beals [5] and J.-M. Bony and J.-Y. Chemin [9].

Definition 4.1. Let 9 be a Hormander metric and Mag-admissible weight function. We define

H(M,g):= {u E S'(JRn) I Au E L2(JRn) for all A E 1]i(M,g)} .

Remark 4.2. By virtue of[9, Theoreme 6.4]' H(M, g) is a Hilbert space and we have S(JRn) '---+ H (M, g) '---+ S' (JRn) with dense embeddings, where S' (JRn) is endowed with the w*-topology. Moreover, for two g-admissible weight functions Ml and M2 with Ml ::; M2 we have H(M2' g) '---+ H(Ml' g).

Definition 4.3. Let 9 be a split Hormander metric and Mag-admissible weight function. For an open set U ~ JRn we define

H(M,g;U):= {u E H(M,g) I supp(u) ~ U} .

Lemma 4.4. Let 9 be a split Hormander metric of type (s). With an orthonormal (with respect to the Euclidean metric) basis B, we define M:= H;;'~ (cf. 2.4). Let U be an open subset ofJRn. Let CP1,CP2,CP;, E coo(JRn) such that [(x,';) f-+ CPi(X)] E 5(1, g) for i = 1,2,3 and, moreover, CPl = 1 on U and CPi+l = 1 on supp (CPi) for i = 1, 2. Let s ::::: O. Then the following statements hold:

1. (a) u E H(MS,g;U) {=} u E L2(JRn),supp(u) ~ U, andcp2A(cpIU) E L2(JRn) for all A E 1]i(MS, g).

(b) If (Uv)vENo c H(MS,g;U), then we have Uv---+Uo in H(MS,g) {=}

Uv ---+ Uo in L2(JRn) and CP2A(CPluv) ---+ CP2A(CPIUO) in L2(JRn) for all A E 1]i(MS, g).

2. (a) u E H(M-S,g;U) {=} u = CP3A(CP2UJ) +U2 where A E 1]i(MS, g) and Ul,U2 E L2(JRn), and supp(u) ~ U.

224 F. Baldus

(b) For u E H (M- S, g; U) the functions U1, U2 E L2 (JRn) can be chosen depending continuously on u E H(M-S,g). Thus, for (U")"ENo C

H(M-S,g;U), we have u" ---+ Uo in H(M-S,g) -{=} U",i ---+ UO,i in L2(JRn) (i = 1,2).

Proof. 1. Since M 2: 1 we have H (MS,g; U) <;;; H(MS,g) <;;; L2(JRn), where the last

embedding is continuous. For A E IJ!(MS,g) we have 'P2A'P1 E IJ!(MS, g), and thus, 'P2A'P1 maps H (MS, g) continuously to L2(JRn). On the other hand, let U E L2(JRn) with supp (u) <;;; U such that 'P2A'P1 U E L2(JRn) for all A E IJ!(MS,g). Let A E IJ!(MS,g). Then we have

Au = A('P1U) = 'P2A('P1U) + (1- 'P2)A('P1U) ,

where 'P2A('P1U) E L2(JRn) by assumption and (1- 'P2)A'P1 E lJ!(l,g), thus (1 - 'P2)A('P1U) E L2(JRn). This implies Au E L2(JRn) and consequently u E

H(MS,g; U). Now let (U")"ENo C H(MS,g;U) with u" ---+ Uo in L2(JRn) and 'P2A('P1U,,) ---+ 'P2A('P1UO) in L2(JRn) for all A E IJ!(MS, g). We choose A E w(MS,g) as an isomorphism H(MS,g) ---+ L2(JRn ). Then the assumptions imply that Au" ---+ Auo in L2(JRn) and thus u" ---+ Uo in H (MS, g).

2. Let U1 E L2(JR?n) (<;;; H(M-S,g)) and A E IJ!(MS, g). Then we have 'P2A('P1Ud E H(M-S,g). On the other hand, let U E H(M-S,g;U). Let Js E IJ!(MS,g) such that there exists J- s := J;l E IJ!(M-S, g). Then we have

U Js ('P2Ls('P1U)) + Js ((1 - 'P2)Ls('P1U))

'P3Js('P2 Ls('P1U)) + ((1 - 'P3)Js'P2) Ls('P1U) -------- ' , =:v

=:w

Since J- s'P1 E IJ!(M-S,g) and u E H(M-S,g) we have U1 E L2(JRn). Moreover, we have (1- 'P3)Js'P2 E lJ!(l,g) and (1- 'P2)J- s 'P1 E IJ! (M-2s,g) and thus v = (1 - 'P3)Js('P2U1) E L2(JRn) and W E L2(JRn).

o Definition 4.5. Let (U, U, 9, g) be an admissible quadruple of type (p). Let 8 be a g-admissible basis (cf. 2.7). Let r < rg and s E JR. Let U' <;;; Ur be an open subset. Applying the notation from 2.6, we define H (H;,B' g; U') := H (H;s,B' gs; U'). Moreover, we endow H (H;,B,g; U') with the topology induced by H (H;s,B,gs).

Remark 4.6. By virtue of 4.4 and 3.2, the definition of H (H;, g; U') is independent of the choice of gs and 8.

Theorem 4.7. Let U1 <;;; JR?n be open, let (U2, U2, 92, g2) be an admissible quadruple of type (p), and let r;, : U1 ---+ U2 be (U2,U2,92,g2)- compatible of type (p). Let M2 := H;;'~B2 (82 a g2-admissible basis such that 7- 1 (82 ) is gl-admissible, 7 as

in 2.7. There exist constants K s, r s > 0 with r s < r 92 and K sr s < r 91 such that the following statements hold:

1. ,,-1(U2,rs) <;;; Ul,Ksrs' 2. Let gl := X*g2, Ml := X* M 2, and Bl := 7-lB2 (for the notation cf. 2.6 and

2.7). Then there exists a constant C > 0 such that for all x E U l and all ~ E IRn the estimate C-lMl(X,~) S; H;;;~Bj(X,O S; CMl(X,~) holds.

3. Let U~ <;;; U2,rs and U{ := li-l(U~). Let s E IR. Then we have

(Ii- l )* (H (Mf, gl; UD) = H (M2' g2; U~) .

Proof The proof of the first two assertions is elementary. Part 3 is a consequence of 3.13 and 4.4. 0

Now we introduce generalized SG-compatible manifolds and admissible structures.

Definition 4.8. Let M be a CCXl-manifold of dimension nand 9 a metric on M. 1. Let (X, Ii) be a chart of M, Ii: M ~ X ---+ Ii(X) <;;; IRn, and let X <;;; X be an

open subset. We call (X, X, Ii) an admissible chart if (Ii(X), Ii(X), (Ii- l )*glx) is an admissible mode13 (cf. 2.5).

2. We call A a generalized SG-compatible atlas of M if the following conditions are satisfied:

• A is finite and consists of admissible charts (Xl, Xl, lid, ... , (Xm' X m , lim) .

• M = U·-l X J·• J- , ... ,m

• For each i,j E {1, ... ,m}, (lij(Xi n Xj),lij(Xi n Xj),(lijl)*g) is an

admissible model and the diffeomorphism lij oliil : Iii (Xi nXj ) ---+ lij (Xin Xj) is (lij(Xi n Xj), Iij(Xi n Xj), (lijl)*g)-compatible.

Moreover, we call (M, A, g) a generalized SG-compatible manifold.

Remark 4.9. For i = 1,2 let (M(i),A(i),9(i)) be a generalized SG-compatible

manifold of dimension ni, A(i) = {(X}i), X?), liji)) I j E {1, ... ,mi }}' On M :=

M(l) x M(2) we define (g(1) Q9 g(2)) (Xj,X2) (tl' t2) := g~~) (tl) + g~;) (t2) Moreover,

let A consist of all charts (xP) x xP), xiI) x XY), (Ii?), Iij2))) , i E {1, ... ,md, j E {1, ... , m2}' Then (M, A, 9) is a generalized SG-compatible manifold of dimension n] + n2.

Definition 4.10. Let (M, A, 9) be a generalized SG-compatible manifold with atlas A= {(Xj,Xj,lij) I j = 1, ... ,m}.

1. We call a metric g on the cotangent bundle T* Mover M (M, A, g)-admissible of type (p) if the following conditions are satisfied:

3In the following, we will omit the restriction of metrics and functions to chart regions when applying pull-backs.

226 F. Baldus

• For all j E {l, ... ,m}, (K:j(Xj),K:j(Xj ), (K:j1)*9, (xj1)*g) is an admissible quadruple of type (p), where we define Xj : T* Xj ----> T*]Rn ::: ]R2n,

T;Xj :3 ~ f---+ (K:j(x),tdK:j(X)-l~) . • For each i,j E {I, ... , m}, (K:j (Xi n X j ), K:j(Xi n X j ), (K:j1 )*9, (xj1 )*g)

is an admissible quadruple of type (p) and the diffeomorphism K:j 0

K:-;1: K:i(XinXj ) ----+ K:j(XinXj) is (K:j(XinXj),K:j(XinXj),(K:j1)*9,

(Xj1)* g)-compatible of type (p). If (M, A, 9) is a generalized SG-compatible manifold and g is (M, A, 9)admissible of type (p), then we call (M, A, 9, g) an admissible structure of type (p).

2. Now let (M, A, Q, g) be an admissible structure of type (p). Let M : T* M ----+

(0,00). We call M an (M, A, g) -admissible weight function of type (p), if for each (X, X, K:) E A, (X- 1)* M is a (K:(X), K:(X), (X-1 )*g)-admissible weight function of type (p).

For abbreviation we introduce some notation which will be used in the following.

Definition 4.11.

1. Let (M, A, 9) be a generalized SG-compatible manifold, A = {(Xj, Xj, K:j) 1

j E {l, ... ,m}}. For i,j E {l, ... ,m} we define Uj := K:j(Xj ), Uj :=

K:j (Xj ), 9j := (K:j1)*9, Ui,j := K:j (Xi n Xj), Ui,j := K:j (X, n Xj), 9i,j :=

(K:j1 )*9, and K:i,j := K:j 0 K:-;1 : K:i (Xi n X j ) ----+ K:j (Xi n Xj). Moreover, let rg := min0=1 {rgj , rg"j} and Cg := maxJ=l Cg.1' For each i, j E

{I, ... , m} there exists a constant Ci .j > 0 such that for 0 < P < rg / Ci,j

we have K:i:J (Ui,j,P,gi.J <;;; Uj,i,G.i,jp,g),i' More precisely, we suppose Ci,j (i, j = 1, ... , m) being chosen sufficiently large such that for x, Y E Uj,i with the property that x + t(y - x) E Uj,i for all t E [0,1] the estimate

9 i ,j'''i,j(X)(K:i,j(X) - K:i,j(Y)) :::; Ci ,j9j,i,x(X - y)

holds. We define CA := maxi,'j=l Ci,j' Furthermore, let r A < rg be fixed. ~ ~ -1 0

For (X,X,K:) E A we define Xg := X := K: (UrA,gJ (cf. 3.1), where U := K:(X).

2. Let (M,A,9,g) be an admissible structure of type (p), A = {(Xj,Xj,K:j) Ij E {I, ... , m}}, and let M be an (M, A, g)-admissible weight function of type (p). We suppose that the constants introduced for (U, U, 9) are also applicable for (U,U,R1(g)). For i,j E {l, ... ,m} we define gj := (Xj1)*g,

gi,j := (Xj1)*g, and M j := (Xj1)* M, Mi,j := (Xj1)* M. Moreover, let rg :=

min7,j=l {r gj , r 9i,j} and Cg := maxJ=l Cgj . Thus, for an admissible structure (M,A,9,g), we have rg:::; rg and Cg 2: Cg (cf. 2.7). Finally, for (X,X,K:) E

A, let )(g := )(R1 (g).

Definition 4.12. Let (M, A, 9) and (M, A, 9) be generalized SG-compatible manifolds. We write A :::; A if A = {(Xj, Xj, K:j) 1 j = 1, ... , m} and A = {(Xj, Xj, K:j)

Ij = 1, ... , m} with Xj <;;;; Xj. In this case we choose the constants in 4.11 simultaneously for (M, A, 9) and (M, A, 9).

Lemma 4.13.

1. Let (M, A, 9) be a generalized SG-compatible manifold. We apply the notation from the preceding numbers. Then

is a generalized SG-compatible atlas, A :::; A, and (M, A, 9) is a generalized SG-compatible manifold. In particular, if (M,A,9,g) is an admissible structure of type (p), then (M,A,9,g) also is an admissible structure of type (p) .

2. Let Q be another metric on M such that (M, A, Q) is a generalized SGcompatible manifold with 9 :::; Q. We suppose that r A < min{rg, rg } and let

A:={(Xj,Xj.g,lij) I jE{l, ... ,m}}. Then (M,A,Q) also is a general

ized SG-compatible manifold and we have A:::; A. Proof. This proof is a generalization of [41, Lemma 3.3]. For details we refer also to [2, 2.8.7]. [J

We proceed with the definition of a canonical scale of Sobolev spaces on generalized SG-compatible manifolds. As a preparation, we first introduce some notations concerning cut-off functions and canonical weight functions which we will apply in the following.

Definition 4.14. Let (M, A, 9) be a generalized SG-compatible manifold with atlas A = {( Xj , X j , Ii j) 1 j = 1, ... , m}.

1. (a) Let (X, X, Ii) E A. Let ° < r < r A (cf. 4.11). We call a function rp : M ---+ [0,1] an (X, X, Ii, 9, r)-admissible cut-off function (or (X, X, Ii, 9)admissible or (X, X, Ii)-admissible for short) if the following conditions are satisfied:

• supp(rp) <;;;; Ii-I (Ii(X)rt (<;;;; X) . • rp 0 Ii-I E S (1, (Ii- 1 )*9).

(b) We call a partition of unity (rpi)i=l ... ,1 on M (M,A,9)-admissible if for each i E {I, ... , l} there exists ji E {I, ... , m} such that rpi is an (Xii' X ji , liji )-admissible cut-off function. More precisely, for ° < r < r A

we call (rpi)i=1..,1 (M, A, 9, r )-admissible if for each i E {I, ... , l}, rpi is an (Xji' X ji , liji' 9, r)-admissible cut-off function. If (M,A,9,g) is an admissible structure, then we call an (X,X,Ii,

R 1 (g),r)-admissible cut-off function (X,X,Ii,g,r)-admissible or (X,X,Ii, g)-admissible and an (M, A, Rdg), r )-admissible partition of unity (M, A, g, r)-admissible or (M, A, g)-admissible.

228 F. Baldus

2. Let (X, X, K,) E A and let zp be an (X, X, K,)-admissible cut-off function. We call an (X, X, K)-admissible cut-off function 'ljJ an associated cut-off function if'ljJ = 1 on supp (zp).

Lemma 4.15. We apply the notation from above. Then there exists an (M, A, 9)admissible partition of unity with associated system of cut-off functions.

Proof. Let j E {l, ... ,m}. Let 0 < P1 < P2 < P3 < P4:= r/2 and 0 < P such that Pi + 3Cg :::; Pi+1 (i = 1,2,3). Let ¢j be the function zp from 3.2 with (K,j(Xj ), Kj(Xj ), (K,j1)*9IAJ instead of (U, U, 9). We define ¢j := ¢jOK,j. In particular, ¢j is an (Xj, X j , Kj)-admissible cut-off function and ¢j = 1 on Xj. We define

!.pj := 'L~jl <Pi' By virtue of 3.6, Zpj is an (Xj, X j , K,j )-admissible cut-off function. 3.2 yields the existence of 'ljJj for j = 1, ... ,m. 0

Definition 4.16.

1. We apply the notation from above. Let (!.pi)i=l, ... ,1 be an (M, A, g)-admissible partition of unity. For x E M and ~ E r; M we define

I

hg(x,O := L h gji (xd x , ~)) . ZPi i=l

and I

Hg(x,O := L H gjJ3 ji (Xji (x, 0) . ZPi(X), i=l

where Bji is a gji-admissible basis (cf. 2.7). 2. Let M be an (M, A, g)-admissible weight function of type (p). We call M

h-;l-adapted, respectively, H;;l- adapted, if with suitable constants C > 0 and N E N the estimate C- 1 h r; :s: M :s: C h -; N , respectively, C- 1 H:: :s: M :s: CH;;N, holds.

Remark 4.17. If (X, X, K) E A, then there exists a constant C > 0 such that for all x E X and all ~ E r;x the estimate C-1hg(x,~) :s: h(X-l)'g(X(x,~)) :s: Chg(x,~) holds. If hg is defined with another (M, A, g)-admissible partition of unity, then, with a suitable constant C > 0, we have C- 1h g :s: hg :s: Chg. Moreover, hg is an (M, A, g)-admissible weight function of type (p). The same remark holds for Hg .

Moreover, up to multiplicative constants, the definition of Hg does not depend on the choice of Bji for i = 1, ... , l.

Definition 4.18. Let M be a Coo-manifold with a complete atlas A. If to every coordinate system (X"' K,) E A we are given a distribution u" E 'D'(K(X,,)) such that

((XK' K,), (XFi' ti;) E A), we call the system a distribution u in M and write (K- 1 )*u := UK' The set of all distributions in M is denoted by V'(M).


Remark 4.19. Let (Xj, K,j)j=l, ... ,rn be a system of charts of M such that M = U;:l Xj' Let (CPi)i=l..,1 be a partition of unity subordinate to (Xj )j=l, ... ,rn (i.e.,

L.:;;=1 CPj = 1 and for all i E {I, ... , l} there exists ji E {I, ... , m} such that supp (CPi) C Xj,}. Then the following statements are equivalent:

U E D'(M) ¢=} CPiU E D'(M) for i = 1, ... , l

¢=} CPiU E D'(XjJ for i = 1, ... , l

¢=} (K,j,l)*(CPiU) E D'(K,JiCXjJ) for i = 1, ... , l .

In the last equivalence we have applied [28, Theorem 6.3.4].

Definition 4.20. Let (M, A, g, g) be an admissible structure of type (p). Let H g

be as in 4.16. For s E JR we define H (Hg,g;M,A) as the set of all U E D'(M) such that

(K,-l)*(cpU) E H (x- 1 )*(Ht), (X- 1 )*g; K,(Xg))

for each (X, X, K,) E A and each (X, X, K,)-admissible cut-off function cP, where we have applied the notation from 4.10, 4.11, and 4.14.

Remark 4.21. Obviously H (Hg,g;M,A) is a vector space and we have

for s' ::; s.

Lemma 4.22. We apply the notation of the preceding definition. Let u E D'(M), (X, X, K,) E A, and let cP be an (X, X, K,)-admissible cut-off function. If we have

(K,-l)*(cpU) E H (x- 1 )*(Ht), (X- 1 )*g; K,(Xg)) ,

then cpu E H (Hg, g; M, A).

Proof. This is a consequence of 4.7. o

Corollary 4.23. We apply the notation from the preceding definition and from 4.14. Let (CPi)i=l, .. ,1 be an (M,A,g)-admissible partition of unity. For u E D'(M) we have u E H (Hg, g; M, A) if and only if for all i = 1, ... , l,

(K,j,l)*(CPiU) E H (Xj,l)*(Ht), (Xj,l)*g; K,ji(Xji,g))

Corollary 4.24. Let (M, A, g, g), (M, A, g, g), and (M, A', g, g) be admissible structures of type (p). If A::; A and A ~ A', then we have H (Hg,g;M,A)

H (Hg,g;M,A) = H (Hg,g;M,A') for each s E JR.

Proof. This is a consequence of the preceding corollary together with 4.4. 0

230 F. Baldus

Definition 4.25. Let (M, A, Q, g) be an admissible structure of type (p). Let r, s E

IR. We call a linear operator P: H (H;,g;M,A) -+ H (H;,g;M,A) bounded in charts if for all (X, X, ""), (X, X, K,) E A, all (X, X, "" )-admissible cut-off functions 'P, and all (X, X, K,)-admissible cut-off functions cp the operator

H ((:\:-1)* (H;), (X- 1 )* g; K,(X g)) -+ H ((x- 1 )* (H;), (X- 1 )* g; ",,(Xg))

u f-7 (",,-1)* ['PP (cp. K,*u)]

(which is well defined by virtue of 4.22) is continuous.

Remark 4.26. 1. Let (M, A, Q, g) be an admissible structure of type (p). Let r, s, t E IR. More

over, let P: H (H;,g; M, A) -+ H (H;, g; M,A) and Q : H (H;,g; M, A) -+

H (H~,g;M,A) be bounded in charts. Then QP : H (H;,g;M,A) -+

H (H~, g; M, A) is bounded in charts, too. 2. If P : H (H;, g; M, A) -+ H (H;, g; M, A) is bounded in charts, then P is

uniquely determined by its actions on Co(M).

Lemma 4.27. Let (M, A, Q, g) be an admissible structure of type (p), let (X, X, ""), (X, X, K,) E A, and let 'P be an (X, X, ",,)-admissible, cp be a (X, X, K,)-admissible cut-off function. Let r, s E IR and let P : Co(M) -+ D'(M) be an operator such that

H ((x- 1 )* (H;), (X- 1 )* g; K,(Xg)) -+ H ((x- 1 )* (H;), (X- 1 )*g; ",,(Xg))

u f-7 (",,-1)* ['PP (cp. K,*u)]

is continuous and extends to a continuous map S' (IRn) -+ S' (IRn). Then 'PPcp : H (H;, g; M, A) -+ H (H;, g; M, A) is bounded in charts.

Proof. This follows from 4.22 and the closed graph theorem. o Definition 4.28. Let (M, A, Q, g) be an admissible structure of type (p). By £(g; M, A)-= we denote the set of all operators P : Co(M) -+ V'(M) which have for each s E IR an extension Ps mapping H (H;, g; M, A) bounded in charts to

H (Hs+s' g. M A) for all s' E IR 9 " , .

Lemma 4.29. Let (M, A, Q, g) be an admissible structure of type (p). Let (X, X, "") E

A and let 'P be an (X,X,,,,,)-admissible cut-off function with associated cut-off function 1jJ (cf. 4.14). Let P : Co(M) -+ V'(M), let s' E IR, and let p E

S ((x- 1 )*(Hf), (x- 1 )*g) such that

Pu = ",,* [(('P 0 ",,-1) . p) (x, D)(",,-l)* (1jJu)]

for U E Co(M). Then, for each s E IR, the operator P extends to an operator

Ps :H(H;,g;M,A) -+H(H;-S',g;M,A)

which is bounded on charts.

Proof. The assertion is a consequence of 4.27. D

As canonical domain of the pseudo-differential operators to be introduced in the following section we define the space SCM).

Definition 4.30. Let (M, A, 9) be a generalized SG-compatible manifold. We define SCM) as the set of all functions u E eOO(M) such that, for all (X, X, "') E A and all functions cp E COO(M) with the properties that

• supp (cp) <;;;; X and • cp 0 ",-1 is a function on JRn with all derivatives of order?: 0 being at most of

polynomial growth,

we have (",-1 )*(cp. u) E S(JRn).

5. Pseudo-differential operators on manifolds

Following the usual approach, we define w(M, g; M, A, 9) - up to negligible terms in W (g; M, A, 9) - 00 - as the set of all operators ego (M) ----+ V' (M) with representations as pseudo-differential operators in local coordinates (in the corresponding classes). One important result and tool in the following is the conclusion from one local representation to those in other coordinates (cf. 5.12). In the proof we mainly apply the cut-off functions from 3.2 and furthermore Theorem 3.12 for the transformation of the part of the operator with Schwartz kernel being supported near the diagonal. In this theorem we also calculate the principal symbol of the operator (cf. also section 6). Moreover, we discuss mapping properties of pseudo-differential operators in the scale H (H;,g;M,A) (8 E JR) of Sobolev spaces (cf. 5.14), and prove that the composition of two pseudo-differential operators is such an operator, too (cf. 5.17). We have decided to describe the operators in W(g; M, A, 9)-00 with the help of mapping properties of iterated commutators with distinguished vector fields and multipliers as operators of order -00 in the scale of Sobolev spaces H (H;,g;M,A) (8 E JR) (cf. 5.4). This description is motivated by characterizations of pseudo-differential operators as in [1, 3.6.6]. Nevertheless, we can prove that for an appropriate choice of the cut-off functions the remainder terms have integral representations with rapidly decreasing kernels (cf. 5.18, 5.19; cf. also 6.13).

Definition 5.1. Let (M, A, 9) be a generalized SG-compatible manifold. We define Mc.A := U(X.x.I<)EA X \ X.

Definition 5.2. Let (M, A, 9, g) be an admissible structure of type (p). 1. By Vrn (M, A, 9) we denote the set of all operators of the form

",* [MXj(",-l)*(¢. u)]

where U E COO(M), j E {1, ... ,n}, (Mxjf)(x) = xjf(x) for f : JRn ----+

C, (X, X, "') E A, and ¢ E eOO(M) is a function satisfying the following properties:

232 F. Baldus

• There exists an atlas A such that (M, A, Q, g) is an admissible structure of type (p), too, A ::; A, and 1> is an (X,X,/i,Q)-admissible cut-off function, where (X, X, /i) E .A.

• d1> = 0 on M \ Me,A. 2. By Vd(M, A, Q) we denote the set of all operators of the form

/i* [D Xj (/i-I)*(1>' u)]

where we have applied the notation from l. 3. We define V(M, A, Q) = Vm(M, A, Q) U Vd(M, A, Q).

Lemma 5.3. Let (M,A,Q,g) be an admissible structure of type (p).

1. For u E '[)'(M), rp E S(M) and, V E V(M, A, Q) we have Vu E ,[)'(M) and Vrp E S(M).

2. Let u E ,[)'(M) such that VI ... VNU E H(l,g;M,A) for all N E No and all VI"'" VN E V(M,A,Q). Then we have u E S(M). More precisely, ifu E ,[)'(M) such that VI ... VkVI'" Viu E H(l,g;M,A) for all k,l E No and all operators VI ... Vk E V d (M, A, Q) and all operators Vi ... Vi E V m (M, A, Q) , then u E S(M).

Proof. For the proof of Part 2 we apply Sobolev's embedding theorem. 0

Definition 5.4. Let (M, A, Q, g) be an admissible structure of type (p). We define the class of negligible \If(g; M, A, Q)-oo as the set of all operators P : CO'(M) ---7

,[)'(M) with the property that for all N E No and all VI, ... ,VN E V(M,A,Q) the operator4 ad(Vd ... ad(VN)P : CO'(M) ---7 ,[)'(M) is in £(g; M, A)-oo. In the case Q = RI(g) we write \If(g;M,A)-oo.

Lemma 5.5. Let (M, A, Q, g) and (M, A, Q, g) be admissible structures of type (p) such that A ::; A. Let P E \If(g;M,A,Q)-oo. Moreover, let (X,X,/i) E

A and let rp,'ljJ be (X,X,/i)-admissible cut-off functions. Then there exists p E

nqENS(X-I)*(HZ),(X-I)*g) such that

rpP(1jJu) = /i* [(rp 0 /i-I). p) (x, D)(/i- 1 )* ('ljJu)]

for u E CO'(M).

Proof. This is a consequence of the definition together with [I, 3.6.6]. 0

Lemma 5.6. Let (M, A, Q, g) and (M, A, Q, g) be admissible structures of type (p) with A ::; A. Let (X,X,/i) E A and let rp be an (X,X,/i,g)-admissible cut-off function with associated cut-off function 'ljJ (cf. 4.14). Let P : CO'(M) ---7 D'(M), let s E JR, and let pES (X- 1 )*(H;), (X- 1 )*g) such that

Pu= /i* [«rpO/i-I). p) (x,D)(/i- I )* ('ljJu)]

4As usual, we define ad(V)P:= VP - PV : CO'(M) -> D'(M) and apply this notation also to unique extensions.

for u E cO" (M). Then, for all N E N and all VI , ... , V N E V (M, A, 9), there exist SN E JR. and p(VI , ... , VN ) E S ((X-I)*(HZ-SN)IT*X, (X-I)*9IT*X) such that

ad(VI) ... ad(VN )Pu = K,* [(('PI 0 K,-I) . p(VI , ... , VN)) (x, D)(K,-I)* (-zPIU)]

for u E C(f(M) and for all (X, X, A, g)-admissible cut-off functions 'PI with associated cut-off functions -zPI such that 'PI = 1 on supp ('P) and -zPI = 1 on supp (-zP).

Proof. For one commutator this follows from the behavior of the remainder term in the asymptotic expansion of the product symbol. More precisely, in the case of an operator in Vrn(M, A, Q) one uses an approximation argument. Then the complete assertion follows inductively. For a detailed proof we refer to [2, 2.10.10]. D

Corollary 5.7. We apply the notation from 5.6. Let P : qj(M) ----+ D'(M), and p E nqEN S ((X- I )*(H$), (X- I )*g) such that

Pu = K, * [( ( 'P 0 K, -1) . p) (x, D)( Ii -1)* (-zPu)]

for u E C(f(M). Then we have P E 1JI (g;M,A,Q)-oo.

Proof. This is a consequence of 5.6, 4.29, and the fact that £(g; M, A)-OO <;;; £(g;M,A)-oo. D

Lemma 5.8. Let (M, A, Q, g) be an admissible structure of type (p). Moreover, let (X, X, Ii), (X, X,;;;;) E A, let'P be an (X, X, Ii)-admissible cut-off function, let tj; be a (X,X,;;;;)-admissible cut-off function, and let k E S(JR.2n). We define (u E

Co(M))

Pu := K,* ['P 0 Ii-I ·1" k(·, y) . (tj;. U)(;;;;-I(y))dY]

Then we have P E 1JI (g; M, A, 9) -00 .

Proof. The assertion follows with 4.27. D

Now we define pseudo-differential operators, associated to a metric 9 and a weight function M, on M or, more precisely, on the generalized SG-compatible manifold (M, A, 9).

Definition 5.9. Let (M, A, Q, g) be an admissible structure of type (p). Let M be an (M,A,g)-admissible weight function of type (p) which is H;I-adapted (cf. 4.16). By 1JI(M, g; M, A, Q) we denote the set of all operators P : C(f(M) --+

D'(M) such that for all (X,X,Ii) E A, for each (X,X,K"g)-admissible cut-off function 'P, and for each associated cut-off function -zP there exists a symbol p E S((X-I)*M,(X-I)*g) such that

'PP(-zPu) K,* [(('P 0 K,-I). p) (x, D)(K,-I)* (-zPu)] (u E Cgo(M)),

ipP(l ~ -zP) E 1JI(g; M, A, 9)-00 .

In the case Q = RI(g) we write 1JI(M,g;M,A).

Remark 5.10. Obviously 1JI(M, g; M, A, Q) is a vector space and we have W(g; M, A,9)-00 <;;; 1JI(M, g; M, A, 9).

234 F. Baldus

Corollary 5.11. Let P E \It (M,g;M,A,Q). Moreover, let (fJ,,,j; E COO(M) with cp. "j; = 0 such that [(x,~) t--7 (fJ 0 K:- 1 (x)], [(x,O t--7 "j; 0 K:- 1 (x)] E S (1, (X- 1 )* gIT*X) for all (X, X, K:) E A. Then we have (fJP"j; E \It (g; M, A, 9)-00.

Proof. Fixing an (M, A, g)-admissible partition of unity with associated system of cut-off functions, 5.7 yields the assertion. D

The following theorem is the "work horse" in our discussion of the operators in \It(M, g; M, A, Q).

Theorem 5.12. Let (M, A, Q, g) be an admissible structure of type (p). Let M be an (M,A,g)-admissible weight function of type (p) which is H;l- adapted. Let (X, X, K:) E A, let r.p be an (X, X, K:, g)-admissible cut-off function, and let 'lj; be an associated cut-off function. Let P : Co(M) -7 D'(M) be an operator such that there exists a symbol pES ((X- 1 )* M, (X- 1 )* g) with

r.pP('lj;u) = K:* [((r.p 0 K:- 1 ). p) (x, D)(K:- 1 )* ('lj;u)] (u E Cg"(M)),

r.pP(1- 'lj;) E \It(g;M,A,Q)-CXJ.

Then r.pP E \It(M,g;M,A,Q). Moreover, if (X,X,i;,) E A and_(fJ is a (X,X,i;"g)admissible cut-off function with associated cut-off function 'lj;, then there exists pES ((:X- 1)* M, (:X-1)*g) such that for each u E Co(M) we have

(fJr.pP("j;u) = i;,* [(((fJOi;,-l) ,p) (x,D)(i;,-l)* ("j;u)]

where

(fJ( i;, -1 (x)) . p( x,~) = (r.p. (fJ) (i;, -1 (x)) . p ( (K: 0 i;, -1) (x), t d (K: 0 i;, -1 (x) r 1 ~) + r( x, 0

((x,~) E i;,(X) x IRn) with rES ((;X- 1 )*(Mhg), (X- 1 )*g).

Proof. As in the proof of 3.13, applying 3.6, 3.2, 3.8, and 3.12 we show that r.pP'lj; E \It(M,g;M,A,Q) and thus, r.pP E \It(M,g;M,A,Q). For the last assertion we have to apply 3.12 once more. For details we refer to [2, 2.10.27]. D

The preceding theorem allows us to characterize pseudo-differential operators on an admissible structure (M, A, Q, g) of type (p) by the help of a fixed (M, A, g)admissible partition of unity and an associated system of cut-off functions.

Lemma 5.13. Let (M, A, Q, g) and (M, A, Q, g) be admissible structures of type (p) with A ::; .A. Let M be an (M, A, g)-admissible weight function of type (p)5 which is H;l- adapted. Let (X,X,K:) E A, (X,X,K:) E A, X <,;;; X, let r.p be an (X,X,K:)

admissible cut-off junction, let "j; be an (X, X, K:) -admissible cut-off function, let p E S( (X- 1 )* M, (X- 1 )* g), and let P : Cg"(M) -7 D' (M) be an operator such that

r.p P ( "j;u) = K: * [( ( r.p 0 K: -1) . p) (x, D) ( K: -1 ) * ( "j;u ) ]

Then we have r.pp"j; E \It(M,g;M,A,Q).

5Then M also is an (M,A,g)-admissible weight function of type (p).

Proof. Let 'ifJ be an (X, X, K, )-admissible cut-off function with 'Ij; = 1 on supp (rp). Then, by virtue of 5.12, we have rpp;[;'ifJ E w(M,g;M,A,9). Moreover, again by virtue of 5.12, we have rpp;[; E w(M, g; M, A, Q) and thus, by virtue of 5.11, rpp;[;(1- 'ifJ) E W(g;M,A,Q)-=. Then 5.5 and 5.7 yield rpP;[;(l- 'Ij;) E w(g;M, A, Q)-=. 0

Remark 5.14. Let (M, A, Q, g) be an admissible structure of type (p). Let s M E lR and let P E W(H;M, g; M, A, 9). Then, by virtue of 4.29, for each s E lR, P extends to an operator H (H;, g; M, A) ----+ H (H;-SM, g; M, A) which is bounded in charts. Moreover, P extends canonically to an operator SCM) ----+ SCM).

Lemma 5.15. Let (M,A,Q,g) be an admissible structure of type (p). Let s E lR, P E W(H;,g;M,A,Q), N E 1'1, and let V1, ... ,VN E V(M,A,9). Then there exists SN such that ad(Vl)'" ad(VN)P E W(H;-SN,g;M,A,9).

Proof. This is a consequence of 5.6 and 5.12. o Corollary 5.16. Let (M, A, Q, g) be an admissible structure of type (p). Let s E lR, P E W(H;,g;M,A,9), and Q E w(g;M,A,9)-=. Then we have PQ,QP E

W(g; M, A, Q)-=.

Theorem 5.17. Let (M, A, Q, g) be an admissible structure of type (p). For i = 1, 2 let Mi be an (M, A, g) -admissible weight function of type (p) which is H; 1_ adapted. Moreover, for i = 1, 2 let Pi E W (Mi' g; M, A, 9). Then we have PI P2 E

IJ!(MIM2' g; M, A, 9).

Proof. Let (rpi)i=l. .. ,l be an (M, A, g)-admissible partition of unity with associated system of cut-off functions ('ifJi)i=l,.,l' By virtue of 5.11 and 5.16 it suffices to consider rpiH'ifJirpkP2'ifJk for i, k E {I, ... , l}. With the notation from 4.14, let 'Pk be an (Xjk , X jk , K,jk' g)-admissible cut-off function with associated cut-off function ;[;k such that ;[;k = 1 on supp (rpk) and 'Pk = 1 on supp (;[;k). Then we have

'PiPl ('ifJi'Pk P2'ifJk) = ('Pk'PiPl'ifJi;[;k + (1- (h)'PiPl'ifJi;[;k) rpkP2'ifJk .

By virtue of 5.11 and 5.16 we have ((1- 'Pk)'PiPl'ifJi;[;k) ('PkP2'ifJk) E \lI(g;M,A,

9)-=. The remaining part ('Pk'PiPl'ifJi;[;k) ('PkP2'ifJk) is seen to be in IJ!(M1M 2,g;

M,A,Q) when being considered in local coordinates (cf. 5.12). 0

We finish this section with a refined discussion of the smoothing terms appearing in the calculus. More precisely, we show that we can restrict ourselves to operators having kernel representations in local coordinates with rapidly decreasing kernels when admitting only special cut-off functions.

Theorem 5.18. Let (M, A, Q, g) be an admissible structure of type (p). There exists 0< 1'0 < l' A and for all (X, X, K,) E A there exists a function;;; E C=(M) such that for 0 < l' < 1'0 and for all (X,X,K"g,r)-admissible cut-off functions rp the following statements hold:

236 F. Baldus

• ~ = 1 on supp(cp) and supp(~) <;;; K- 1 (K(X)p,(x-1)*gr (<;;; X) where X :=

K- 1 (K(X) p,(",-l )*Q rand p, p > 0 are sufficiently small 6 .

• [(x,.;) ...... ~(K-1(X))l E s (1, (X-1)*g). • VP E w(gjM,A,g)-oo, V(X,X,~) E A, Wp (X,X,~,g)-admissible cut-off

junction, 3k1 , k2 E S(1R2n ), Vu E COO(M),

cpP ((1- ~)(<pu))

<j;(1 - ~)P(cpu)

K* [cp(.) 'In k 1 (·,y)· (<j;. U)(~-l(Y))dY] ,

~* [<j;O 'In k2 (·,y)· (cp. U)(K-1(Y))dY] .

Proof. Let A = {(Xj,Xj , Kj) I j = 1, ... , m}. We apply the notation from 4.11. Let o < p < r A/2. For j E {1, ... , m} we define Xj := Kjl (Kj(Xj)p,Qjr and Xj :=

Kjl (Kj(Xjhp,Qjr. Moreover, let A:= {(Xj,Xj,Kj) I j = 1, ... ,m}. By virtue

of 4.13, (M, A, g) is a generalized SG-compatible manifold and thus, (M, A, g, g) is an admissible structure of type (p). Let r A be the corresponding constant (cf. 4.11) and let 0 < p < r A' For j E {1, ... , l} let Xj := Kjl (Kj(Xj)p,gjr. We fix

j E {1, ... , l} and let",:= Kj, X := Xj, X := Xj, X := Xj' and X:= Xj' Then we

have X <;;; K- 1 (K(X)p,(x-1)*gr <;;; X <;;; X <;;; X <;;; X. By virtue of 3.2 there exist

an (X,X,K,g)-admissible cut-off function cp with supp(cp) <;;; X, an (X,X, ""g)admissible cut-off function </J with supp (</J) <;;; X, </J = 1 on X, and an (X, X, K, g)admissible cut-off function ~ with supp (~) <;;; X and ~ = 1 on supp (</J). Moreover, let (X, X, ~) E A, let <j; be a (X, X, ~, g)-admissible cut-off function, and let <1 be

a (X, X,~, g)-admissible cut-off function with </J = 1 on supp (<j;) U X. We define

¢ := (1 - </J) . </J. Then we have 1 - </J = 1 ~n supp (1 - ~), in particular on

supp ((1 -~) . <j;), and <1 = 1 on supp (<j;), in particular on supp ((1-~) . <j;). Thus

¢ = 1 on supp((1-:;j). <j;), ¢ = 0 on supp(cp), and ¢ is a (X,X,~,g)-admissible cut-off function. Moreover, we have d¢ = 0 on M \ Me,A' By definition, for all N E No and all VI"'" VN E V(M, A, g), the operator

is bounded. For j E {l, ... ,n} and U E COO(M) let

V1(j)u .- K* [MXj(",-l)*(</J' u)] , V?)u := K* [DXj(K-1)*(</J. u)] ,

V?)u .- ~* [MXj(~-l)*(¢.u)] , V2(j)u := ~* [DXj(~-l)*(¢.U)]

6In particular, there exists an atlas .4 such that (M,.4, g, g) is an admissible structure of type (p), A::;.4, (X,,:r, K) E.4, and;jj is an (X,,:r, K,g)-admissible cut-off function.

s (M, g)-calculus on Manifolds 237

(Id + ~ (ad (Vl(jl) ) 2) =1 (Id + ~ (ad (V2(jl) ) 2) =2

X (Id + ~ (ad (V2(jl)r) m2 (Id + ~ (ad (Vl(jl) r) m1 (~p (1 _~)~)

(Id+ ~ (V?l) 2) =1 (Id + ~ (vyl) 2) =2 (~p (1 _~)~)

x (Id+ ~ (V1(j)r) m1 (Id+ ~ (VYlr) m2 .

Thus, the operator L2(JRn) ----> L2(JRn),

u 1-+< Mx >2m1 < Dx >2m2 (~-1)* [cpp (( 1 -,;j;) 0A;* [< Mx >2m1 < Dx >2m2 uJ)] ,

is bounded. Hence, by virtue of [1, 3.6.6], the operator

Fu := (~-1)* [cpp (( 1 -,;j;) 0A;*u)]

(u E S(JRn)) is in n 111 (M-q1 M-q2 gl,l) where q1,q2EN 1 2'

1,1 '= Ildxll; + Ild~lI; M ( C) .=< > and g(x,t;)' < x >2 < ~ >2' 1 x,... x,

Since nQ1 ,Q2EN S (M1Q1 M:;Q2 ,gl,l) = S(JR2n), this yields the assertion concerning

cpP(l - ,;j;)0. For 0(1 - ,;j;)pcp the proof is analogous. 0

Corollary 5.19. Let (M, A, Q, g) be an admissible structure of type (p). Let M be an (M, A, g)-admissible weight function of type (p) which is H;;l- adapted. Let (CPi)i=l, ... ,1 be an (M,A,g,r)-admissible partition of unity, where 0 < r < ro (cf. 5.18).

1. For i E {I, ... , l} let ,;j;i E COO(M) be the function from 5.18 corresponding to CPi. Then: '<IP E 1I1(M,g;M,A,Q), 3Pi E S((Xj.1)*M,(Xj.1)*g), '<Iv E

{I, ... , l}, 3ki ,v E S(JR2n), '<Iu E CQ"(M)

CPi' ~;i [pi(x,D)(~j.1)*(,;j;i' u)] , (5.1)

I

?;CPi' ~;i [Ln ki,v("y), (CPv' u) (~.i.,l(y)) dY]. (5.2)

2. Let ,;j;i' i = 1, ... , l, be the functions from part (a). If P : CO'(M) -+ D'(M) has a representation (5.1), (5.2), then we have P E 111 (M, g; M, A, 9).

Proof. The first assertion follows with 5.18 and for the second assertion we apply 5.8 and 5.13. 0

238 F. Baldus

Corollary 5.20. Let (M,A,Q,g), (M,A,Q,g), and (M,A',Q,g) be admissible structures of type (p) with A :::::: A and A ~ A' . Moreover, let M be an (M, A, g)admissible weight function of type (p)7 which is H;;I-adapted. Then we have \I!(M, g; M, A, Q) = \I!(M, g; M, A, 9). If M is an (M, A', g)-admissible weight function of type (p) which is H;;I-adapted, then we have \I!(M, g; M, A, 9) = \I!(M, g; M, A', 9).

Proof. The fact that \I!(g;M,A,Q)-CXl ~ \I!(g;M,A,Q)-CXl yields \I!(M,g;M,A, Q) ~ \I!(M, g; M, A, Q). Conversely, 5.19 together with 5.8 and 5.13 yields \I!(M, g; M,A,Q) ~ \I!(M,g;M,A,9). The equation w(M,g;M,A,9) = \I!(M,g;M,A' , Q) is a consequence of 5.19 together with 5.8 and 5.13, too. 0

Corollary 5.21. Let (M, A, Q, g) and (M, A, Q, g) be admissible structures of type (p) such that 9 :::::: g. Let M be an (M, A, g)- and(M, A, g)-admissible weight function of type (p) which is H;;I_ and H;;I-adapted. Then w~ have \I!(M, fl.; M, A, Q) ~ w(M, g; M, A, 9) . -

Proof. By virtue of 4.13 there exists an atlas A such that (M, A, Q, g) is an admissible structure of type (p), A:::::: A, and for each (X,X,K) E A, each (X,X,K,fl.)admissible cut-off function is an (X, X, K, g)-admissible cut-off function where (X, X, K) E A. Then the assertion is a consequence of 5.19 together with 5.8, 5.13, and 5.20. 0

6. Symbol-homomorphism and *-structure

In this section, we prove the existence of a symbol-homomorphism mapping an operator P to its principal symbol O"p,M(P) with the property that O"p,M1 M 2(PI P 2 ) =

O"p,Ml (Pd' O"p,M2 (P2) (cf. 6.5 and 6.6). Furthermore, we introduce an invariant measure (Xc on M such that H(1, g;

M, A) = L2(M, (Xc), chosen in such a way that P E W(1, g; M, A, 9) implies P* E \I!(1, g; M, A, 9) (cf. 6.14). The density (Xc is induced by a so-called (M, A, g)admissible Riemannian metric G (cf. 6.7, 6.10). With the help of this measure we can write the remainder term of a pseudo-differential operator as an integral operator with rapidly decreasing kernel (cf. 6.13).

Definition 6.1. Let (M, A, Q, g) be an admissible structure of type (p) and let M be an (M, A, g)-admissible weight function of type (p). We define S (T* M, A; M, g) as the set of all functions a E CCXl(T* M) such that we have

(x-I)*a E S ((X- I )* M, (X-I)*g)

for all (X,X,K) E A, where

x: T* X -+ T*]Rn ~ ]R2n, T;X 3'; f-+ (K(x)/dK(x)-I.;).

7Then M also is an (M, A, g)-admissible weight function of type (p).

8(M, g)-calculus on Manifolds 239

Theorem 6.2. Let (M, A, Q, g) be an admissible structure of type (p) and let M be an (M, A, g) -admissible weight function of type (p) which is H; I-adapted.

1. To each operator P E w(M,g;M,A,9) there exists p E 8(T*M,A;M,g) with the property that for (X,X,K) E A and for an (X,X,A, g)-admissible cut-off function cP with associated cut-off function 'l/J there exist symbols

q E 8 (X-1 )* M, (X- 1 )*g) and r E 8 (X- 1 )*(Mhg), (X- 1 )*g)

such that

cpP( 'l/Ju) = K* [«cpo K-1) . q)(x, D)(K-l)* ('l/Ju)]

foru E Co(M), and for (x,~) E K(X) x]Rn we have

cp(K-1(X)) ·q(x,~) = cp(K-1(X)) 'P(K-l(x),tdK-l(X)-l~) +r(x,~).

2. If Pl,P2 E 8 (T* M, A; M, g) have the properties of p, then

PI - P2 E 8(T*M,A;Mhg,g).

3. If for i = 1,2, Mi is an (M,A,g)-admissible weight function of type (p) which is H;l-adapted, Pi E w(Mi,g;M,A,Q), Pi E 8(T*M,A;Mi,g) is the corresponding symbol of Part 1, and if c E <C, then

is such a symbol for the operator cPI + P2.

Proof. Let (cpi)i=I .... ,1 be an (M, A, g)-admissible partition of unity with associated system of cut-off functions ('l/Ji)i=l"",l. We apply the notation from 4.14. For i E {1, ... ,l} there exists qi E 8(X~1)*M,(X~1)*g) such that we have for u E Co(M):

CPiP('l/JiU) = Kji [((CPi 0 K~l). qi)(x,D)(K~I)* ('l/JiU)] .

We define Pi := Xji (CPi 0 K~I) . qi) (E 8 (T* M, A; M, g)). By virtue of 5.12, Pi

has the asserted properties with CPiP instead of P. Then p := L:~=I Pi has the asserted properties. The other assertions are obvious. D

Definition 6.3. The theorem induces a well-defined linear mapping

w(M,g; M, A, Q) --+ 8 (T* M, A; M, g) /8 (T* M, A; Mhg, g) , P f--+ CTp,M(P)

of an operator P to its principal symbol CTp,M(P).

Lemma 6.4. Let (M,A,Q,g) be an admissible structure of type (p) and let M be an (M, A, g) -admissible weight function of type (p) which is H; I-adapted.

1. For P E w(M,g;M,A,Q) we have

CTp,M(P) = 0 if and only if P E w(Mhg,g;M,A,Q).

240 F. Baldus

2. Let pES (T* M, A; M, g). Then there exists P E 1l!(M, g; M, A, Q) such that p has the properties of 6.2 1. More precisely, P can be chosen such that the following statement holds: For each (X, X, K,) E A and each (X, X, K" g)admissible cut-off function cp, there exists an (X, )(g, K" g)-admissible cut-off function fj5 (cf. 4.11, 4.13) such that fj5 = 1 on supp(cp) and cpP = cpPfj5, Pcp = fj5P'f!.

Proof. The first assertion is obvious. To prove Part 2, let pES (T* M, A; M, g). We apply the notations from 3.2 with (Uj,Uj , RI(gj)) (cf. 4.11) instead of (U,U,Q), the constants p, PI, ... ,P4 being chosen simultaneously for all j E {I, ... , m}. Moreover, we suppose that P4 < r A and 2CA..;c;P < TA erA the constant from

4.11 corresponding to A in 4.13). Let 2CA..;c;P < ro < TA· Let (CPj,Z)ZEUj •P2 ,gj'

('l/Jj Z)zEU . (j = 1, ... , m) be the corresponding families and let , ],P2,9)

Let ('f!j)j=I .... ,rn be an (M,A,g,pd-admissible partition of unity with associated system of cut-off functions ('l/Jj)j=I, ... ,rn (cf. 4.15). Furthermore, for j = 1, ... ,m and u E CaCM), we define

Pj '- fu . (cpj,z . (Xjl)*p) #KN 'l/Jj,z IR I (gj)zI I/2dz, ],P2,9)

Pju .- K,j [pj(x,D)(K,jl)*('l/Jj ·u)] .

By virtue of 5.12, 'f!j . p plays the role of p in 6.2 1. with 'f!jPj instead of P. Let P := L';=I 'f!jPj . Then P has the asserted properties. For details we refer to [2, 2.11.5]. 0

Corollary 6.S. Let (M, A, g, g) be an admissible structure of type (p) and let M be an (M, A, g)-admissible weight function of type (p) which is H;;I-adapted. Then the following statements hold:

1. The mapping

1l!(M,g;M,A,Q) -+ S(T*M,A;M,g)/S(T*M,A;Mhg,g) ,

P f-+ ap,M(P)

is surjective. 2. Ker(ap,M) = w(Mhg,g;M,A,g). This induces a bijective linear mapping

1l!(M,g;M,A,g)/1l!(Mhg,g;M,A,g) -+ S(T* M,A;M,g)/ S(T* M,A;Mhg,g).

Theorem 6.6. Let (M, A, g, g) be an admissible structure of type (p) and let M I , M2 be (M,A,g)-admissible weight functions of type (p) which are H;;I-adapted. For Pi E W(Mi' g; M, A, Q) (i = 1,2) we have

ap,M1M2(PIP2) = ap,M1 (Pd' ap,M2(P2)

which belongs to S(T*M,A;MIM2,g)/S(T*M,A;MIM2hg,g).


Proof. Taking into account the behavior of the remainder term in the asymptotic expansion for the product symbol of the calculus on ]Rn, this assertion is proved directly. For details cf. [2, 2.11.7]. 0

Definition 6.7.

1. Let (M, A, g) be a generalized SG-compatible manifold. We call a Riemannian metric G on M (M, A, g)-admissible if we have

(K-1)*Gt (6,6) = (G(K)(x)6, 6) (x E K(X), 6, 6 E ]Rn)

for all (X, X, K) E A with a positive definite matrix G(K)(X) such that

[x 1--4 G(K)(X)] , [x 1--4 (G(K) (x)) -1] E S (1, (K- 1 )*glx; £(]Rn)) .

2. If (M, A, g, g) is an admissible structure of type (p), then we call an (M,A, R 1(g))-admissible Riemannian metric (M,A,g)-admissible.

Lemma 6.8. If (M, A, 9) is a generalized S G- compatible manifold, then there exists an (M, A, g)-admissible Riemannian metric.

Proof. Let A:= {(Xj, Xj, Kj) I j E {I, ... , m}} and let ('Pi)i=l, ... ,1 be an (M, A, 9)admissible partition of unity. Applying the notation from 4.14, for x E M and 6,6 E TxM we define Gx(6, 6) := :L~=1 'Pi (x) (dliji (x)6, dKji (x)6)· This yields an (M, A, g)-admissible Riemannian metric. 0

Remark 6.9. If G is an (M, A, g)-admissible Riemannian metric, then, for all (X,X, Ii) E A, we have

[x I--4IG(K)(x)1 := det(G(K) (x))] , [x I--4IG(K)(x)I-1] E S (1, (K- 1 )*glx)

and thus [x 1--4 IG(K)(x)1 1/ 2] , [x 1--4 IG(K)(x)I- 1/ 2 ] E S (1, (K- 1)*glx).

Remark 6.10. It is well known that each Riemannian Coo-manifold (M, G) has a canonical non-vanishing density Qo defining an invariant integral on M.

Remark 6.11. Let (M, A, g, g) be an admissible structure of type (p) and let G be an (M,A,g)-admissible Riemannian metric. Then we have H(l,g;M,A) = L2(M,Qo). In particular, H(l,g;M,A) is a Hilbert space.

Lemma 6.12. Let (M,A,9,g) be an admissible structure of type (p). Let G be an (M, A, g)-admissible Riemannian metric. Let K E SCM x M) (cf. 4.9) and K : CaCM) ----* 'D/(M) be defined by

Ku(x) := 1M K(x,y)u(Y)QO(Y) .

Then we have K E \[1(g; M, A, g)-oo.

Proof. Applying an (M, A, g)-admissible partition of unity, the assertion follows from 5.8. 0

242 F. Baldus

Theorem 6.13. Let (M, A, g, g) be an admissible structure of type (p). Let M be an (M,A,g)-admissible weight function of type (p) which is H;;l- adapted. Let G be an admissible Riemannian metric. Let 1"0 as in 5.18, let 0 < 1" < 1"0, and let (<Pi)i=l ... ,1 be an (M, A, g, r)-admissible partition of unity. Let (:J;i)i=l, ... ,1 be an associated family of functions as in 5.19. Then, applying the notation from 4.14, for an operator P : CO'(M) ----+ V'(M) the following statements are equivalent:

1. P E w(M,g;M,A,Q). 2. 3K E S(M x M), Vi E {l, ... ,l}, 3Pi E S((Xj,l)*M,(Xj,l)*g) , Vu E

CO'(M),

Pu 1

L <PiP (:J;i U) + j KC, y)u(y)cxc(y) , where i=l M

<PiP (:J;i U ) KJi [((<Pi 0 Kj,l). Pi) (X,D)(Kj,l)*(:J;iU)]

Proof. To prove "1. =} 2.", let ki,v E S(]Rn x Il~n) as in 5.19 (i,l/ E {I, ... , l}). For

x, y E M we define K(x, y) := I:~,V=l Ki,v(x, y) where

Ki,v(x, y) := 'Pi(X) . ki,v (Kji (x), Kjv (y)) . <pv(y) IG(KOjv)(Kjv (y))1- 1/ 2 .

The implication "2. =} 1." is a consequence of 5.13 and 6.12. o

Theorem 6.14. Let (M, A, g, g) be an admissible structure of type (p). Let M be an (M, A, g)-admissible weight function of type (p) which is H;;l- adapted. Let G be an (M, A, g)-admissible Riemannian metric. For P E w(M, g; M, A, Q) there exists an operator P* E w(M, g; M, A, g) with the property that for u, v E S(M) the equation

(Pu, v) L2(M,a g ) = (u, P*v) L2(M,a g )

holds. The operator P* is called the formal adjoint (with respect to (., "h 2 (M,a g »)'

Proof. The assertion is a consequence of 6.13 together with the corresponding result from the global calculus on ]Rn (cf. [28, 18.5]), where we have to take into account 5.12 and 5.20. 0

7. Examples of algebras of pseudo-differential operators on manifolds

Example. Let M be an SG-compatible manifold in the sense of E. Schrohe [41] with an atlas A = {(Xj,Xj,Kj) I j = 1, ... ,m}. For j E {l, ... ,m} let Uj := Kj(Xj )

and Xj : T*Xj ----+ T*]Rn c::: ]R2n, T;Xj :1 ~ 1--+ (K(x),tdK(X)-l~). Let

where 0::; P1 ::; 1,0 < P2 ::; 1,1 - P2 ::; 6 ::; P2, 6 < 1 and let

where 0 0, V(x,~) EUj x lR,n,

Moreover, let g = R 1(g[1,1,O]). Then the following statements hold:

1. (M, A, g) is a generalized SG-compatible manifold. 2. (M, A, g, gl) and (M, A, g, g2) are admissible structures of type (p). 3. (M, A, R 1(g[1,0]), g[l,O]) (0::; 6 < 1) is an admissible structure of type (p). 4. If P1 > 0, then (M, A, gI) is an admissible structure of type (p), too.

We continue with examples of anisotropic metrics on products of manifolds.

Example. Let M(1) and M(2) be two compact manifolds. For i = 1,2 let g(i) be a continuous metric on M(i) and let A(i) such that (M(i),A(i),g(i)) is an SG-compatible manifold. Let M, A, and g be as in 4.9. Moreover, let "'j1,12 .-

( ",(1) ",(2)) and X· . : T* (x(1) x X(2)) ---+ lR,n, +n2 J1 ' J2' J1,J2 J1 J2 '

(x,~) f-+ ("'jd2(x),td"'j1'h(X)-1~)

(ji E {I, ... , md, i = 1,2). Fors 0 < P1, P2 ::; 1, 1 - min{p1' P2} ::; 61 ::; P1, 61 < 1, 1- min{p1' P2} ::; 62 ::; P2, 62 < 1, and x, ~ E lR,n, t], T1 E lR,n 1 , and t2, T2 E lR,n2 let

With the help of a partition of unity we can construct a metric 9 on T(T* M) such

that'iji E {l, ... ,mi} (i = 1,2), :3C, Vx E "'h,jz (xg) x xg)) , V~ E lR,n 1+n2 ,

C- 1 g(~:~).01,02(-)::; (Xj,~h)*g(x,O(-)::; C g(~:B,01,02(-) .

Moreover, (M, A, g, g) is an admissible structure of type (p).

Example. Let (M(1),A(1),g(1)) and (M(2),A(2),g(2)) be two SG-compatible manifolds of dimension n1 and n2, respectively. Moreover, for 0 < P1, P2 ::; 1,

80bviously 'h = PI = 1/2, 'h = P2 = 2/3 fulfill the assumptions.

244 F. Baldus

1- min{Pl,P2}:::; 01 < PI, 1- min{PI,P2}:::; 02 < P2, and x,~ E ll~n, tl,Tl E lRn1 ,

and t2, T2 E lRn2 let

.- <x, ~>281 Iltlll; + <x, ~>282 Il t211;

+ IITIII; + II T211; <x, ~>2Pl <x, ~>2P2

We apply the notation analogous to that in the preceding example. There exists a metric 9 on T(T* M) such that Vji E {I, ... , mi} (i = 1,2), ::lC, "Ix E

Ii .. (X(l) XX(2)) VCElRn, +n2 J1,J2 J1 J2'" ,

C- l g(~::),81,82(.) :::; (Xh~12)*g(x,e)(-) :::; C g(~:B,81,82(.) .

Moreover, (M, A, 9, g) is an admissible structure of type (p).

Remark 7.1. Let M = Mo U MI U M2 U aMI u aM2, where Mo,Ml,M2 are n-dimensional submanifolds, aMI, aM2 are connected (n - I)-dimensional submanifolds, Mo is relatively compact, aMo = aMI u aM N , for j E {I, ... , N}, M j is diffeomorphic to aM j x (1, 00), and let ° :::; Pi :::; 1 (i E {O, 1,2}), ° < P2 :::; 1, 1- P2 :::; 0 < P2. Applying the notation from the preceding example, with the help of a partition of unity we can construct a metric 9 on M such that the following statements hold:

• (M, A, 9, g) is an admissible structure of type (p) . • ::lC > 0, Vi E {O, 1, 2}, "Ix E Mi, V~ E T;M, C-lgi~i,82,8]:::; g(x,e) :::;

Cg[Pi,P2,8] (x,e) .

In particular, this example shows that situations appear where different Hormander metrics occur in different local models, i.e., in the admissible quadruples.

8. Topological properties of the algebras of pseudo-differential operators

In this section we give an outlook and present results concerning the topological properties of the algebras of pseudo-differential operators constructed so far. More precisely, in order to prove these results, we have to impose stronger conditions on the admissible structures than we have done so far. But the following results are true for all examples given in Section 7 if we always assume that 0 < p, respectively, 01 < PI and 02 < P2· We omit the proofs and refer instead to [2, Sections 3.8, 5.4, and 5.5]. In fact, the conditions concerning the admissible structures imposed in [2] exclude the anisotropic examples. But in the situations occuring there the results can be proved with minor modifications. The most important modification consists of working out the characterizations with V(MI' AI, 91) U V(M2' A2, 92) instead of V(M, A, 9).

In the following, let 9 be one of the metrics in the examples in Section 7 and let M be an admissible weight function which is in local coordinates of the


form (x,~) f-+ (§(x,,o(T)r, where s E JR, T E JR2n \ {O}, and § is the local model corresponding to g.

The first theorem states the existence of order reducing operators. Due to this result we can restrict ourselves in the following to the algebra of operators of order zero.

TheoremS.I. There exist operators P E 1I1(M,g;M,A,Q) and Q E 1I1(1/M,g;M, A,9) such that PQ = Id = QP.

Remark 8.2. In particular, P is an isomorphism H(M, g; M, A) ...... H(l, g; M, A) where the Sobolev space H(M, g; M, A) can be defined as the space of all u E

UsEIRH(H;,g;M,A) such that Au E H(l,g;M,A) for all A E 1I1(M,g;M,A,Q). Since H(l, g; M, A) = L(M, ac) (cf. Section 6), H(M, g; M, A) is a Banach space.

For weight functions M Beal's- and Corde's-type characterizations for 1I1(M, g;M,A,Q) via commutators with operators in V(M,A,Q) can be proved. These, together with the existence of order reducing operators, yield the following two theorems:

Theorem S.3. 111(1, g; M, A, Q) is spectrally invariant in £ (L2 (JR)) , i.e.,

1I1(1,g;M,A,9) n £ (L2(JR))-1 = 1I1(1,g;M,A,Q)-1.

Theorem S.4. For kENo there exists a Banach algebra 1I1k(g; M, A, Q) such that 1I1k+l(g;M,A,Q) '----' 1I1 k(g;M,A,Q), 1I1o(g;M,A,9) = £ (L2(JR)) , and

1I1(1,g;M,A,Q) = n 1I1k(g;M,A,Q) . kENo

In particular, 111(1, g; M, A, 9) is a submultiplicative Frechet algebra which is continuously embedded in £ (L2(JR)).

Summing up, 6.14, 8.3, and 8.4 yield that 111(1, g; M, A, Q) is a submultiplicative 1I1*-algebra in the sense of B. Gramsch [18], [19].

Acknowledgement I am very grateful to Professor B. Gramsch for proposing to me to generalize the S(M, g)-calculus to manifolds and to investigate spectral invariance and submultiplicativity as well as for valuable support and helpful suggestions. I wish to thank Professor E. Schrohe for several hints, explicitly for explaining to me how to simplify an argument in the proof of 4.13 which allowed me to dispense with an additional assumption concerning the admissible manifolds. Moreover, I wish to thank Dr. R. Lauter for many fruitful discussions and some good advice. Finally, I thank Dr. J. Lutgen, Dr. J. M011er, and Dr. O. Caps for useful comments and for correcting my English.

246 F. Baldus

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Submultiplikativitiit der durch Kommutatoren definierten Frechetalgebren. Diplomarbeit, Fachbereich 17-Mathematik, Johannes Gutenberg-Universitiit Mainz, 1996.

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[9] J.-M. Bony and J.-Y. Chemin. Espaces fonctionnels associes au calcul de WeylHormander. Bull. Soc. Math. France, 122:77~118, 1994.

[10] J.-M. Bony and N. Lerner. Quantification asymptotique et microlocalisations d'ordre superieur I. Ann. Sci. Ecole Norm. Sup. 4e serie, 22:377~433, 1989.

[11] H.O. Cordes. A global parametrix for pseudodifferential operators over IRn. Preprint No. 90, SFB 72, Bonn, 1976.

[12] H.O. Cordes. On some C* -algebras and Frechet* -algebras of pseudodifferential operators, volume 43 of Proc. Symp. in Pure Math. - Pseudodifferential operators, pages 79-104. Amer. Math. Soc., Providence, Rhode Island, 1985.

[13] H.O. Cordes. The technique of pseudodifferential operators, volume 202 of London Mathematical Society, Lecture Note Series. Cambridge University Press, Cambridge - London - New York, 1995.

[14] H.O. Cordes and S.H. Doong. The Laplace comparison algebra of spaces with conical and cylindrical ends. In H.O. Cordes, B. Gramsch, and H. Widom, editors, Pseudodifferential operators, volume 1256 of Lect. Notes Math., pages 55~90, Berlin - Heidelberg - New York ~ London - Paris - Tokyo, 1987. Proc. Conf., Oberwolfach/Ger. 1986, Springer-Verlag.

[15] N. Dencker. The Weyl calculus with locally temperate metrics and weights. Ark. Mat., 24:59-79, 1986.

[16] Y.V. Egorov and B.-W. Schulze. Pseudo-differential Operators, Singularities, Applications, volume 93 of Operator Theory, Advances and Applications. Birkhiiuser Verlag, Basel - Boston - Berlin, 1997.

[17] K.O. Friedrichs. Pseudo-Differential Operators, An Introduction. Lecture Notes, Courant Inst. Math. Sci. New York University, 1968.

[18] B. Gramsch. Some homogeneous spaces in the operator theory and \]i-algebras. In Tagungsbericht Oberwolfach 42/81 - Funktionalanalysis: C* -Algebren, 1981.


[19J B. Gramsch. Relative Inversion in der Storungstheorie von Operatoren und wAlgebren. Math. Annalen, 269:27-71, 1984.

[20] B. Gramsch. Oka's principle for special Frechet Lie groups and homogeneous manifolds in topological algebras of the microlocal analysis. In Banach algebras 97 Proceedings, pages 189-204, Berlin - New York, 1998. de Gruyter.

[21] B. Gramsch and W. Kaballo. Multiplicative decompositions of holomorphic Fredholm functions and w*-algebras. Math. Nachr., 204:83-100, 1999.

[22] B. Gramsch, J. Ueberberg, and K. Wagner. Spectral invariance and submultiplicativity for Frechet algebras with applications to pseudo-differential operators and w*-quantization. In Operator theory: Advances and Applications, vol. 57, pages 71-98, Birkhauser, Basel, 1992.

[23] B. Helffer. Theorie spectrale pour des operateurs globalement elliptiques, volume 112 of Asterisque. Societe Mathematique de France, 1984.

[24] L. Hormander. Pseudo-differential operators. Comm. Pure Appl. Math., 18:501-517, 1965.

[25] L. Hormander. Pseudo-differential operators and hypoelliptic equations, volume X of Pmc. Symp. in Pure Math. - Singular Integrals, pages 138-183. Amer. Math. Soc., Providence, Rhode Island, 1966.

[26] L. Hormander. Fourier integral operators I. Acta Math., 127:79-183, 1971. [27] L. Hormander. The Weyl calculus of pseudo-differential operators. Comm. Pure

Appl. Math., 32:359-443, 1979. [28] L. Hormander. The analysis of linear partial differential operators, vol. 3. Springer

Verlag, Berlin - Heidelberg - New York, 1985. [29] J.J. Kohn and L. Nirenberg. An algebra of pseudo-differential operators. Comm.

Pure Appl. Math., 18:269-305, 1965. [30] H. Kumano-go and K. Taniguchi. Oscillatory integrals of symbols of pseudo

differential operators on ]Rn and operators of Fredholm type. Pmc. Japan Acad., 49:397-402, 1973.

[31] R Lauter. Holomorphic functional calculus in several variables and w* -algebras of totally characteristic operators on manifolds with boundary. PhD thesis, Johannes Gutenberg-Universitat Mainz, 1996, Shaker Verlag, Aachen, 1997, 241 pages.

[32] R Lauter and V. Nistor. Analysis of geometric operators on open manifolds: a groupoid approach. Heft 108, Universitat Munster, SFB 478 Geometrische Strukturen in der Mathematik, May 2000.

[33] RB. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidean space. In M. Ikawa, editor, Spectral and Scattering Theory, volume 162 of Lecture Notes in Pure and Applied Mathematics, pages 85-130, New York, 1994. Marcel Dekker Inc. Proceedings of the Taniguchi International Workshop held in Sanda, November 1992.

[34] RB. Melrose. Geometric scattering theory. Cambridge University Press, 1995. [35] L. Nirenberg. Pseudo-differential operators. Amer. Math. Soc. Symp. Pure Math.,

16: 149-167, 1970. [36] C. Parenti. Operatori pseudo-differenziali in]R" e applicazioni. Ann. Mat. Pura Appl.

(4), 93:359-389, 1972. [37] B.A. Plamenevskij and V.N. Senichkin. Pseudodifferential operators on manifolds

with singularities. Funct. Anal. Appl., 33(2):154-156, 1999.

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[38] B.A. Plamenevskij and V.N. Senichkin. A class of pseudodifferential operators in lR,7n

and on stratified manifolds. Mat. Sb., 191(5):725-757, 2000. [39] Yu. Safarov. Pseudodifferential operators and linear connections. Proc. London

Math. Soc., 74(3):379-416, 1997. [40] E. Schrohe. Komplexe Potenzen elliptischer Pseudodifferentialopemtoren. PhD the

sis, Johannes Gutenberg-Universitat Mainz, 1986. [41] E. Schrohe. Spaces of weighted symbols and weighted Sobolev spaces on manifolds.

In H.O. Cordes, B. Gramsch, and H. Widom, editors, Pseudo-differential operators, volume 1256 of Lect. Notes Math., pages 360-377, Berlin - Heidelberg - New YorkLondon - Paris - Tokyo, 1987. Proc. Conf., Oberwolfach/Ger. 1986, Springer-Verlag.

[42] E. Schrohe. A \If. -algebra of pseudo-differential operators on noncompact manifolds. Arch. Math., 51:81-86, 1988.

[43] E. Schrohe. Boundedness and spectral invariance for standard pseudo-differential operators on anisotropically weighted Lp-Sobolev-spaces. Integral equations operator theory, 13(2) :271-284, 1990.

[44] E. Schrohe. Frechet algebra techniques for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance. Math. Nachrichten, 199:145-185, 1999.

[45] M.A. Shubin. Pseudodifferential operators in lR,n. Sov. Math. Dokl., 12, Ser. Nl, pages 147-151, 1971.

[46] M.A. Shubin. Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin - Heidelberg - New York - London - Paris - Tokyo, 1987.

[47] M.A. Shubin. Spectral theory of elliptic operators on non-compact manifolds. In Methodes semi-classiques, Volume 1, Ecole d'Ete, Nantes 1991, volume 207 of Asterisque, pages 35-108. Societe Mathematique de France, 1992.

[48] J. Ueberberg. Zur Spektralinvarianz von Algebren von Pseudo-Differential Operatoren in der LP -Theorie. Manuscripta Math., 61:459-475, 1988.

[49] A. Unterberger. Encore des classes de symboles. In Sem. Goulaouic-Schwartz 1977-1978, Ecole Poly technique, Paris, 1977, 17 pages.

[50] A. Unterberger. Oscillateur harmonique et operateurs pseudo-differentiels. Ann. Inst. Fourier, 29-3:201-221, 1979.

[51] A. Unterberger and J. Bokobza. Les operateurs de Calderon-Zygmund precises. C. R. Acad. Sci. Paris, 259:1612-1614, 1964.

[52] A. Unterberger and J. Bokobza. Sur les operateurs pseudo-differentiels d'ordre variable. C. R. Acad. Sci. Paris, 261:2271-2273, 1965.

[53] L. Waelbroeck. Le calcul symbolique dans les algebres commutatives. J. Math. Pures Appl., 33(9):147-186, 1954.

[54] H. Widom. A complete symbolic calculus for pseudodifferential operators. Bull. Sc. math.,2e serie, 104:19-63, 1980.

F. Baldus Fachbereich Mathematik Johannes Gutenberg-Universitat Mainz Staudingerweg 9 D-55099 Mainz, Germany e-mail: [email protected]


Spectral Invariance and Submultiplicativity for the Algebras of SCM, g)-pseudo-differential Operators on Manifolds

F. Baldus

Abstract. For appropriate triples (M,g,M), where M is an (in general noncompact) manifold, g is a metric on T* M, and M is a weight function on T* M, we developed in [5] a pseudo-differential calculus on M which is based on the S(M,g)-calculus of L. Hormander [30] in local models. Here we prove that the algebra of operators of order zero is a submultiplicative w* -algebra in the sense of B. Gramsch [21] in £ (L 2 (M)). For the basic calculus we generalized the concept of E. Schrohe [40] of so-called SG-compatible manifolds. In the proof of the existence of "order reducing operators" we apply a method from [4], and the proof of spectral invariance and submultiplicativity uses methods and results from B. Gramsch, J. Ueberberg, and K. Wagner [26] building on Beals' -type characterizations.

1. Introduction

Already in 1967, R.T. Seeley proved in connection with a holomorphic functional calculus for pseudo-differential operators that the algebra of classical pseudodifferential operators of order zero on a compact manifold M is spectrally invariant in HD(M) (d. [45, Theorem 5]; cf. also [46]). Since the seventies great effort has been made to prove more and more general results concerning spectral invariance for algebras of pseudo-differential operators in 1:- (L2(M)) and 1:- (L2(JRn)) (d. R. Beals [7], [8], J. Dunau [20], F. Bruyant [13]' H.O. Cordes [15], [16], B. Gramsch [22], B. Gramsch, J. Ueberberg, and K. Wagner [26], E. Schrohe [41], [43], J.-M. Bony and J.-Y. Chemin [11], J. Sjostrand [49]) or, more generally, on LP(JRn) (d. J. Ueber berg [50], cf. also [2]) and weighted Sobolev spaces over LP(JRn) (d. E. Schrohe [42]). Spectral invariance was analyzed for boundary value problems by E. Schrohe [44]. On the other hand, as e.g., H. Widom [55] and E.B. Davies, B. Simon, and M. Taylor [18] pointed out, spectral invariance

2000 Mathematics Subject Classification. Primary 35S05; 58J40 Secondary 47 A60; 47 AlO; 47L15; 46F05. Key words and phrases. pseudo-differential operators on manifolds; Weyl-Hormander calculus; SG-calculus; spectral invariance; submultiplicativity.

250 F. Baldus

fails in slightly different situations. B. Gramsch [21], [22] stressed the importance of spectral invariance for Frechet operator algebras in the perturbation and homotopy theory of Fredholm functions related to pseudo-differential analysis. For further results in this direction we refer to B. Gramsch [23] and B. Gramsch and W. Kaballo [24]. Moreover, spectral invariance plays a decisive role in [4].

Originally, the calculus of pseudo-differential operators on manifolds was based on symbols which satisfy local estimates, i.e., estimates which are only uniform for x in a compact set (cf. [53], [54], [28], [29]). On ]Rn symbols satisfying global estimates were considered (cf. L. Hormander [29], M.A. Shubin [47], C. Parenti [37], H. Kumano-go and K. Taniguchi [32], R. Beals and C. Fefferman [9], R. Beals [6], H.O. Cordes [14], [17], B. Helffer [27]). In connection with the Weyl calculus, L. Hormander [30] (cf. also [31, Section 18.5]) presented a vast extension of these classes: To a metric g on T*]Rn and a weight function M on T*]Rn

he associated the symbol class S(M, g) and developed the corresponding pseudodifferential calculus. For essential contributions to the S(M, g)-calculus we wish also to refer to A. Unterberger [51], [52], R. Beals [8], N. Dencker [19], J.-M. Bony and N. Lerner [12], J.-M. Bony and J.-y' Chemin [11], J.-M. Bony [10]. These data g and M also make sense on the cotangent bundle of a manifold, but, for a generalization of the S(M, g)-calculus, we have to make assumptions concerning both the manifold, expressed by the existence of a suitable atlas, and the data M and g. In [30] L. Hormander investigated already the behavior of operators associated to metrics not too far from the standard p, 6-metric under certain canonical transformations, introduced Fourier integral operators, and proved a generalization of Egorov's theorem.

In [5] we generalized the concept of E. Schrohe (cf. [39], [41], [44]) of socalled SG-compatible manifolds which are appropriate in order to transfer the SGcalculus of M.A. Shubin [47], C. Parenti [37], and H.O. Cordes [14]. In so doing, we developed an S(M, g)-based pseudo-differential calculus on manifolds M, at least for appropriate combinations of (M, g, M).

Having provided the basic calculus on manifolds - including composition, mapping properties in the canonical scale of Sobolev spaces, and the existence of a symbol homomorphism - in a quite general setting, in this article we impose more restrictive assumptions in order to prove spectral invariance and submultiplicativity for the algebra of operators of order zero. Thus, it is a submultiplicative I}I*-algebra in the sense of B. Gramsch [21] in I: (L2 (M)).

For the proofs of spectral invariance and submultiplicativity, the existence of "order reducing operators" is of essential importance. To show their existence we apply ellipticity-arguments within the S(M, g)-calculus. We used this method already in [4]. Moreover, in the proof of spectral invariance we apply Beals'-type characterizations with commutator methods for the operators in our calculus on manifolds. Here methods and results from B. Gramsch, J. Ueberberg, and K. Wagner [26] playa decisive role (cf. 4.13,4.15). In connection with submultiplicativity we apply a method from [26] starting from commutator-characterizations which was also used by B. Gramsch and E. Schrohe [25] and in [2]. By the way, it is an open

Spectral Invariance and Submultiplicativity 251

problem whether every \I! * -algebra is submultiplicative (d. B. Gramsch, List of open problems, p. 552 in [1]). For the commutative case we refer to W. Zelazko [56, Theorem C], for the non-commutative case cf. also [56, Theorem 3]. For an application of submultiplicativity in connection with non-abelian cohomology and Oka's principle we refer to B. Gramsch [23] and B. Gramsch, W. Kaballo [24].

Our approach to S(M, g)-pseudo-differential calculus on manifolds developed so far allows already - at least in simple situations - to piece different local models together (cf. 6). To go further in this direction, e.g., to piece different p,8-type calculi together, in [3] we defined algebras of pseudo-differential operators with the help of commutator methods (cf. [26]) and proved spectral invariance in L (L 2 (M)). Besides modifications of the characterizations in 3.20, for the proof of spectral invariance we have to admit conditions for the commutators with pseudodifferential operators of sufficiently small order in the definition of the algebras. In the case of the algebras constructed in [5], these conditions are automatically fulfilled. All these conditions make the construction rather technical. Thus, we decided not to present them in this article but to refer the reader to [3, Sections 5.6 and 7.5]. Here we restrict ourselves to a slightly different approach using methods from [26] which we describe in Section 7. Since their methods are similar, at this point we wish to refer to results of R. Lauter [33], [34], [35] and F. Mantlik [36] in connection with manifolds with singularities.

Instead of explaining the organization of this paper at this point, we refer to brief introductions at the beginning of each section.

This paper is part of the author's Ph. D. thesis [3] written under the supervision of Professor B. Gramsch at the University of Mainz, Germany.

2. Admissible coordinates

There is a natural way to generalize 9 and M and thus, S(M, g) to cotangent bundles over manifolds. But, on the other hand, it is well known that the S~8-classes are invariant under (local) diffeomorphisms - and thus can be considered on the cotangent bundles of manifolds- if and only if 0 ::; 1 - P ::; 8 ::; p ::; I, 8 < 1. Therefore, for an extension of the S(M,g)-calculus, we have to make assumptions concerning both the manifold, expressed by the existence of a suitable atlas, and the data M and g. In [40] E. Schrohe transfers a calculus of M.A. Shubin [47], C. Parenti [37], and H.O. Cordes [14], which is based on global estimates, to so-called SG-compatible manifolds (cf. also [39], [41], [44]). The local charts are assumed to satisfy conditions which allow us to prove invariance of the pseudodifferential calculus with respect to the changes of coordinates. Here we generalize these conditions in order to treat classes of operators associated to more general calculi on IRn. Apart from some technical conditions concerning the metric, e.g., a generalization of the assumption 1 - P ::; 8, we require that the diffeomorphisms have differentials which are matrix-valued symbols in the calculus and that they extend to larger domains which are chosen in such a way that the existence of

252 F. Baldus

admissible cut-off functions and partitions of unity can be proved. The conditions concerning the manifold M with atlas A depend on two metrics g and 9 on T* M and M, respectively, which determine the pseudo-differential calculus. In the SGcalculus we have 9 = R 1 (g), but, as the examples in Section 6 show, in some cases it is reasonable to consider 9 i= R1 (g). Consequently, we describe admissible (in order to develop a pseudo-differential calculus) structures by the data (M, A, 9, g). In contrast to most of the already existing calculi, our approach allows us to patch different local models together, provided that we have compatible changes of coordinates (cf. 6).

In this section we introduce the underlying local models, metrics and weight functions on these models, and diffeomorphisms between them. The assumptions imposed on these data are more restrictive than in [5]. For the formulation of the conditions we need the following weight functions:

Definition 2.1. Let U be an open subset of]Rn and g a metric on U x ]Rn.

1. For x E U and ~ E ]Rn let h9(X,~) ( 9(x,f.) (T») 1/2

.- SUPTEIR2n\{o} gO (T) where (x,O

a (T) a(W,T)2 g(x/;) := SUPWEIR2n\{o} 9(x,<;)(W)'

2. For x E U and~, t E ]Rn, t i= 0, we define

Definition 2.2. 1. Let U, U be open subsets of]Rn with U ~ U, and let 9 be a smooth metric

on U. We call (U, U, Q) an admissible model if the following conditions are satisfied:

• 3rg > 0, Vx E U, Vy E ]Rn, 9x(x - y):::; r~ ===;. y E U. • 3Cg > 0, Vx, y E U, 9x(x-y) :::; Cg1 ===;. Cg19x(-) :::; 9 y (') :::; Cg9x(-) .

• 3C> 0, \;jt1, t2 E ]R \ {O}, Vx E U, ~ Il t 211; < C. IitI1l2 gx(t2) -

2. Let U1 be an open subset of]Rn, let (U2 , U2 , 92) be an admissible model, and let", : U1 ---> U2 be a Cexo-diffeomorphism. '" is called (U2, U2, 92)-compatible if the following statements hold:

• (U1, U1, ~h) is an admissible model, where U1 := ",-1 (U2) and 91 := ",*92, i.e., 91,x(t) = 92,t«x) (",'(x)(t)).

• [U1 3 x f---+ d",(x)] E S (1, 91; £ (]Rn)) and [U2 3 Y f---+ d (",-1) (y)] E S (1, 92; £ (]Rn)).

• 3C> 0, Vx E U1, Vy E U2, 11",(x)112:::; C(x), 1I",-1(y)ii2:::; C(y).

Definition 2.3. We call 'split' a Hormander metric g on ]R2n with g(x,t,) = JI(:~~f2 + J~~~~f2 ((x,~) E ]R2n) of type (t) if the following conditions are satisfied:

• There exists a constant C > ° such that

VX,~,t,T E ]Rn, g(x,O)(t,O):::; Cg(x,~)(t,O), g(x,O)(O,T):::: C- 1g(x,t,)(0,T)

• 3Cg > 0, VX,y,f;, E JRn , g(x,O)(x-y,O)::; C;;l ===;. C;;l g(x,E,)(-) ::; g(y,E,)(-) ::; Cgg(x,f;) U .

• VK > 0, ::ICK > 0, Vx, f;, E JRn, VA E £(JRn)-l with IIAII.C(lRn) < K and

IIA-111.C(lRn) ::; K, C"K1g(x,E)U::; g(x,A~)U::; CKg(x,E,)(-) . • 3C > 0, ::1O::; i5 < 1, VX,f;, E JRn , Vt,T E JRn

Moreover, with a vector t E JRn \ {O} the following assumptions hold:

• 3C> 0, ::IN E N, Vx, f;, E JRn, (f;,)::; CHg,t(x, f;,)-N . • VT E JR2n , ::IC > 0, ::IN E N, VX,f;, E JRn , C-1H g,t(x,f;,)N::; g(x,E,)(T) <

CHg,t(x, f;,)-N .

Definition 2.4.

1. Let (U, U, g) be an admissible model and let 9 be a smooth metric on U x lK,n. We call (U, U, g, g) an admissible quadruple of type ('Ij;), if the following conditions are satisfied: We start with the condition concerning g:

• There exists C > ° such that we have yx(t) ::; C Iltll; for all x E U and all t E JR, and gx(t) ::; C(x)-21Itll; for all x E U \ U and all t E JR.

We continue with conditions concerning g: • There exists a Hormander metric gs of type (t) on JR2n such that with

a suitable constant C s > ° for all x E U and all f;, E JRn the estimate C;lgs.(x,~)(-) ::; g(x,~)(-) ::; Csgs,(x,E)U holds.

• ::IC> 0, ::IN E N, Vx E U, Vf;, E JRn, (f;,)::; Chg(x, f;,)-N . • VT E JR2n, ::IC > 0, ::IN E N, Vx E U, Vf;, E JRn, C-1hg(x,f;,)N <

g(x,~)(T) ::; Chg(x, O-N . Next, concerning both 9 and g, we suppose that the following conditions hold:

• For all x E U we have gxu ::; R 1(g)xU, where R1(g)x(t) := g(x,O) (t,O).

• ::IC> 0, Vx E U, Vt,f;, E JRn, gx(t) g(x.~)(O,f;,)::; Cg(x.E,)(t, 0). 2. We apply the notation from the preceding point. We call (U, U, g, g) an ad

missible quadruple of type ('Ij;)s, if additionally the following condition is satisfied:

• There exists C > 0 such that g(x,E)(t,T)::; CII(t,T)II; for all (x,f;,) E U x JRn and all t, T E JRn.

Remark 2.5. Let (U, U, g) be an admissible model. In the following we will suppose that Cg 2: 1 and Tg ::; 1/ VCQ. If (U, U, y, g) is an admissible quadruple of

254 F. Baldus

type ('l/J), we suppose that Cg :2: Cg, Cg is chosen such that for x, y E U and f"ry E JR",

g(x,t;)(x ~ y, f, - ry) :::; C;l ===} C;lg(x,t;)(·) :::; g(y,ry)(') :::; C9g(x,t;)(') ,

and rg :::; min{C;1/2,rg}.

Definition 2.6. Let (U, U, 9, g) be an admissible quadruple of type ('l/J) and let M : U x JR" -+ (0, (0) be a continuous function. We call M a (U, U, g)-admissible weight function of type ('l/J), if

• For each split Hormander metric g8 as in 2.4 there exists a gs-admissible weight function M : JR2n -+ (0,00) and a constant (; > ° such that for all (x, f,) E U x JRn we have (;-1 M(x, f,) :::; M(x, f,) :::; (;M(x, f,).

• VK > 0, :3CK > 0, Vx E U, Vf, E JRn, VA E £(JRn)-l such that IIAIIL(IRn ) :::; K

and IIA-11IL(IRn) :::; K, C1/ M(x, f,) :::; M(x, Af,):::; CK M(x, f,) .

• VX,y E U, Vf, E JRn, g(x,O)(x - y,O) :::; C;l ===} Ci}M(x,~):::; M(y,f,) :::; CMM(x,f,) .

Definition 2.7. Let U1 be an open subset of JR", let (U2, U2, 92, g2) be an admissible quadruple of type ('IjJ), and let fi : U1 -+ U2 be a (U2,U2,92)-compatible diffeomorphism. fi is called (U2 , U2 , 92, g2) -compatible of type ('l/J) if (U1 , U1 , 91 ,gd is an admissible quadruple of type ('l/J), where U1 := fi- 1 (U2 ), 91 := fi*92 , gl := X*g2'

X: U1 X JRn -+ U2 X ~n, (x,£,) f-+ (fi(X),tdfi(x)-lf,).

If we replace the definitions in [5] concerning the local models by the preceding definitions, we obtain more restrictive conditions on generalized SG-compatible manifolds and admissible structures which we now call admissible structures of type ('l/J). We omit to repeat the precise definition in this article and apply the notation introduced in [5] in connection with the basic calculus.

3. Characterizations of pseudo-differential operators on manifolds

In this section we prove characterizations of operators in \II(M, g; M, A, Q), expressed by mapping properties of iterated commutators with operators in V(M, A,9) between appropriate Sobolev spaces (cf. 3.20). In the proof of 3.20 we make use of the corresponding result on ~n (cf. [2, 3.6.6]). In order to formulate and prove these characterizations, we first have to introduce Sobolev spaces H (M, g; M, A), associated to general weight functions of the calculus, and endow these spaces with a suitable topology such that pseudo-differential operators are continuous maps between appropriate spaces (cf. 3.14). As in the case of the scale of spaces associated to H g , we define the Sobolev spaces with the help of our local models (cf. 3.5). In Section 4 we will prove the existence of "order reducing operators" with the help of which H (M, g; M, A) become Banach spaces (cf. 4.21).

Lemma 3.1. Let 9 be a Hormander metric of type (t), let U <:;; U <:;; JRn be open subsets, and let Q be a metric on U such that (U, U, Q, glUXlRn) is an admissible

quadruple of type ('IjJ). Let t E JRn \ {O}. Let 0 < p < rg/2 and let D := (Up,gt, l:{ := (U2p,g r· Then (U, l:{, Q) is an admissible model. Let rg be the corresponding constant from 2.2, let 0 < P < rg, and let [1 := (Up,gr. Moreover, let cp,;j; E Coo (JRn) be functions satisfying the following properties.'

• [(x,O f--+ cp(x)], [(x,';-) f--+ ;j;(x)] E 8 (l,g). • supp ( cp) <:;; D. • supp (;j;) <:;; [1, ;j; = 1 on l:{.

Then, for all s E JR and all p E 8 (H~,t, g), there exists q E S(JR2n) such that

(1 - ;j;)p(x, D)cp = q(x, D) .

Proof There exists a function ¢ E 8(1, Q) with supp (¢) <:;; l:{ and ¢ = 1 on

D. Then we have (1 - ;j;)p(x, D)cp E nfLEN W (H:'t, g). For j E {I, ... , n} and u E S(JRn) let

£(L2(JRn ))

v1(j)u .

V(j)u .-2

3 (Td 4 ~ (OO(l7,(j»)),) ,h, (Id + ~ (ad(V,{j»)),) ""

x (Id +-t, (OO(V,(j»))') m, (Td +- t, (ad(V,(j))),) m, (1 - (fi)l'(x, D)'P

(Mx)2ml (Dx)2m2 (1 - ;j;)p(x, D)cp(Mx)2rnl (Dx)2m2 .

By virtue of [2, 3.6.6] this yields

(1 - ;j;)p(x, D)cp E n h d fi 1,1._ Ildxll~ + IldEII~ M ( C)._ () d M ( C)._ (C) ( C werewe e neg(x.I;)·- (Xl2 (E)2' 1 x,." .- x ,an 2 x,." .-." x,." E

JRn). Since n fLl ,fL2EN 8 (M;fLl M;;fL2, g1,1) = S(JR2n), this yields the assertion. 0

Lemma 3.2. Let (U, U, Q, g) be an admissible quadruple of type ('IjJ). Let t E JRn \ {O}. Let M be a (U, U, g) -admissible weight function of type ('IjJ) such that with suitable constants C > 0 and N E N for all (x,';-) E U x JRn the estimate

C- 1 Hg,t(x, .;-)N :::; M(x,';-) :::; CHg,t(x, O-N

256 F. Baldus

holds. Let 0 < p < rg/2 and let ip,7./J be functions as in 3.1 with ip = 1 on Up / 2,g. Finally, let gs be as in 2.4 and let M be as in 2.6. We define M N,9"t :=

min(M, H;;:~) (which is a gs-admissible weight function). 1. For u E S'(JRn) with supp (u) <:;; Up / 2,g we have

U E H (MN,9s,t, g) ¢=} VA E W(MN,9"t, gs), -J;A(ipu) E L2(JRn) .

2. If (Uv)vENo <:;; H(MN,9s,t,g) with supp(uv) <:;; Up / 2,g, then we have

Uv ---+ Uo in H (MN,9s,t, gs) ¢=} VA E w(MN,9s,t, gs), -J;A(ipuv) ---+ -J;A(ipuo) in L2(JRn).

Proof. In Part 1 we apply 3.1. To prove Part 2, let (UV)VEN C H (MN,9s,t,g) with supp(uv ) <:;; Up / 2,g such that -J;A(ipuv) ---+ 0 in L2(JRn) for all A E W(MN,9s,t,gS)' There exist constants C > 0 and if E N such that (x, t:,)~2N :::; CMN,9s.t(X, t:,) for all x,t:, E JRn. Thus, by assumption and by virtue of [27, Proposition 1.7.5, Theoreme 1.8.11 and Corollaire 1.9.5], there exists q E S (MN,9s,t, gs) such that

(x, Dx)2N q(x, D) = Id. Since (x, Dx)2N := (1 + ~7=1 (x; + D~j)) N is a local

operator, we obtain

---+ 0 in S' (JR.n ) .

Thus, applying 3.1 once more, for A E w(MN.9s ,t,gs) we have

Auv = ~A(ipuv) + (1 - -J;)A(ipuv ) ---+ 0 in L2(JRn ) . ~' ,

-+0 in L2 (IRn) -+0 in S(lRn)

This yields Uv ---+ 0 in H (MN,9s,t, g). D

Definition 3.3. Let (U, U, g, g) be an admissible quadruple of type (7./J) and M as in 3.2. With the notation from 3.2, for an open set U' <:;; Up / 2 ,g we define H(M, g; U') as the space of all U E H(MN,9s.t,gs) with supp(u) <:;; U' and endow H(M,g; U') with the topology inherited by H (MN ,9s,t, gs).

Remark 3.4. By virtue of 3.2, the definition of H(M, g; U') is independent of the choice of gs, M, and t.

Definition 3.5. Let (M, A, g, g) be an admissible structure of type (7./J) and let M be an (M, A, g)-admissible weight function of type (7./J) which is H;l-admissible. We define H(M,g;M,A) as the set of all U E V'(M) such that (li~l)*(ipU) E

H ((X~l)* M, (X~l)* g; Ii(X9)) for all (X, X, Ii) E A and all (X, X, Ii, g)-admissible

cut-off functions ip.

Definition 3.6. Let (M,A,g,g) be an admissible structure of type (7./J). Then we define H(g; M, A)~OO := UsEIR H (H;, g; M, A).

Lemma 3.7. Let (M,A,Q,g) be an admissible structure of type ('lj;) and let M be an (M, A, g)-admissible weight function of type ('lj;) which is H;;l- admissible. Let u E H(g; M, A)-oo. Then we have

UEH(M,g;M,A) ~ VPEW(M,g;M,A,Q), PUEH(l,g;M,A)

Proof. The direction "===?" is proved elementary. For "~" we apply 3.2. D

Remark 3.8. Let (M, A, Q, g) be an admissible structure of type ('lj;) and let M, M' be (M,A,g)-admissible weight functions of type ('lj;) which are H;;l-admissible. Then, if P E W(M,g;M,A,g), we have P(H(M',g;M,A)) <;;; (M'jM,g;M,A).

Definition 3.9. Let (M,A,Q,g) be an admissible structure of type ('lj;) and let ('Pi)i=l .... ,1 be an (M, A, g)-admissible partition of unity. For u E H (1, g; M, A) we define

I

Il u II H(1,9;M.A) := L 11(~;:l)*('Piu)II£2(lRn) ;=1

Remark 3.10. 1. Let (X,X,~) E A and let 'P be an (X,X,~,g)-admissible cut-off function.

There exists a constant C > 0 such that for all u E H (1, g; M, A) the estimate

C-1 11'Pu IIH(1.9;M,A)

:=; 11(~-l)*('Pu)II£2(lRn) :=; C II'Pu IIH(1,9;M,A) :=; C IluIIH(1,9;M,A)

holds. 2. In particular, up to multiplicative constants, the norm 11'IIH(l,g;M,A) is inde

pendent of the choice of the (M, A, g)-admissible partition of unity. 3. Moreover, a sequence (UV)VEN <;;; H(l,g;M,A) is convergent, respectively, a

Cauchy sequence, with respect to 11'IIH(1,9;M,A)' if and only if for all (X, X,~) E A and all (X, X,~, g)-admissible cut-off functions 'P the sequence ((~-1 )*('Puv)) is convergent, respectively, a Cauchy sequence, in L2(JRn).

4. Consequently, (H(l, g; Mm, A, 11'IIH(1,9;M,A)) is a Banach space and CO' (M) is dense in H (I, g; M, A).

5. Let G be an (M, A, g)-admissible Riemannian metric. There exists C > 0 such that for u E H(l,g;M,A) = L2(M,ac;) the estimate

C-1 Il u II H (l,g;M,A) :=; IluIIU(M,Qc) :=; C IluIIH(1,9;M.A)

holds.

Corollary 3.11. Let (M, A, Q, g) be an admissible structure. Then, for a linear operator P : H (1, g; M, A) --+ H (1, g; M, A), the following statements are equivalent:

1. P is bounded in charts. 2. P E .c (H (1, g; M, A), 11'IIH(l,9;M,A))'

258 F. Baldus

Definition 3.12. Let (M, A, g, g) be an admissible structure of type ('ljJ) and let M be an (M, A, g)-admissible weight function of type ('ljJ) which is H;l-adapted. We endow H (M, g; M, A) with the coarsest topology such that for each P E w(M, g; M, A, g) the operator

P: H (M, g; M, A) ---+ (H (1, g; M, A), 11·IIH(l,g;M.A»)

is continuous.

Remark 3.13. 1. On H (1, g; M, A) this topology coincides with the topology induced by

11·11 H(l,g;M,A)' 2. Let (M,A,9,g) be an admissible structure of type ('ljJ) such that A :::; .A.

Then M is (M,.4, g)-admissible of type ('ljJ), too. We have H(g; M, A)-OO

= H(g;M,A)-OO and H(M,g;M,A) = H (M,g;M,.4). Moreover, by def-

inition, the topologies on H (M, g; M, A) and H (M, g; M,.4) coincide.

3. The topology on H(M, g; M, A) is the projective topology with respect to the mappings in w(M, g; M, A, g). Thus, this topology is induced by the semi-norms pp(u) := IIPuIIH(l.g;M,A) (P E w(M, g; M, A, 9)). Hence, sets

of the form niEIPi-l(Vi), where I is a finite index set, Pi E w(M,g;M,A, g), and Vi S;;; H(l,g;M,A) is an open set, form a base of topology (cf. [38, Chapter II 5]).

Lemma 3.14. Let (M, A, g, g) be an admissible structure of type ('ljJ) and let M, M' be two (M, A, g)-admissible weight functions of type ('ljJ) which are H;l-adapted. Let PEW (M, g; M, A, 9). Then the mapping P : H (M', g; M, A) ---+ H(M'IM, g; M, A) (cf. 3.8) is continuous.

Proof. Let V S;;; H(M'IM,g;M,A) be open. Hence, there exists an index set A, for each A E A a finite index set 1;.., and, for all A E A, i E 1;.., an open set U>.,i S;;; H (1, g; M, A) and operators B>.,i E W (M' 1M, g; M, A, g) such that V = U>'EA V>. where V>. := niEI" B):,;(U>.,;). This yields

A-l(V) = U A-l(V>.) = U n A-l(B):,;(U>.,i)) = U n (B>.,iA)-l(U>.,i) . >'EA

o Corollary 3.15. If M:::; M', then we have H(M',g;M,A) '---+ H(M,g;M,A).

Lemma 3.16. Let (M, A, g, g) be an admissible structure of type ('ljJ) and let M be an (M, A, g)-admissible weight function of type ('ljJ) which is H;l-adapted.

1. Let (UV)VEN S;;; H (M, g; M, A). Then the following statements are equivalent: (a) U v ---+ 0 in H (M, g; M, A). (b) For all (X,X,K:) E A and for all (X,X,K:,g)-admissible cut-off functions

rp we have (K:- l )*(rpuv ) ---+ 0 in H ((X- l )* M, (X- l )*g; K:(Xg)).

2. Let (X, X,~) E A and let rp be an (X, X, Ji,g)-admissible cut-off function.

Moreover, let (UV)VEN <;;; H((X-l)*M'(X-l)*g;~(Xg)), Uv ---+ 0. Then we

have (rp . Ji*uV)VEN <;;; H (M, g; M, A) and rp' ~*uv ---+ ° in H (M, g; M, A).

Proof. 1. For the implication "===}" we apply 3.2. For the inverse implication note that

for (Uv)vEN <;;; H(M,g;M,A), in order to show that Uv ---+0 in H (M, g; M, A), we have to prove that Puv---+O in H(l,g;M,A) for all PEw(M,g;M,A,Q).

2. In addition to the last observation we apply 3.7.

D

Remark 3.17. If P : H (M, g; M, A) ---+ H (M', g; M, A) is continuous, then P is uniquely determined by its action on CO'(M).

Definition 3.18. Let (M, A, Q, g) be an admissible structure of type (1jJ). Let j E

{l, ... ,m} and let tj E lRn with IItjl12 = 1. Let <l>j(x,O:= (gj,(x,~)(tj,0))1/2 and

Wj(x,~) := (gj,(x.~)(0,tj))1/2 ((x,~) E ~j(Xj) x ]Rn). Now let (rp;)i=l,.l be an (M, A, g)-admissible partition of unity. For x E M and ~ E T;M we define

I

Mg(l,O)(x,~) '- Lrpi(X)' <l>j,(Xji(X,~)), i=l

I

Mg(O, l)(x,~) '- L rpi(X) . Wj; (Xl; (x, ~)). i=l

For ml,m2 E.No let Mg(ml,m2):= Mg(l,o)mlMg(O, 1)m2.

Remark 3.19. 1. For (X, X,~) E A there exists a constant C> ° such that for all x E X, all

~ E T; X, and all t E lR with IItl12 = 1 the estimates

C- 1 (Mg(1,0)(x,~))2 < ((X-l)*9IT*X)x(x,~) (t,O) < C(Mg(1,0)(x,~))2

C-1 (Mg(O, 1)(x,O)2, < ((X-1)*91T*x)x(x,O (0, t) < C (Mg(O, l)(x, ~))2

hold. If Mg(l, 0) and Mg(O, 1) are defined with another (M, A, g)-admissible partition of unity, then, with a suitable constant C > 0, we have

C- 1 Mg(l,O)

C- 1 Mg(O, 1)

:s; Mg(l,O):S; CMg(l, 0) ,

:s; Mg(O,l) :s; CMg(O, 1) .

Finally, Mg(l,O) and Mg(O, 1) are (M,A,g)-admissible weight functions of type (1jJ). By assumption, they are H;l-adapted and there exists a constant C> ° such that Mg(O, 1) :s; C.

Theorem 3.20. Let (M, A, Q, g) be an admissible structure of type (1jJ) and let M be an (M, A, g)-admissible weight function of type ('ljJ) which is H;l-adapted.

260 F. Baldus

1. Let P : CO'(M) --> D'(M). Then the following statements are equivalent: (a) P E 'l1(M,g;M,A,9). (b) For alll, kENo with l :S k, all VI' ... ' Vi E Vrn(M, A, Q), all Vi+I, ... ,

Vk E Vd(M,A,9), all permutations 1f of{I, ... ,k}, and all (M,A,g)admissible weight functions M' of type ('l/J) which are H;; I-adapted the operator

has a continuous extension to an operator

H (M', g; M, A) --> H (M' M-I Mg(k - l, l)-I, g; M, A)

(c) For alll,k,kl ,k2 E No with l:S k, all VI'.·.'Vi E Vrn (M,A,9), all Vi+I, ... ,Vk E Vd(M,A,9), and all permutations 1f of {I, ... ,k} the operator

ad (V7r(I)) ... ad (V7r(k)) P : CO'(M) --> D'(M)


H (Mg(k l ,k2)-I,g;M,A) --> H (M-IMg(kl,k2)-IMg(k -l,l)-I,g;M,A).

2. More precisely, as far as the implication ===} "(a) ===} (b)" is concerned: If P E 'l1(M,g;M,A,Q), l,k E No with

l :S k, VI,.·. , Vi E Vrn(M, A, Q), Vi+I, ... , Vk E Vd(M, A, Q),

and 1f is a permutation of {I, ... , k}, then we have

ad (V7rCI)) ... ad (V7rCk)) P E 'l1 (MMg(k -l,l),g;M,A,Q) .

3. More precisely, as far as the implication "(c) ===} (a)" is concerned, there exist finite subsets

W rn (M,A,9) <;;; Vrn(M,A, 9) and Wd(M,A,Q) <;;; Vd(M,A, 9)

such that the property that for alll, k, kl' k2 E No with l :S k, all VI'···' Vi E Wrn(M, A, 9), all Vi+I, .. ·, Vk E Wd(M, A, Q), and all permutations 1f of {I, ... , k} the operator

ad (V7rCI)) ... ad (V7rCk)) P: CO'(M) --> D'(M)


H (Mg(kl , k2)-1, g; M, A) --> H (M- I Mg(kl , k2)-1 Mg(k - l, l)-I, g; M, A)

implies P E 'l1(M,g;M,A,Q).

Proof. We start with the proof of Part 2. Let (X, X, Ii) E A, let ¢ be as in the definition of V(M, A, 9) in [5], let j E {I, ... , n}, let L j := MXj or L j := Dxj ,

and let Vu := Ii* [Lj (Ii-I)*(¢U)] (u E CO'(M)). By assumption there exists an

atlas A = { (X, X, Ii) I (X, X, Ii) E A} such that (M, A, Q, g) is an admissible

structure of type ('l/J), A:S A, and supp (¢) <;;; X where (X, X, Ii) E A. Let i.p be an (X, X, Ii, g)-admissible cut-off function which equals I on X. Moreover, let ;j;

be a corresponding function as in [5, Theorem 5.18] applied with A instead of A. For U E CaCM) we have

ad(V)Pu V(cpPu) - P(~Vu)

V(cpP(~u)) + V(cpP((1 - ~)u)) - ~P(~Vu) - (1 - ~)P(~Vu) V(~P(~u)) - ~P(~Vu) + V(cpP((1 - ~)u)) - (1 - ~)P(cpVu),

where VcpP(1 - ~), (1 - ~)PcpV E 'It (g; M, A, 9)-00 . A precise study of the behavior of the remainder term in asymptotic expan

sions of the product symbol in the calculus on JRn shows that ad(V) (~p~) E

'It(MMg(mv),g;M,A,9) where mv := (0,1) if L j = MXj and mv := (1,0) if L j = Dxj . For details in this proof we refer to [3, 3.8.19]. Together this yields ad(V)P E 'It (M Mg(mv), g; M, A, 9) and thus, by induction, Part 2.

For the proof of Part 1 it remains to show that (c) implies (a). Let P be an operator which satisfies (c). We apply the notation from [5, 5.18 and 5.19]. There exists an atlas .4 such that (M,.4, 9, g) is an admissible structure of type (c), A ::::: .4, and for i E {I, ... , l}, CPi and ~i are (Xji' Xj" K:j" g)-admissible cut-off functions where (Xj;, Xji , K:jJ E A. By virtue of [2, 3.6.6], for each i E {I, ... ,l} there exists a symbol Pi E S ((Xj,l)* M, (Xj,l)* g) such that we have

for u E CaCM). Moreover, applying the notation from 3.18 and 2.2, and the methods from the proof of 3.1, for m, m1, m1, m2, m2 E No, i, J-l E {I, ... , l}, and all A > 0 the operator

L2(JRn ) 3 U f-+ (x, Dx)2rn(Mx)2rn1 (Dx)2rn2

(K:j,1)* [CPiP ((1- ~i) . CPfL' (K:j,J* [(Mx)2m1 (Dx) 2m2 UJ)] E H ( (1 + <l>ji) -(rn2+rn+rrI2) (1 + ~ ji) -(rn1 +m+m,) Mj-;l, gji'S)

is bounded. By assumption, there exist constants C > 0 and 0 ::::: 5 < 1 such that

- 2N - 25-Mji(x,c,) ::::: C(x,f,) ,1 + <Pji(X,f,)::::: C(x,f,) ,1 + 'lt ji (x, f,) ::::: C

for all x, f, inJRn. Since 2(m22~;-;-:l2)5 ---+ 5- 1 as m ---+ 00 there exists mEN such that 2m 2: 2(m2 + m + m2)5 + 2N and thus,

By virtue of [27, Proposition 1.7.5, TMoreme 1.8.11, and Corollaire 1.9.5] there

( ( -) -(rn2+rn+m2) - 1 ) exists q E S 1 + <Pji Mj-; ,gji,s such that q(x, D)(x, Dx)2rn = Id.

262 F. Baldus

Thus, defining

P(ml' ml, m2, m2)(u) := (Mx)2m 1 (Dx)2m2 (K:j,l)*

['PiP ((1 - :(f;i) . 'PI" . (K:j,J* [(Mx)2m1 (Dx) 2m2 uJ)] , for all J E IJ! ( (1 + <l>ji) m2+

m+

m2 Mj" gji. S ) the operator

is continuous and thus, P(ml' ml, m2, m2) E £ (L2(JRn)). As in the proof of 3.1, there exists ki,1" E S(JR2n) such that we have

'PiP ((1 - (h) . 'PI" . U) = K:ji [('Pi 0 K:j,l) 'In ki,I"(-' y) . (cpl" . U)(K:h,l(Y))dY]

for U E CaCM). This yields P E IJ!(M, g; M, A, 9). In all these conclusions, only iterated commutators with a finite number of operators in V(M, A, Q) have to be taken into account. 0

4. Spectral invariance for w(l,g;M,A,Q) in £(L2(M))

In this section, we prove spectral invariance for the algebra 1J!(1, g; M, A, g) in £(H(l,g;M,A)) (= £ (L2 (M,O'c))). More precisely, to do so, we have to assume that (M, A, g, g) is of type ('ljJ) and there exists an admissible structure (M, A, g, g) of type ('ljJ)s such that g is a minorant of g (cf. 4.22). The assumptions

arise from-the conclusion that, with P E 1J!(1, g; M, A, 9) n £ (H (1, g; M, A))-\ also p-1 fulfills the characterizations from 3.20, expressed by the mapping properties of iterated commutators with operators in V(M, A, 9) between appropriate Sobolev spaces. For this conclusion we have to justify the formula ad(V) (P-l) =

_p- 1 (ad(V)P) p- 1 and therefore prove that p- 1 maps SCM) to SCM) and behaves well between the Sobolev spaces which have to be taken into consideration in the characterizations (cf. 4.15 and 4.20). To do so we need "order reducing operators" , the existence of which we can show under the just mentioned conditions (cf. 4.21). We prove their existence in three steps: In the first step, for M 2: 1, we prove the existence of an operator P E IJ!(M, g; M, A, Q) such that P : H (M, g; M, A) --t H (1, g; M, A) is a topological isomorphism (cf. 4.5, 4.6). Then H (M, g; M, A) can be endowed with a norm and, for M, M' 2: 1, we can apply the same proof as before to show the existence of an operator P E IJ!(M,g;M,A,9) such that P : H(MM',g;M,A) --t H(M',g;M,A) is a topological isomorphism (cf. 4.8). This allows us in the third step to prove spectral invariance of 1J!(1,g;M,A,9) in £(H(l,g;M,A)), where (M,A,9,g)

is supposed to be of type (7jJ)8 (cf. 4.16). As essential tools, we make use of results from B. Gramsch, J. Ueberberg, and K. Wagner [26] (cf. 4.13, 4.15) and of the characterizations from 3.20. In particular, for M ~ 1 there exist operators P E 1Jf(M,g;M,A,g) and Q E W(1/M,g;M,A,9) such that PQ = Id = QP. With the help of this result we obtain "order reducing operators" in the general case of weight functions under consideration (cf. 4.21). With these preparations we can - applying 3.20 again - finally prove spectral invariance of w(l, g; M, A, 9) in £ (H (1, g; M, A)) under the already stated assumptions.

For the w* -property in the special case of the SG-calculus we refer to E. Schrohe [41].

The following definitions and the succeeding lemma originate from B. Gramsch [22, Definition 5.1 and Lemma 5.3].

Definition 4.1. Let 8 be a Banach algebra with unit element e and let A be a subalgebra of 8 with e E A.

1. A is called locally spectrally invariant in 8 if there exists an c > 0 such that

{aEAllle- alls <c}<;;;A- l

2. A is called spectrally invariant in 8 if

An 8- 1 = A-I

Definition 4.2.

1. Let 8 be a Banach algebra with unit element e. A continuously embedded Fn§chet sub algebra A of 8 is called a W -algebra if e E A and A is spectrally invariant in 8.

2. If 8 is a C* -algebra with unit element and A is a symmetric l w-subalgebra, then A is called a w* -algebra.

Lemma 4.3. Let 8 be a unital Banach algebra and let A be a locally spectrally invariant subalgebra. Let A be the closure of A in 8. Then A is spectrally invariant in A.

Corollary 4.4. Let 8 be a C" -algebra with unit element and let A be a locally spectrally invariant subalgebra. We define A* := {a E A I a* E A}. Then A* is a symmetric spectrally invariant subalgebra of 8.

Theorem 4.5. Let (M, A, g, g) be an admissible structure of type (7jJ) and let M be an (M,A,g)-admissible weight function of type (7jJ) which is h-;l-adapted and which satisfies M ~ 1. Then there exist operators P E w(M, g; M, A, g) and Q E W(1/M,g;M,A,9) such that PQ,QP E £(H(l,g;M,A))-I.

Proof. With the help of an (M, A, g)-admissible partition of unity, we can assume that M E S(T* M, A; M, g). For j E {1, ... , m} let M j and gj be as in [5] and NIj and gj.8 be as in 2.4, 2.6. Since M is h-;l-adapted there exist N E Nand C > 0

1 I.e., a E A implies a* E A.

264 F. Baldus

such that M j ::::: Ch~~ for all j E {I, ... , m}. We define Mj := min (h~~, NIj ) (which is a gj,s-admissible weight function). We apply the notation from [5, 3.2] with (Uj , Uj , Rl (gj)) instead of (U, U, 9), the constants p, PI, ... ,P4 being chosen simultaneously for all j E {I, ... , m}. Moreover, we suppose that P4 ::::: r A/2 and 2 ICgp < TA. Let (<PJ'Z)ZEU ., (~J'Z)zEU . (j = l, ... ,m) be the corre-Y'--"9 '],P2,9]' ],P2,.9)

sponding families and let r{Jj := Ju <Pj.z IRl(g)zll/2dz. Let (<pj)j=I,.,m be an ],P2,9j

(M, A, g, pd-admissible partition of unity with associated system of cut-off func-tions (~j )j=I .... ,m. For j E {I, ... ,m} let !pj, :(f;j be (Xj, X j , /'\;j, g)-admissible cut-off functions such that lPj 0 /'\;jl = 1 on supp (<pj.z), :(f;j 0 /'\;jl = 1 on supp (~j,z) for all

z E Uj,P2.9j' and, moreover, :(f;j = 1 on supp (<pj). For A > 0 and j E {I, ... ,m} let

((x,~,y) E Uj x]Rn xUj ). This yields an amplitude q)..,j E A((Mj +A)-I,gj) with estimates independent of A > O. For U E S(]Rn) we define

and, for v E CaCM), Q)...jv:= /'\;j [(~h.j(/'\;jl)* (:(f;j. v)]. Let Q).. := 2:7=1 <PjQ)..,j'

Then we have (2)..,j = R(q)..,j)(x,D) and Q)..,j E W (M~)..,g;M,A,9) ~ w (k,g; M, A, 9). Moreover, for (x,~, Y) E Uj x ]Rn x Uj we have

(Mj(x,~) + A)' <pj(/'\;jl(X))' q)...j(x,~,y)

= <pj (/'\;jl (x)) .j <Pj,z(x)~j,z(Y) IRl(gj)zll/2dz Uj.P2,9j

and thus,

R ((Mj + A)' (<pj 0 /'\;,-;1). q)..,j) (x,~)

<pj(/'\;jl(X)) . i <Pj,z(X)~j,z(X) IR 1 (gj)zll/2dz . ],P2,9j

Now let P E w(M, g; M, A, 9) be an operator as in [5, Lemma 6.4] with M instead of p. For A > 0 we have

m

(P + A)Q).. - Id "'i:)¢j(P + A)<pjQ)...j - <pj) j=1

where <Pj is an (Xj, Xj,g, K" g)-admissible cut-off function like Ij5 in [5, Lemma 6.4]. For U E CaCM) and j E {l, ... ,m} we have

<pj(P +.\) ('PjQ)..,ju)

= K,j [((<pj 0 K,;1). (Mj +.\ + Tj) #KN (('Pj 0 K,t)· R(q)..,j)))

(x, D) (K,; 1 ) * (;;Jj . U) ]

(<pj 0 K,;1) . (Mj +.\) #KN (('Pj 0 K,t) . R(q)..,j)) - 'Pj 0 K,;1

(<pj 0 K,;1) . (Mj +.\) #KN (('Pj 0 K,;1) . R(q)..,j))

- R ((<pj 0 K,;1). (Mj +.\). ('Pj 0 K,;1) . q)..,j)

- R ((<pj 0 K,;1). M j · ('Pj 0 K,;1). q)..,j)

-(<pj 0 K,;1) . .\. ('Pj 0 K,t) , R (q)..,j)

(<pj 0 K,;1). M j #KN (('Pj 0 K,;1). R(q)..,j))

-R ((<pj 0 K,;1) . M j · ('Pj 0 K,;1). q)..,j)

(<pj 0 K,;1). M j #KN (('Pj 0 K,;1). [(q)..,j)jrj - R 1(q)..,j)])

- [(<pj 0 K,;1). M j · ('Pj 0 K,;1) . (q)..,j)jr,

-R1 ((<pJ 0 K,;1). M j · ('Pj 0 K,;1). q)..,j)]

(<pj 0 K,;1) . M j #KN (('Pj 0 K,;1) . (q)..,j)jrj)

-(<pj 0 K,;1). M j . ('Pj 0 K,;1) . (q)..,j)lr j

- (<pj 0 K,;1). M j #KN (('Pj 0 K,;1). R 1(q)..,j))

+R1 ((<pj 0 K,;1) . Mj . ('Pj 0 K,;1). q)..,j) .

Each of the three terms is in S ( MeL . hg' ,gJ' s) with estimates independent )..+Mj J,8 ,

of'\ > O. Moreover,

with estimates independent of .\ > O. Together we have proved that

T)..,j '- (<pj 0 K,;1) . (Mj +.\ + Tj) #KN (('Pj 0 K,;1). R(q)..,j)) - 'Pj 0 K,;1

E S (.\:~ .hgj,s,gj,s)

with estimates independent of .\ > O.

266 F. Baldus

Now let j E {I, ... , m} and 0 < c :::; 1/ N. By the definition of M j we have

hgj,s :::; M j- I/N and thus, for X E ]R2n, the estimate

hgj,s (X) . Mj(X) < ~(X)I-E < ~ Mj(X)+>' - Mj(X)+>' - >.C

holds. This yields for X E ]R2n, kENo, I :::; k, TI' ... , Tz E ]R2n \ {O},

IOTl ••. OT1r>.,j(X) I TI~=I vi gj,s,x(Tj )

<

where the constant C(k) > 0 can be chosen independently of >.. Hence,

Ilr>.,jllk,S(1,9js) :::; C(k) (~) c .

Defining

R>.,j .- r>.,j(x,D), R>.,j := Koj [R>.,j(KojI)* (~j .u)] (uECg"'(M)),

with suitable constants independent of>. > 0 we obtain (u E Cg"'(M))

II R>.,j(KoJ-:-I )* (~j . u) II :::; C2 (~) E IlullH(l g'M A)

L2(JR") 3.10 A " ,

Moreover, noting that 11'IIH(l,g;M,A) = 11'IIH(1,9;M,A)' we have

IIR>.,juIIH(1,9;M,A) 3io C2 C3 (~) E IluIIH(1,9;M,A)

Since Cg"'(M) is dense in H(I,g;M,A) (cf. 3.10) this implies Tn

II(P + >.)Q>. - Idll.c(H(l,g;M,A» < L II<pj(P + >')i.pjQ>.,j - i.pjll.c(H(1,9;M,A» j=1

t,IIR>.,jll.c(H(1'9;M,A» :::; C4 (~)c . Thus, for>. > 0 sufficiently large, we have

(P+>')Q>. E .L:(H(I,g;M,A))-1.

Analogously, we show that

Q>.(P+>') E .L:(H(I,g;M,A))-l

for>. > 0 sufficiently large. o

Corollary 4.6. We apply the notation from the theorem. Then P : H (M, g; M, A) ----t H(I,g;M,A) is invertible and p- I : H(I,g;M,A) ----t H(M,g;M,A) is continuous. In particular, H (M, g; M, A) is a Banach space.

Remark 4.7. We apply the assumptions and notation from the preceding corollary and define

IluIIH(M,g;M,A) ;= IIPuIiH(l,g;M,A) .

Then, for all (X, X, K,) E A and all (X, X, K" g)-admissible cut-off functions <p, there exists a constant C > 0 (which depends on P) such that for all u E H (M, g; M, A) the estimate

holds.

C-111<puIIH(M,g;M,A) < 11(K,-1)*(<pu)IIH((x-1)*M,(x-1)*g;l«xg ))

< C II<puIIH(M,g;M,A) ::; C21IuIIH(M,g;M,A)

Corollary 4.8. Let (M, A, Q, g) be an admissible structure of type ('IjJ) and let M, M' be (M, A, g) -admissible weight functions of type ('IjJ) which are h-;/ -adapted and which satisfy M, M' ;:::: l. Then there exist operators P E IJi(M, g; M, A, Q) and Q E lJi(l/M,g;M,A,g) such that PQ,QP E £(H(M',g;M,A))-l. In particular, P ; H(M'M,g;M,A) --+ H(M',g;M,A) is invertible and p-1 ;

H (M', g; M, A) --+ H (M'M, g; M, A) is continuous.

Proof. Applying 4.7, this is proved as in 4.5 and 4.6. D

Lemma 4.9. Let (M, A, Q, g) be an admissible structure of type ('IjJ) and let M be an (M, A, g) -admissible weight function of type ('IjJ) which is H; I-adapted. Let Z E V(M,A,g) U IJi(M,g;M,A,g) and D(Z) ;= {u E H(l,g;M,A) I Zu E

H(l,g;M,A)}. Then we have S(M) ~ D(Z) and Z ; D(Z) --+ H(l,g;M,A) is closed.

Proof. Let (UV)VEN ~ D(Z) and u,v E H(l,g;M,A) such that U v --+ U in H(l,g;M,A) and Zuv --+ v in H(l,g;M,A). Consideration in local coordinates shows that Zu = v E H (1, g; M, A) and thus, in particular, u E D(Z). D

In the following we will apply results from [26]. As a preparation, we will repeat some notation introduced in [26] in a more general situation.

Definition 4.10. Let (M, A, Q, g) be an admissible structure of type ('IjJ). Moreover, let G be an (M, A, g)-admissible Riemannian metric. With the * -operation induced by C")L2(M,aG)' where (Xc is the corresponding density, £(H(1,g;M,A)) is a C* -algebra.

• For 0 < T::; 1 let ZT;= Vm(M,A,g) U lJi(h;;-T,g;M,A,Q) and let Z;= Zl. • For Z E Z let

D(Z) {u E H(l,g;M,A) I Zu E H(l,g;M,A)},

T(Z) {A E £ (H (1, g; M, A)) I A(D(Z)) ~ D(Z)} .

• For Z E Z, A E T(Z), and u E D(Z) let 6z(A)u ;= ZAu - AZu. • For Z E Z let D(6z) ;= A(Z) be the set of all operators A E T(Z) such

that 6z(A) ; D(Z) --+ H (1, g; M, A) extends continuously to an element of £ (H (1, g; M, A)).

268 F. Baldus

• Let lEN and Z 1, ... ,Zl E Z. For kEN and ]1, ... ,] k E {I, ... , l} let

D(Oz ... oz ):={AED(Oz. ) loz ... oz (A)ED(oz· ),I<J.l<k-l}. Jk 31 31 Jp. 31 JI-'+1 --

Moreover, for n E N let

'110 := wo(Zl, ... , Zl) := .c (H (1, g; M, A)),

Wn(Zl, ... ,Zl):= {A E '110 I A ED (OZik" .OZh) ,

k :::; n, ]1, ... ,]k E {I, ... ,l} }

By 'lin (Zr) (0 < r :::; 1) we denote the set of all operators PEW 0 such that P E Wn(Zl, ... , Zl) for alll E N and all Zl,"" Zl E Zr and define W(Zr) := nnEN wn(Zr)'

• For Zl,"" Zl E Z, n E No let W~(Zl"'" Zl) := {A E Wn(Zl, ... , Zl) I , A* E Wn (Zl, ... , Zl)}. Analogously, for 0 < r :::; 1 let

W~(Zr) .- {A E wn(Zr) I A* EWn(Zr)} and

W*(Zr) := {A E w(Zr) I A* E W(Zr)} = n w~(Zr)' nEN

• For n E N let Wn(Zl, ... , Zl) be the closure of Wn (Zl, ... , Zl) in .c (H(I, g; M,A)).

Remark 4.11.

1. If A E Wn(Zl, ... , Zl) and] E {I, ... , l}, then OZj (A) E Wn-1(Zl, ... , Z!). 2. For all n, lEN and all Zl, ... ,Zl E Z we have

Lemma 4.12. Let (M,A,Q,g) be an admissible structure oJ type ('ljJ). Let 0 < r:::; 1 and A E w(Zr)'

1. Let M be an (M, A, g)-admissible weight Junction oj type ('ljJ) which is h-;l_ adapted and which satisfies M ;::: 1. Then AIH(M,g;M,A) : H (M, g; M, A) --t H (M, g; M, A) is well defined and continuous.

2. A (S(M)) ~ S(M).

Proof.

1. By assumption, there exist N E Nand C > 0 such that 1 :=; M :::; Ch-;Nr. By virtue of 4.8, for 1/ E {I, ... , N} there exists an operator

Zv EW(M1/N ,g;M,A,Q)

such that

is an isomorphism. By assumption we have A E W N (Z 1, ... , Z N ). For u E H (M, g; M, A) (~ D(ZN)) we have ZNAu = bzN(A)u + AZNu. Inductively we obtain

Zl ... ZN Au = L OZVl ... OZVl (A) (Z111 ... ZI1N_1 U)

where the sum runs over all 0 ::; l ::; N, 1 ::; VI < ... < VI ::; N, 1 ::; J-ll < ... < J-lN-1 ::; N, {VI, ... , VI}n{J-ll,"" J-lN-z} = 0. This yields Zl ... ZNAu E H (1, g; M, A) and therefore, since Zl ... ZN : H (M, g; M, A) ----> H(l, g; M, A) is an isomorphism, we have Au E H (M, g; M, A). Moreover, since H(M,g;M,A) '----+ H(l,g;M,A) (cf. 3.15) and A E £(H(l,g;M,A)), by the closed graph theorem the operator A1H(M,g;M,A) : H (M, g; M, A) ---->

H (M, g; M, A) is continuous. 2. Let V1, ... ,Vk E Vd(M,A,9), V1, ... ,Vt E Vrn (M,A,9). By assumption,

there exists N E N such that VI ... Vk E \[! (h;;-N, g; M, A; 9). Let A E

W (h;;-N,g;M,A;9) be an isomorphism A : H (h;;-N,g;M,A) ----> H(l,g; M,A) as in part 1. As before, for u E SCM) we have AV1 ... VtAu E H (1, g; M, A). Moreover, VI ... VkA-1 E £ (H (1, g; M, A)) and thus, VI ... Vk1ll ... VtAu = VI ... VkA-1AVl ... VtAu E H(l,g;M,A). Then Sobolev's embedding theorem yields Au E SCM).

o Theorem 4.13. Applying the notation from 4.10, for l, n E Nand Zl, ... ,Zl E Z we have

W n ( Z 1, ... , Zl) - 1 n W n ( Z 1, ... , Zl) = \[! n ( Z 1, ... , ZI)-l

Proof. Cf. [26, Theorem 2.10].

Corollary 4.14. For l, n E Nand Zl,"" Zl E Z we have

£ (H (1, g; M, A))-l n W~(Zl" .. ,Zl) = W~(Zl" .. ,ZI)-l

Thus, in particular, for 0 < T ::; 1 we have

£ (H (1, g; M, A) ) -1 n W * ( ZT) = W * ( ZT ) - 1

o

Proof. This is a consequence of the preceding theorem together with 4.4. 0

Corollary 4.15. Let (M,A,9,g) be an admissible structure of type ('Ij.;). Furthermore, let P E w(l, g; M, A, 9) n £ (H (1, g; M, A))-l.

1. Let M be an (M, A, g)-admissible weight function of type ('Ij.;) which is h;;-l_ adapted and which satisfies M ::::: 1. Then (P- 1 )IH(M,g;M,A) : H(M, g; M, A) ----> H (M, g; M, A) is well defined and continuous.

2. p-1 (S(M)) ~ SCM). 3. For u E SCM) and Z E V(M, A, 9) we have

[Z, p-l]U = _p-l[Z, P]P- 1u.

Proof. This is a consequence of 4.11,4.12, and 4.14. o

270 F. Baldus

Corollary 4.16. Let (M,A,9,g) be an admissible structure of type ('I/;)s'

1. We have

W(I, g; M, A, 9) n'( (H (1, g; M, A))-1 = W(I, g; M, A, 9)-1 .

2. Let M be an (M, A, g) -admissible weight function of type ('I/;) which is h-;; I-adapted and which satisfies M 2: 1. Then there exist operators P E

w(M, g; M, A, 9) and Q E W(I/ M, g; M, A, 9) such that PQ = Id = QP. In particular, H(M,g;M,A) and H(I/M,g;M,A) are Banach spaces.

Proof. Part 1 is a consequence of the preceding corollary together with 3.20. Then Part 2 follows together with 4.5. 0

The following proof is an adaption of methods from R. Beals [7] (cf. also [50]).

Lemma 4.17. Let (M, A, 9, g) be an admissible structure of type ('I/;) and let (M, A, 9, flJ be an admissible structure of type ('I/;) s such that fl. ::; g. Let M be an (M,A,g)- and (M,A,fl)-admissible weight function of type ('I/;) which is h-;;l_ and h-;;l-adapted and which satisfies M 2: 1. Then, for each P E W(I,g;M,A,9)

n ,( (if (1, g; M, A))-1 and all t E JR., the operator P : H (Mt, g; M, A) ----> H(Mt, g; M, A) is invertible.

Proof. For t > 0 let At E W(Mt ,fl.;M,A,9) (<;;; W(Mi,g;M,A,9)) such that there exists A_ t := A; 1 E W (M- t , fl.; M, A, 9). For t 2: 0 the assertion is already proved in 4.15. By assumption, there exist So > 0 and C > 0 such that MSo ::; Ch-;;l.

First we show surjectivity of P: H(M-t,g;M,A) ----> H(M-t,g;M,A) for all t 2: O. Let to 2: 0 be chosen such that P is surjective on H (M- t , g; M, A) for all 0 ::; t ::; to. Let s E [0, so]. Then we have [As, P] E W(I, g; M, A, 9). Let t E [0, to] and v E H(M-t-S,g;M,A).

==:;. A-sv E H (M-t,g;M,A)

==:;. 3u E H (M-t,g;M,A) , Pu = A_sv

==:;. v = AsPu = [As, P]u + PAsu ==:;. v - PAsu = [As,P]u E H (M-t,g;M,A)

==:;. 3w E H (M-t,g;M,A) , Pw = v - PAsu

==:;. v = Pw + PAsu = P(w + Asu)

where w+Asu E H(M-t-S,g;M,A). To show injectivity of P on H (M- t , g; M, A) for all t 2: 0, let to 2: 0 be

chosen such that PEL (H (M- t , g; M, A))-1 for all 0 ::; t ::; to. For s E [0, so] we have [P,A- s] E W (M- 2S,g;M,A,9). Let t E [0, to] and u E H(M-t-S,g;M,A)

with Pu = O.

===} PA-su = PA_su-A_sPu = [P,A-slu E H(M-t+s,g;M,A)

===} ::Jw E H (M-HS,g;M,A) (<;;; H (M-t,g;M,A)) , Pw = PA_su

===} P(w-A_su)=O where W-A_sUEH(M-t,g;M,A)

===} A_su = w E H (M-t+s,g;M,A)

===} u E H (M-t,g;M,A)

and thus, by the choice of t, u = o. o Definition 4.18. Let (M, A, Q, g) and (M, A, Q, fl.) be admissible structures of type (1/J). We call fl. a minorant of 9 if

• fl.(x.fJ :::; g(x.O for all x E M and all ~ E T;M . • hg, Mg(l, 0), and Mg(O, 1) are (M, A, g)-admissible weight functions of type

(1/J). -

Remark 4.19. If fl. is a minorant of g, then hg, Mg(l,O), and Mg(O, 1) are h;/-adapted. -

Corollary 4.20. Let (M, A, Q, g) be an admissible structure of type (1/J) and let (M,A,Q,g) be an admissible structure of type (1/J)s such that 9 is a minorant of g. Let M be an (M, A, g)- and (M, A, g)-admissible weight function of type (1/J) which is h-;l_ and h-;l-adapted. Then~ for each P E W(l,g;M,A,Q) n £'(H(l,g;M, A))-l, the ~perator P : H(M,g;M,A) -+ H(M,g;M,A) is invertible. Moreover, H (M, g; M, A) is a Banach space and there exists a constant C > 0 such that for all u E S(M) the estimate IIp-1UIIH(M.9;M.A) :::;

C IluIIH(M.9;M.A) holds.

Proof. By assumption, there exist C > 0 and N E N such that Mh{j :::; C. 4.16 yields operators

such that

A (Mhf) E W (Mhf,fl.;M,A,Q) (<;;; 1}i (Mhf,g;M,A,Q)),

A (M-1h;N) E W (M-lh;N, g; M, A, Q) ,

A (hf) E W (hf, g; M, A, Q) , and

A (h;N) E W (h;N,g;M,A,Q)

A (Mhf) A (M-lh;N) = Id = A (M-lh;N) A (Mhf) ,

A (hf) A (h;N) = Id = A (h;N) A (hf) .

By virtue of 4.17, A (h-;N) PA (h{j) E W(l, g; M, A, Q) n £, (H (1, g; M, A)) -1. Applying 4.17 again, we have

A (Mhf) A (h;N) PA (hf) A (M-lh;N)

E w(l, g; M, A, Q) n £, (H (1, g; M, A))-l

272 F. Baldus

Since A (Mh{;') A (h"iN) H (M, g; M, A) ---+ H (1, g; M, A) is an isomorphism,

this yields P(M) E £(H(M,g;M,A))-l, where P(M) denotes the extension of P: S(M) ---+ S(M) to H (M, g; M, A). Thus, there exists a constant C > 0 such that for all u E H (M, g; M, A) the estimate

iiP(M)-lUiiH(M,9;M,A) :::; C IluIlH(M,g;M,A)

holds. o

Remark 4.21. In the proof of the preceding corollary we have shown - under the assumptions of the corollary - that there exist "order reducing operators", i.e., operators A(M) E "iJ!(M, g; M, A, Q) and A(M- 1 ) E "iJ!(M- 1 , g; M, A, Q) such that A(M)A(M-l) = Id = A(M- 1 )A(M). In particular, A(M) : H (M, g; M, A) ---+

H (1, g; M, A) is an isomorphism and thus, H (M, g; M, A) is a Banach space and S(M) is dense in H (M, g; M, A).

Theorem 4.22. Let (M,A,Q,g) be an admissible structure of type ('lj;) and let (M,A,Q,~) be an admissible structure of type ('lj;)s such that ~ is a minomnt of g. Then we have

"iJ!(I,g;M,A,Q)n£(H(I,g;M,A))-l = "iJ!(I,g;M,A,Q)-l.

Proof. This is a consequence of the preceding corollary together with 4.15 and 3.20. 0

5. Submultiplicative Frechet topology for W(l, g; M, A, Q)

So far, we have not endowed "iJ!(I, g; M, A; Q) with any topology. Under the assumptions in 4.20 we proved that H (M, g; M, A) is a Banach space (cf. 4.21). Together with 3.20 this allows us to define a submultiplicative Fnkhet topology on "iJ!(1, g; M, A; Q) (d. 5.5). This result generalizes [2, Satz 3.8.5] in the case of ]Rn to manifolds. On the other hand, in [2] we proved submultiplicativity for all "iJ! (1, g)-classes in the Weyl-Hormander calculus without technical restrictions in contrast to the current situation. The method which we apply was developed by B. Gramsch, J. Ueberberg, K. Wagner [26] to prove submultiplicativity of 1J!~,o and applied by B. Gramsch, E. Schrohe [25] to prove submultiplicativity of Boutet de Monvel's algebra. In [1, p. 552] B. Gramsch formulates the question whether every "iJ!* -algebra is submultiplicative as an open problem. For the commutative case we refer to W. Zelazko [56, Theorem C], for the non-commutative case cf. also [56, Theorem 3]. B. Gramsch [23] and B. Gramsch, W. Kaballo [24] applied submultiplicativity in connection with non-abelian cohomology and Oka's principle.

Definition 5.1. Let (M, A, Q, g) be an admissible structure of type ('lj;) and (M, A, Q,~) an admissible structure of type ('lj;) s such that ~ is a minor ant of g.

1. For x E M and ~ E T; M we define Mg(O, O)(x, 0 := 1. Applying notation from 3.18, for k 1 , k 2 , h, b E No we define

L g(k1 , k2, h, l2) :=

12 (H (Mg(k1, k2)-1, g; M, A) , H (Mg(k l , k2)-1 Mg(ll, l2)-I, g; M, A)) .

2. We fix finite sets Wm(M,A,9) ~ Vm(M,A,Q) and Wd(M,A,9) ~ Vd(M,A,9) as in 3.20. For kENo let PdM, A, g, g) be the set of all operators such that for all k 1 , k 2, ll, l2 E No with kl + k2 + h + l2 ::; k, all VI"'" Vi, E Wd(M,A,9), Vi,+I, ... ,Vi , +12 E Wm(M,A,Q), and all permutations 7r of {I, ... , II + l2} the operator

ad (V7r(l))'" ad (V7r (1,+12)) P: Cg<'(M) ---+ D'(M)

has an extension to an operator in Lg(kl , k2, h, l2)' Moreover, for kENo let

1IPIIpdM.A,9.g) := max IIPll cg (kl,k2,hh)

where the maximum is taken over all kl , k2, ll, b E No with kl + k2 + II + l2 ::; k, all Vl, ... ,Vi, E Wd(M,A,Q), Vi I +1, ... ,Vi,+1 2 E Wm(M,A,9), and all permutations 7r of {I, ... , II + l2}.

Remark 5.2. Theorem 3.20 implies W(l,g;M,A,Q) = nkENo PdM,A,9,g).

Lemma 5.3. Let (M,A,9,g) be an admissible structure of type ('ljJ) and (M,A, g,!}) an admissible structure of type ('ljJ)s such that[L is a minorant ofg. For each kENo, PdM, A, g, g) is a Banach space.

Proof. Let kENo and let (AV)VEN be a Cauchy sequence in

(PdM,A,9,g), 11·llpdM.A,9,9))·

Let kl' k2' h, l2 E No with kl + k2 + II + l2 ::; k, VI"'" Vi, E Wd(M, A, g), Vi,+I, ... ,Vi , +1 2 E Wm(M,A,9), and let 7r be a permutation of {l, ... ,ll + l2}. Then, by definition of 11·llpdM.A.9.g)' (ad(V7r(J))'" ad(V7r(l, +12))Av) vEN is a Cauchy sequence in Lg(k1 , k2' ll, b) and thus, converges to an operator A(kl' k2, 7r, VI,.'" Vi d 12 ) E L g(k1 , k2, ll, l2)' For all kl' k2 E No with kl + k2 + II + l2 ::; k the operators A(kl , k2, 7r, VI, ... , Vi, +12) coincide on SCM) and thus, coincide with an operator A(7r, VI"'" Vi ,+1 2 ) : SCM) ---+ D'(M) with continuous extensions in L g(k1 ,k2,ll,b) for all kl,k2 E No with kl + k2 + h + 12 ::; k. Let A := A(0). Thus, Av ---+ A in L(H (1, g; M, A)). Consideration in local coordinates shows that ad(V7r(I)) ... ad (V7r(l I +1 2 ) )Au = A(7r, VI, ... , Vi, +12 )u. This yields A E Pk(M,A,9,g) and Av ---+ A in PdM,A,9,g). 0

Lemma 5.4. Let (M, A, g, g) be an admissible structure of type ('ljJ) and (M, A, g, g) an admissible structure of type ('ljJ)s such that 9 is a minorant of g. For kENo there exists Ck > 0 such that we have -

IIPQllpdM,A,9,g) ::; Ck IIPllpdM,A,9,g) IIQllpk(M,A,9,g)

for all P, Q E W(l, g; M, A, g).

274 F. Baldus

Proof. Let M 1 , M2 be two (M, A, g)-admissible weight functions of type ('lj;) which are H;--l-adapted. For P E W(M1 ,g;M,A,9), Q E W(M2,g;M,A,9), lEN, and VI"'" Vz E V(M,A,9), we have

ad(Vd ... ad(Vz)(PQ) = L [ad(Vi,) ... ad(ViJP] [ad(Vj,) ... ad(Vjl_JQ]

as an operator S(M) ----+ S(M), where the sum runs over all ° :S v :S l, 1 :S i 1 < ... < iv :S l, 1 :S jl < ... < jl-v :S l with {iI, ... , iz} n {jl, ... ,jl-v} = 0. 0

Theorem 5.5. Let (M, A, 9, g) be an admissible structure of type (1jJ) and (M, A, 9, g) an admissible structure of type ('lj;)s such that 9 is a minorant of g. For kENo let wk(g;M,A,9) be the closure ofW(I,g;M,A;-9) in

(Pk(M, A, 9, g), 11· llpd M ,A.Q,9)) .

Then the following statements hold:

1. For kENo, wk(g;M,A,9) is a Banach algebra. 2. W(I,g;M,A,9) = nkEN wk(g;M,A,9).

In particular, endowed with the obvious topology induced by Part 2, W(I, g; M, A, 9) is a submultiplicative Frechet algebra.

Proof. Part 1 is a consequence of 5.3 and 5.4 and the continuous embedding of Pk(M, A, 9, g) in L(H (1, g; M, A)). Part 2 is a consequence of

W(I,g;M,A,9) <;;;; wk(g;M,A,9) <;;;; Pk(M,A,9,g)

(k E No) and 5.2. o

6. Examples of algebras of pseudo-differential operators on manifolds

Example. Let M be an SG-compatible manifold in the sense of E. Schrohe [40] with an atlas A = {(Xj,Xj, K:j) I j = 1, ... ,m} For j E {I, ... , m} let Uj := K:j(Xj ) and Xj : T*Xj ----+ T*]Rn c::::' ]R2n, T;;Xj :;, ~ f---+ (K:(x),tdK:(x)-lO. Let

p',p2,8 '= (~)28 IIdxll2 + Ild~ll; g(x,!;,) . (x)2p, 2 (~)2P2

where ° :S PI :S 1, ° < P2 :S 1,1 - P2 :S 8 < P2 and let

gP,8 := (x ~)281IdxI12 + Ild~ll; (x,~) , 2 (x, ~)2p

where ° 0, V(x,O E Uj x ]Rn,

C;lg(~:B,8(.) < (Xjl)*gl,(x,O(-)

C;lg(~~~)(-) :S (Xj l)*g2,(x,O(-)

Moreover, let 9 = R I (g[l,l,OJ). Then the following statements hold:

1. (M, A, 9) is a generalized SG-compatible manifold. 2. (M,A,9,gl) and (M,A,9,g2) are admissible structures of type ('ljJ).

Moreover, defining fl.1 := g[1,1,oJ and fl.2 := g[l,OJ, then (M, A, g, fl.) is an admissible structure of type ('lj;)s and 9 is a minor ant of gi, i = 1,2.

-2

Example. Let M = M o U M1 U M2 U 8M 1 U 8M2, where Mo,M1,M2 are n-dimensional submanifolds, 8M I , 8M 2 are connected (n - I)-dimensional submanifolds, M o is relatively compact, 8Mo = 8M I U 8M N , for j E {I, ... , N}, M j is diffeomorphic to 8M j x (1, (0), and let ° :s: Pi :s: 1 (i E {a, 1, 2}), ° < P2 :s: 1, 1- P2 :s: 8 < P2. Applying the notation from the preceding example, with the help of a partition of unity we can construct a metric 9 on M such that the following statements hold:

• (M, A, g, g) is an admissible structure of type ('lj;). • ::lC > 0, Vi E {a, 1, 2}, Vx E M i , V~ E T;M, C- 1gi:i,t)2,8J :s: g(x,f;) <

Cg[Pi,P2,8J (x,i;) .

• (M, A, g, g[1,1.0J) is an admissible structure of type ('lj;)s and g[1,1,oJ is a minorant of g.

In particular, this example shows that situations appear where different Hormander metrics occur in different local models, i.e., in the admissible quadruples.

The following anisotropic examples are excluded by the assumptions which we impose on admissible structures of type ('lj;). But they fulfill the assumptions of quadruples which we call of type (p) and for which we introduced the general S(M,g)-pseudo-differential calculus in [3]. Moreover, the results stated below can be shown with minor modifications of the proofs which we have presented so far. The most important modification consists of working out the characterizations with V(M1' AI, gl) U V(M2' A 2 , g2) instead of V(M, A, 9).

Example. Let M(l) and M(2) be two compact manifolds. For i = 1,2 let g(i) be a continuous metric on M(i) and let A(i) be such that (M(i),A(i),g(i)) is an SG-compatible manifold. Let M, A, and 9 be the obviously defined objects on

the product. Moreover, let Kj"j2 := (K)~),K)~)), and Xh,h : T* (-{'X) x xg)) ---+

JR.n 1+n2 , (x,~) f--+ (Kh'h(X),tdKjl'12(X)-l~) (ji E {l, ... ,m;},i = 1,2). For ° < P1,P2:S: 1, 1 - min{Pl,P2} :s: 81 < PI, 1- min{P1,P2} :s: 82 < P2, and x,~ E JR.n, t1, T1 E JR.n" and t2, T2 E JR.n 2 , let

Pl,P2"l,,82(t t T T)'= (t\28 1 lit 112 + (t\202 1It 112 + II T 111; + II T 211; g(x,f;) 1, 2, 1, 2· 1..,/ 1 2 1..,/ 2 2 (02Pl (02P2

With the help of a partition of unity we can construct a metric 9 on T(T* M) such

that Vji E {I, ... ,m;} (i = 1,2), ::lC, Vx E Kj,,12 (.t'j,1) x Xj~2)), V~ E JR.n , +n2 ,

C- 1 gPl.P2,01,82 (.) < (X-:- 1 )*g (.) < C gPl,P2,8 1 .82 (.) . (x,i;) - Jl,J2 (x,i;) - (x,i;)

276 F. Baldus

For weight functions which are in local coordinates of the form (x,~) f---+ (~)S, s E JR, we have order reducing operators, and w(l, g; M, A, 9) is a submultiplicative w*algebra in £ (L2(JRn)).

Example. Let (M(1), A(1), 9(1)) and (M(2), A(2), 9(2)) be two SG-compatible manifolds of dimension n1 and n2, respectively. Moreover, for 0 < P1,P2 :::; I, 1-min{p1' P2} :::; 61 < PI, 1- min{p1' P2} :::; 62 < P2, and x, ~ E JRn , t1, T1 E JRn 1 , and t2, T2 E JRn2, let

(x, ~)2(h Ilh II; + (x, ~)282 Ilt211;

+ II T 111; + II T 211; (X,02P1 (x, ~)2P2

We apply notation analogous to that in the preceding example. There exists a metric 9 on T(T* M) such that

Vji E {I, ... , mil (i = 1,2), ::lC, Vx E "'j1,h (x2) x Xj~2)) , V~ E JRn l+n 2,

C- 1 g(~::),81,82(.):::; (X;~h)*g(x,~)(-):::; C g(~:f).81.82(-) .

For weight functions which are in local coordinates of the form (x,~) f---+ (x, ~)S, s E

JR, we have order reducing operators, and w(l, g; M, A, 9) is a submultiplicative w*-algebra in £ (L2(JRn)).

7. Generalized algebras of pseudo-differential operators

Combining the methods from [26], Chapters 2 and 3, we can define submultiplicative w* -algebras which cover different pseudo-differential calculi in different local models when we make choices concerning the operators to commute with. This procedure uses arguments analogous to those presented in Sections 4 and 5. More precisely, we fix a Riemannian metric on M and let .6. be the Friedrichs' extension of the corresponding Laplace-Beltrami operator starting from Cif'(M) if M is non-compact. Let c > 0 (e.g., c :::; 1/2) and A := (Id - .6.)"'. In the case of the classes W~,8 on JRn with 0 :::; 6 :::; P :::; I, 6 < I, we can take c := 1- 6 (cf. [50,4.1])

and thus, in the case of manifolds of the form M = M1 U M1 U M2 U 8M 1 where

• M j (j E {I, ... , N}) is diffeomorphic to JRn \ {x E JRnlllxl12 :::; I}, • M j (j E {I, ... , N - I}) is relatively compact, • 8M j ='" 8M j U 8Mj+1, j E {I, ... , N - I},

we may think of 0 < c :::; min{C1' c2} with C1 := 1 - 61 and C2 := 1 - 62 where o :::; 61 :::; PI :::; I, 61 < I, and 0 :::; 62 :::; P2 :::; I, 62 < 1. The invariance with respect to compactly supported diffeomorphisms for 1-P :::; 6 :::; P is not needed. Moreover, let N E N and, for j = 1, ... , N, let mj E COO(M) be a real-valued function such that, (e.g.) with a suitable constant C > 0, Imj(x)1 :::; C (dist(x,xo) + I), where Xo E M is fixed and possibly using the Levi-Civita connection for decay estimates


of "derivatives" of mj. Starting with the operators A and M j : 'P 1-+ mj' 'P, with the methods from [26, Chapter 2] we can define a submultiplicative \(1* -algebra Ao in £ (L2(M)) such that there exists a dense subspace 1) of L2(M) with CO"(M) ~ 1)

which is invariant under all operators in Ao and under A and M j , j = 1, ... ,N (cf. 4.10-4.16). The commutator conditions occurring in the definition of this algebra Ao are formulated without order shift. However, Ao might not even be pseudolocal.

With the methods from [26, Chapter 3] Ao is refined with the help of commutator conditions with order shift. For this purpose, we consider the scale of Sobolev spaces HJ.. (M), S E JR, associated to A and define commutator conditions within this scale of spaces. On different non-compact parts of the manifold (e.g., Ml and M 2 ) we might require commutator conditions with different shift properties for vector fields supported on these parts. This procedure maintains the \(I-property since all operations can be considered on V ~ ns HJ.. (M) (cf. 4.17). Moreover, we can endow the algebra Aoo obtained in this way with a submultiplicative Frechet topology (cf. [26, (3.5) and Theorem 3.8]).

Acknowledgement. I am very grateful to Professor B. Gramsch for proposing to me to generalize the S(M,g)-calculus to manifolds and to investigate spectral invariance and submultiplicativity as well as for valuable support and helpful suggestions. Moreover, I wish to thank Dr. R. Lauter for many fruitful discussions and some good advice. Finally, I thank Dr. J. Lutgen, Dr. J. M011er, and Dr. O. Caps for useful comments and for correcting my English.

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[51] A. Unterberger. Encore des classes de symboles. In Sem. Goulaouic-Schwartz 1977-1978, Ecole Poly technique, Paris, 1977. 17 pages.

[52] A. Unterberger. Oscillateur harmonique et operateurs pseudo-differentiels. Ann. Inst. Fourier, 29-3:20l-221, 1979.

[53] A. Unterberger and J. Bokobza. Les operateurs de Calderon-Zygmund precises. C. R. Acad. Sci. Paris, 259:1612-1614, 1964.

[54] A. Unterberger and J. Bokobza. Sur les operateurs pseudo-differentiels d'ordre variable. C. R. Acad. Sci. Paris, 261:2271-2273, 1965.

[55] H. Widom. Singular integral equations in £P. Transactions Am. Math. Soc., 97:131-160, 1960.

[56] W. Zelazko. Concerning entire functions in Eo-algebras. Studia Math., 110:283-290, 1994.

F. Baldus Fachbereich Mathematik Johannes Gutenberg-UniversiUit Mainz Staudingerweg 9 D-55099 Mainz, Germany e-mail: [email protected]


Domain Perturbations and Capacity in General Hilbert Spaces and Applications to Spectral Theory

Andre Noll

1. The significance of the bottom eigenvalue in mathematical physics

In this section we explain the significance of the bottom eigenvalue of a self-adjoint operator in various areas of mathematical physics, so this will be a motivation for investigating and estimating the bottom eigenvalue. The main role is played by the Laplacian because this is one of the most important self-adjoint operators in mathematical physics as it plays a fundamental role in quantum mechanics, theory of heat, theory of vibrations and other areas. ___ Qn the other hand, higher-order differential operators are important as well. For instance the dynamics of the clamped plate is described by the bi-potential equation

6 2u = 0, where 6 2 is the biharmonic operator, subject to Dirichlet boundary conditions. Another example of a relevant higher-order differential operator is the analysis of the vertices of incompressible fluids which also leads to the biharmonic equation.

Most of the known estimates for the bottom eigenvalue only apply to secondorder differential operators and some of them [Szn98], [Tay79], [Oza81], [MR84], can only handle the Laplacian. This is due to the fact that in the case of secondorder differential operators there is an interplay between analysis and stochastics via the theory of Dirichlet forms. One can prove that for certain second-order differential operators, see Appendix C.l for a precise statement, there is a stochastic process associated to the operator. It turns out that the existence of such a process is closely related to positivity preserving properties of the semigroup (and the resolvent) and to the maximum principle, [BH86, FOT94, MR92, DC]. This allows us to treat analytic problems, e.g., estimating eigenvalues, with stochastic methods or by using the powerful tools of potential analysis and vice versa.

It is known that in contrast to the second-order case there is no such interplay for higher-order differential operators. Therefore the proofs of the eigenvalue estimates which rely on stochastic processes or the maximum principle typically do not carryover to the higher-order case.

282 Andre Noll

In order to include higher-order differential operators one needs a more general method. The results for self-adjoint operators in abstract Hilbert spaces (i.e., not necessarily L2-spaces) which are described in Section 5 turn out to be an appropriate tool in estimating eigenvalues of a wide class of self-adjoint operators. For instance they are sufficient to handle eigenvalue estimates for higher-order differential operators.

1.1. Preliminaries on the Dirichlet Laplacian on a bounded domain

Let Q be an open and bounded subset of IRd. The following Poincare inequality is valid for all u E H(5(rl),

r lul 2dx ::; Co. j l\7uI 2dx, Jo. 0.

where Co. is a geometric constant and may be taken four times the diameter of Q, see [Joh7l], page 99. From the functional analytic point of view this inequality simply states that the (negative) Dirichlet Laplacian -~n in L2(Q) is a strictly positive operator, i.e., there is a constant c> 0 such that

(-~nu,u) ~ cllull 2

for all u E dom(H). Here -~o. is defined as the unique self-adjoint operator which is associated to the closure of the quadratic form

f[u] := ll\7u I2dX,

with u in the initial form domain C~ (Q), the set of smooth functions with compact support in Q, see Section 2.

The bounded ness of Q implies that the resolvent (-~o. - z) -1, z in the resolvent set of -~n, is a compact operator in L2(Q), see [RS78], p. 255. Since compactness of the resolvent is equivalent to discreteness of the spectrum, 0"( - ~o.)

consists only of isolated eigenvalues and each of them is of finite multiplicity. If additionally Q is connected, i.e., ~o. is the Dirichlet Laplacian on a bounded domain, it is known that the bottom eigenvalue }.1 is simple, see [Goe77]. In fact this non-degeneracy of the ground-state holds in great generality for secondorder elliptic differential operators even in the more general case of Riemannian manifolds. This can be proved with the aid of a Perron-Frobenius argument which uses the positivity preserving property of the semigroup (ett:;." )t>o. A proof can be found in [Dav89], Theorem 5.2.1 and Proposition 1.4.3. In contrast to the secondorder case, higher-order operators may have a degenerate bottom eigenvalue. For example the ground-state of the Bi-Laplacian ~2 on a punctured disc turns out to be degenerate, see [CD92], or the survey article [Dav97].

1.2. Interpretation of the bottom eigenvalue for the heat equation

With the notion of the previous section, look at Q as a body in Euclidean space which carries an initial heat distribution f E L2(Q). As time goes by the heat will

Domain Perturbations and Capacity in General Hilbert Spaces 283

distribute itself among 0. According to the laws of physics the time evolution of the system is given by the heat equation

a d a2 at u(t,·) = L ax2 u(t,·) = Doou(t, .), u(O,·) = f(-), t > O. (1.1)

j=l J

The choice of Dirichlet boundary conditions describes the physical situation that the boundary of ° is held on temperature zero. The unique solution of the heat equation (1.1) is given by the semigroup (e t 6.")t>o, i.e., we have

u(t,·) = et6." f, t > O.

Let us arrange the eigenvalues of -Don in increasing order

o < Ai < A2 :S A3 :S ... ,

where each Aj has to be repeated according to its multiplicity. Let (<Pi )iEN be a corresponding sequence of normalized eigenfunctions.

The heat content stored in the system at time t and with initial distribution f is

Q(t,j):= i u(t,x)dx = J et6.<lfdx = 1 fe-Ajt(<Pi,f)<Pi. (1.2) o 0 0 i=l

If f == 1, i.e., in the beginning the body is at constant temperature 1, then Q(t) := Q(t, 1) will be a decreasing function of t because of the Dirichlet boundary conditions. This can also be read off from equation (1.3) below. It is therefore natural to ask for the speed of decay of Q(t). From (1.2) we get

Q(t) = 12 et6."ldx = L ~ e-Ajt(<pj, l)<pj = ~ e- Ajt 112 <Pj dxl2 (1.3)

From this equality we immediately deduce the estimates 2 oc

e-A)t 112 <Pi dxl :S Q(t) :S 101 ~ e-A,t = 10Itr(et 6.,,),

where tr( et6.,,) denotes the trace of the semi group (e t6. n ). Clearly e- A1 t in the above sum is the most significant term. Moreover the norm of et6." as an operator in L 2 (0) equals e- A1t by the functional calculus. Putting all together we agree that the bottom eigenvalue Ai measures how quickly the system cools out.

1.3. The significance of the bottom eigenvalue in quantum mechanics The states of a quantum mechanical system are described by the non-zero vectors of a separable Hilbert space 1{ over the complex field. Two vectors u, v E 1{

describe the same state if u = AV for some scalar A i= O. Hence we may always assume that the vector u E 1{ has unit length.

Physical quantities which can be determined experimentally are called observables and it is a basic postulate of quantum mechanics that each observable corresponds to a unique self-adjoint operator in 1{. One very important observable

284 Andre Noll

is the energy. The corresponding self-adjoint operator H is called the Hamiltonian or the Schrodinger operator. Since arbitrarily high energies should be possible, the Hamiltonian turns out to be always an unbounded operator. Hence the domain dom(H) of H is a dense subspace of 'H and we always have dom(H) =I- 'H (otherwise H would be bounded because of the closed graph theorem). The energy of a state u E dom(H) is given by the expectation value

E(u) = (u, Hu)

and is a real number since H is self-adjoint. Observe that E(u) = E(AU) for each scalar A of modulus 1.

Assume that at time t = 0 the system is in state Uo E 'H. Then at time t the state of the system is represented by some vector u(t) E 'H and it is again a postulate of quantum mechanics that the 'H-valued function t f--> u(t) satisfies Schrodinger's equation

where n = 1.055.10-34 Js is the Planck constant. Clearly n enters only as a physical constant without mathematical relevance. We shall therefore assume from now on that n = 1.

The solution of the Schrodinger equation is given by the unitary group (e-itH)t>o, i.e., we have

u(t) = e-itHuo, t 2 o.

If Uo is an eigenvector of H with eigenvalue A, then clearly

(1.4)

Since Uo and e-itAuo describe the same state, the eigenvectors of H are often called bound-states of the system. The converse can also be proved easily: If the state is time-independent, then it must be an eigenvector of the Hamiltonian. From equation (1.4) we see that

E(u(t») = E(uo) = Alluol12 = A.

Thus the energy of a bound-state equals the corresponding eigenvalue of the Hamiltonian. Of particular physical interest is the ground-state of the system (if it exists), i.e., the state that minimizes the energy. This minimal energy is called the groundstate energy and is exactly the bottom eigenvalue Al of H. One can compute Al explicitly in the case of the hydrogen atom, for the quantum mechanical harmonic oscillator and some other systems, see [AGHKH88] for a comprehensive treatment of solvable models in quantum mechanics. Nevertheless it is far from obvious to compute the ground-state energy of more complicated systems. It is therefore of physical interest to have estimates for AI.


1.4. Domain perturbations in mathematical physics Let n be a "pitcher filled with liquid", by which we simply mean an open and bounded subset of JRd. Let -~o be the Dirichlet Laplacian in L 2 (n), defined as the Friedrichs extension of the Laplacian, initially defined on C;:O (n), cf. Section 2.1. As described in Section 1.2, the semigroup (et6n )t>o determines the solution u(t,·) of the heat equation

8 d 82

-8 u(t,·) = L 8 2 u(t,·) = ~ou(t, .), u(O,·) = j, t > O. (1.5) t x

j=1 J

Moreover u(t,·) behaves asymptotically like ce-tAl(D) as t ----> 00, where )q(n) is the bottom eigenvalue of -~o. Hence the size of Al(n) is a measure of the efficiency of cooling.

Now let K be a compact subset of n and consider the Dirichlet Laplacian -~n\K in L2(n \ K) which may be defined by the same method we used to define -~o, see Figure 1. Physically speaking, -~O\K describes the situation that the

FIGURE 1. Domain perturbations

"pitcher" n is held on temperature zero in K. Thus K models a cooler maintained at temperature zero. It is clear for physical reasons as well as from the fact that the form domain of -~n\K may be viewed as a subspace of the form domain of -~Sl' that the bottom eigenvalue Al (n \ K) of -~n\K is greater than or equal to Al (n). If K consists of many small pieces one may look at K as crushed ice. From the mathematical point of view it is an interesting problem to determine the improvement in cooling efficiency resulting from the extra cooler K, i.e., the question is how the difference Al (n\K) -A(n) depends on K. Crude guesses might be that the surface of K measures this improvement of cooling efficiency but this turns out to be wrong as was proved in [Tay76J. See also [RT75aJ, [Rau75J, [RT75b], as well as [Sim79], Section 22 for a general survey on the crushed ice problem. In fact the capacity of the set K determines the shift of the bottom eigenvalue. This is a surprising fact because the capacity of a ball of radius T in JR3 is given by some constant times T, and the above result yields

COT::; Al (n \ K) - A(n) ::; Cl T

286 Andre Noll

in this particular case. This means that the radius of the ball is the decisive quantity rather than its surface area.

Let us finally mention that there are many other physically relevant questions which lead to the same mathematical problem: Give estimates on the shift of the bottom eigenvalue of a self-adjoint operator which is restricted to a smaller domain. A further example is the "fireman's pole problem" which consists in describing the effect of the presence of a perturbation for the propagation of sound. Another example is to explain rigorously why a cloud of small conductors sprayed into the air appear solid on a radar screen.

2. Defining of operators by quadratic forms

In this section we recall the method of defining self-adjoint operators by the quadratic form technique. The first subsection summarizes without proofs the necessary facts on this rather standard topic whereas Section 2.2 emphasizes several advantages of this method. In particular we give an easy example of operators H and V in L2(IR) for which dom(H) n dom(V) = 0, but the intersection of the according form domains happens to be dense in L 2 (IR).

2.1. The one-to-one correspondence between quadratic forms and self-adjoint operators

Let H be a real or complex Hilbert space. If B is a bounded operator in H, the sesquilinear form

feu, v) := (u, Bv), u, v E H

satisfies, by Cauchy-Schwarz inequality

If(u,v)l:s; Cilulllvll, (2.1)

where C = IIBII. Conversely, let f be a sesquilinear form on H x H satisfying inequality (2.1). Then for each u E H the linear functional

Fu(v) := feu, v)

is bounded. By the Riesz representation theorem there is a unique vector Bu E H such that

Fu(v) = (Bu, v)

and it is easy to see that the map u f-7 Bu defines a bounded linear operator in H with IIBII :s; C. Moreover B is self-adjoint if and only if f is hermitian, i.e., feu, v) = f(v, u) for all u, v E H.

The situation becomes much more complicated if the operators/forms are no longer bounded, but it turns out that the one-to-one correspondence described above carries over to certain unbounded operators. This allows us to define selfadjoint operators by means of hermitian sesquilinear forms or by quadratic forms since the quadratic form of a hermitian sesquilinear form already determines the sesquilinear form by the polarization identity.


Since all theorems of this section are standard results which can be found in many text books, for instance in [Kat80] or [Wei76], we only state the results without proof.

Definition 2.1. Let F be a dense subspace of H and let E be a hermitian sesquilinear form on F x F.

(a) (E,F) is called semi-bounded from below with bound ry E IR. if

E[u] := E(u, u) 2: ryllul1 2

for all u E F. If, in particular, ry = 0, then (E, F) is called non-negative. (b) (E, F) is called closed if F together with the scalar product

E1-,(u, v) := E(u, v) + (1 - ry)(u, v)

is a Hilbert space.

Now the one-to-one correspondence between forms and operators reads as follows.

Theorem 2.2. (a) Let F be a dense subspace of H and let E be a hermitian sesquilinear form on

F x F which is semi-bounded from below and closed. Then there is a unique self-adjoint operator H in H such that

dom(H) c F and (u,Hv; = E(u,v)

for all u E F and all v E dom(H). One has

dom(H) = {u E F::3v E H, 't:/w E F: £(w,u) = (w,v)}.

In fact this v is unique and satisfies H u = v. (b) Let H be a self-adjoint operator in H such that (u, Hu) 2: ryllul1 2 for all

U E dom(H). Define F:= dom((H - ry)1/2) and

£(u, v) := ((H - ry)1/2U, (H - ry)1/2V) + ry(u, v)

for u, v E F. Then (£, F) is a closed form which is bounded below with bound ~(. Moreover the unique self-adjoint operator associated to (£, F) according to part (a) coincides with H.

In practice it is difficult to determine a dense subspace F that makes (£, F) closed. This leads to the following definition.

Definition 2.3. Let (£, F) be a densely defined hermitian sesquilinear form which is semi-bounded from below. (£,F) is called closable if for any Cauchy sequence (Un)nEN in the pre-Hilbert space (F, £1-,(', .)) with Un ----+ 0 in H we have Un ----+ 0 in (F,£1-,(', .)).

Remark 2.4. Let (F, £1-,(" .)) be the completion of (F, £1-,(" .)). Then the embedding J: (F,£1-,C, .)) '---+ H extends to a bounded linear operator

J: (F, £1-,(', .)) '---+ 1t.

The form (£, F) is closable if and only if J is injective.

288 Andre Noll

The following Lemma justifies the name "closable".

Lemma 2.5. Let (E, F) be a closable form. Then there is a unique smallest closed extension of (E, F) which is called the closure of (E, F).

An important application of the theory of quadratic forms is the Friedrichs extension of a symmetric operator.

Theorem 2.6. Let S be a symmetric operator in H satisfying

(u, Su) 2': I'IIul1 2

for all u E domeS). Then the hermitian sesquilinear form

E(u, v) := (u, Sv), u, v E F:= domeS)

is closable.

By Theorem 2.2 there is a unique self-adjoint operator associated to the closure of (E, F) which is called the Friedrichs extension of S.

2.2. Advantages of the quadratic form technique

Defining a self-adjoint operator via the quadratic form technique has several advantages. We illustrate some of them here: Firstly the sum of two quadratic forms coming from self-adjoint operators HI, H2 again defines a self-adjoint operator HI -i-- H2 if the intersection of the two form domains is dense which often can be checked very easily. HI -i-- H2 is called the form sum of HI and H2 and may differ from the operator sum

dom(HI + H 2) := dom(Hd n dom(H2), (HI + H2)U = Hlu + H 2u,

which may happen to be defined only for the zero vector although the form sum is densely defined. We next give an example for this phenomenon which is taken from [Bra94].

Example. Let (Xn)nEN be an enumeration of the rational numbers in ITt Define for each n E N the function Vn E LI(lR) by

Vn(x) := L 2-j lx - Xj I-I/2e-lx-xjl, x E lR \ {Xl, ... , x n }, j~n

see Figure 2. An easy calculation shows that (Vn)nEN is a Cauchy sequence in LI(lR). Hence the Ll-limit

V:= lim Vn n--oo

belongs to LI (lR) and is a non-negative function. Therefore the form sum - dd:2 -i-- V defines a self-adjoint operator in L2(lRd ). On the other hand

l V(x)2dx = 00


'" . .

FIGURE 2. The function Vn for n = 5 and Xj = j. Vn has integrable poles at each Xj. As n increases, the set of poles becomes dense in IR but the limit limn -+CXl Vn exists as a function in L 1 (IR).

for any non-void open set n c IR. Thus V f (j. L2(IR) for any continuous function f =I- 0, i.e., the only continuous function in the domain of the multiplication operator corresponding to V is the zero function. Since

dom ( - d~2 ) = H2(IR) c C(IR)

by the Sobolev embedding theorem, cf. [Ada78], Theorem 5.4, we have

dom ( - ::2 ) n dom(V) = 0,

i.e., the operator sum - d~2 + V does not define a self-adjoint operator in L 2 (IR).

Secondly there is a concept for monotonicity of quadratic forms which allows one to handle limiting processes in a fairly simple way. To be concrete, let (Ej, Fj) JEN be an increasing sequence of quadratic forms associated to self-adjoint operators H j . This means that the form domains F j are decreasing and for all U E F:= njENFj the sequence (Ej[U])jEN is increasing. Then the limit form

E[uJ := lim Ej [u], J-+CXl

defined for all U E F where this limit is finite, is associated to a self-adjoint operator H if it is densely defined, see [RS80J Theorem S.14. Moreover we have H j -f H in strong resolvent sense. There is no analogous result for (unbounded) operators.

290 Andre Noll

Finally we give another advantage of the quadratic form technique in the case of elliptic differential operators which is the most striking aspect from our point of view. For simplicity we only consider the Dirichlet Laplacian on a bounded set n in ]Rd. One can always define -lln via the closure of the quadratic form

(2.2)

In particular, domains with highly singular boundary such as fractal domains can be treated. For such domains it is difficult to define the operator -lln directly because it is difficult to describe in which sense the functions in dom( -llSl) should vanish on the boundary.

3. Domain perturbations

Considering a formally self-adjoint differential expression (e.g., the Laplacian) in two different Hilbert spaces leads to the question of comparing the spectra of the two resulting self-adjoint operators in terms of these Hilbert spaces. See Section 1.4 for a motivation of this kind of problem in mathematical physics.

In Section 3.1 we only consider local operators in L2-spaces and indicate the difficulties in defining self-adjoint realizations of the restricted operator directly by means of the operator. These difficulties can be overcome by using the quadratic form technique to define the domain perturbations.

This general method is described in Section 3.2 and it is illustrated how the perturbed operators appear in the case of differential and integral operators.

Section 3.3 treats another method of defining domain perturbations in the case of regular Dirichlet forms. With the aid of the associated Hunt process one can define a new process by killing the sample paths when leaving a prescribed set. This new process turns out to be again a Hunt process, hence it gives rise to a unique self-adjoint operator. As we shall see, this operator coincides with the domain perturbations in the sense of Section 3.2.

Finally, in Section 3.4 we collect some simple properties of the perturbed operator which immediately follow from the construction.

3.1. The L2-case

Here we explain the difficulties of directly defining domain perturbations in an L2-space by means of the operator itself. Let (X, A, mo) be a measure space, that is, X is an arbitrary non-empty set, A is a i7-algebra on X and mo is a measure defined on A. If X is a topological space there is always a natural i7-algebra 8 0 on X, namely the smallest i7-algebra that contains all open subsets of X. 8 0 is called the Borel i7-algebra of X. Let 8 be the completion of 8 0 with respect to mo and let m be the extension of mo to 8, that is

8

m(A) {A eX: ::JBI , B2 E 8 0 such that BI cAe B 2},

mo(BI).


Passing from (X, 8 0 , mo) to (X, 8, m) has the advantage that every subset of a nullset is measurable and has of course measure zero. When dealing with a topological space X we always take 8 as the CT-algebra and simply write (X, m) for (X, 8, m).

The set L2(X, m) consisting of all measurable and real-valued (complexvalued) equivalence classes of square-integrable functions on X becomes a real (complex) Hilbert space with respect to the scalar product

(u,v)P(X,m) := [ uvdm, (3.1)

where the bar is meaningless in the real case. We shall omit the subscript in (3.1) if it is clear from the context which underlying measure space is meant. If X is some open subset of lRd and m is the Lebesgue measure, we will further simplify our notation by writing L2(X) for L2(X, m).

Suppose that u is a measurable function on the topological space X. If u is continuous, the support of u is the set

supp(u) := {x EX: u(x) =I- O}.

For functions which are merely measurable this definition of the support can cause trouble as it depends on the representative in the equivalence class of u. For instance the zero function on lR may be written as u(x) = 1 if x is rational and zero elsewhere. We therefore define the support of u for any measurable function u on X slightly differently, namely

supp(u) := X \ U{ u : U C X, U open, u(x) = 0 m-a.e. on U} (3.2)

It is easy to see that the two definitions of the support coincide if u is continuous. Note that the right-hand side of equation (3.2) is exactly the definition of the support of the measure /Llul on 8 which is defined by

/Llul (B):= r lui dm . .IB A (bounded or unbounded) operator H in L2(X, m) is called local if

supp(Hu) c supp(u)

for all u E dom(H). Standard examples for local operators are differential operators which shall be treated in detail below. If Y is any measurable subset of X, the restriction of m to the trace-CT-algebra

8 ' := {Y n B : B E A}

yields a measure m ' on 8 ' . In this way we obtain a Hilbert space L2 (Y, m') which may be viewed as a subspace of L2(X, m) in the obvious way. In what follows we omit the dashes and use also the letter m for the restriction m'.

Suppose now that H is a self-adjoint and local operator in L2(X, m) and Y is an open subset of X. One can try to define a self-adjoint operator H Y in L2(y, m) by starting from the restriction of H to the space of all functions in dom(H) which are contained in L2(y, m), i.e., which have support contained in Y. Clearly this

292 Andre Noll

yields a hermitian operator in L2(Y,m). If X and Yare open subsets oflRd , m is the Lebesgue measure and C;:"'(X) is contained in the domain of H, it is also clear that H Y is even symmetric. Hence H Y is closable. Nevertheless this construction does not yield a self-adjoint operator in L2(y, m) as is seen from the following easy example.

Example. Let X = (-1,1) C IR and let H be the Dirichlet Laplacian in L2(X). Further let Y = (0,1) and denote by H Y the restriction of H to the space of all functions in the domain of H with support in Y. If u E dom(HY ), then all derivatives of u vanish on the boundary points of Y. Consequently the closure of H Y is not a self-adjoint operator, i.e., H Y is not essentially self-adjoint.

The crucial point of this example is of course that the restriction of the operator domain already determines the value of all derivatives of u at the boundary of Y if u E dom(HY). This makes dom(HY ) too small for essential self-adjointness. In order to obtain a self-adjoint operator one has to introduce suitable boundary conditions which turns out to be a difficult procedure in the general case of differential operators of higher order, see [RSS94], Section 2 and [Agr97]. However, in the next subsection we shall see that there is a much simpler way to define domain perturbations which is based on quadratic form techniques and avoids the above-mentioned troubles. Moreover this method of defining self-adjoint operators makes even sense in a general Hilbert space setting.

3.2. The general Hilbert space case

Let (H, (-, .) be an arbitrary real or complex Hilbert space and let H be a selfadjoint operator in H which is semi-bounded from below with spectral bound>' :=

inf a(H). Let (E, F) be the non-negative closed quadratic form which corresponds to H - >. in the sense of Theorem 2.2. We then have F = dom((H - >.)1/2) and

E(u, v) := ((H - >.)1/2U, (H - >.)1/2v) U, v E :F. (3.3)

In what follows we will use the abbreviations

E1(u,v) := E(u,v) + (u,v), E[u]:= E(u,u), E1[U]:= E1(U,U). (3.4)

Since (E, F) is closed, the space (F, E1 (', .» is a Hilbert space. Observe also that E depends on >., a dependence which is suppressed in our notation.

Consider the following situation: Suppose that 9 is a closed subspace of the Hilbert space (F, E1 (', .». Then the form (E + >., Q) is (of course) semi-bounded from below and closed. Moreover it is densely defined in He;; := Q, the closure being taken with respect to the weaker topology of H. Hence there is a unique self-adjoint operator He;; in He;; that corresponds to (E + >., Q) in the sense of Theorem 2.2.

Definition 3.1. We call the operator He;; the domain perturbation of H with respect to the subspace g.

Let us illustrate this construction in the case of the Dirichlet Laplacian.


Example. Let 0 and A be open subsets of IRd with A c 0 and let H = -~n be the Dirichlet Laplacian in ri = £2(0). The form (£,F) which corresponds to H is given by

F = HJ(O), £[u] = llV'u l2 dX for u E F,

see Section 1.1. Clearly the Sobolev space 9 := HJ(A) may be viewed as a closed subspace of F. Hence we can apply the above construction. By definition the Hilbert space rig equals the £2-closure of g, i.e., we have rig = £2(A). Moreover the restriction of (£, F) to 9 is obviously just the form which is associated to the Dirichlet Laplacian in £2(A). Hence Hg = -~A.

In general the space dom(H) n rig is not an invariant subspace for H but nevertheless Hg defined via the quadratic form (£, g) is an operator in rig. It is therefore natural to ask for a characterization of dom(Hg) as well as for the action of Hg on this domain. This question is partially answered by the following proposition.

Proposition 3.2. We have dom(H) n rig c dom(Hg) and Hgu = rrg Hu for u E dom(H) n rig, where rrg E B(ri) denotes the orthogonal projection onto rig.

Proof. By Theorem 2.2 (a) the domain of Hg is given by

dom(Hg) = dom(Hg -,\) = {u E 9 : :3v E rig 'Vw E g: £(u, w) = (v, w)},

and (Hg - ,\)u = v in this case. Therefore let U E dom(H) n rig, put v .rrg (H - ,\)u and let w E 9 be arbitrary. Then

£(u,w) = (H - '\)u,w) = (H - '\)u,rrgw) = (rrg(H - '\)u,w) = (v,w).

Consequently u E dom(Hg) and (HG - '\)u = v = rrg(H - '\)u = rrg Hu - '\u, which completes the proof. 0

Remark 3.3. If H is bounded, then F = dom(H) = ri and the topology induced by the scalar product £1 (., .) coincides with the topology of ri. We conclude that 9 = rig. Therefore Proposition 3.2 gives us

9 = dom(H) n rig c dom(Hg) c g.

Hence dom(Hg) = rig and Hg is exactly the restriction of rrg H to rig.

The following example shows that the domain perturbation of a bounded integral operator in £2(X, A, m) to a subspace of the form £2(y, A, m) is just the integral operator which arises from the truncated kernel.

Example. Let (X, A, m) be a measure space and k : X x X ---+ C be an integral kernel which induces an everywhere defined integral operator K, that is:

(i) For all f E £2(X, A, m) and almost all x E X the function k(x, ·)f(·) belongs to £l(X, A, m).

(ii) For all f E £2(X, A, m) the function Ix k(-, y)f(y)dy belongs to the space £2(X,A,m).

294 Andre Noll

The induced integral operator is then given by

K f = Ix k(·, y)f(y)dy. (3.5)

Because of the first condition the integral in equality (3.5) is well defined for all f E £2(X, A, m) and the second condition guarantees that K maps £2(X, A, m) into itself. One can prove, see [HS78] Theorem 3.10., that every integral operator K with dom(K) = £2(X, A, m) is closed. Hence, by the closed graph theorem, K is bounded. In order to have self-adjoint ness we have to assume additionally that k(x, y) = k(y, x) for all x, y E X.

N ow let Y be a measurable subset of X. Denote the restriction of k to Y x Y by kY . Clearly kY induces a bounded and self-adjoint operator K Y in £2(y, A, m). We claim that K Y coincides with the operator we obtain by the procedure described in the beginning of this section. For the proof first note that :F = £2(X, A, m) since K is bounded. Hence 9 := £2(y, A, m) is a closed subspace of:F and we may define the self-adjoint operator Kg according to Definition 3.1. By Remark 3.3 we have

dom(Kg ) = £2(y, A, m)

and Kgu = rrg Ku for all u E £2(y, A, m), where rrg E B(£2(X, A, m)) is the orthogonal projection onto £2(y, A, m). We write rrgu = XyU with XY being the characteristic function of Y. Putting all these facts together we obtain for u E £2(y, A, m),

Kgu XyKu

Xy Ix k(·, y)u(y)dy

i xyk(-, y)u(y)dy

i kY (-, y)u(y)dy

KYu.

Remark 3.4. The preceding example suggests the following method for estimating the shift of the eigenvalues of differential operators which are subjected to a domain perturbation: Suppose that H is a self-adjoint and semi-bounded differential operator in £2(0,) with 0, an open subset of ]Rd. By the functional calculus the spectrum of the semigroup (e-tH)t>o is given by a(e-tH ) = e-tO"(H). In particular, if a(H) is discrete, then so is the spectrum of e- tH in each interval (c, (0). Therefore the semigroup carries the same spectral information as H and it seems to be easier to analyze the semigroup rather than the differential operator H itself, since then one has to deal only with bounded operators.

If, in addition, the semigroup is ultracontractive, i.e., each e-tH maps £2(0,) boundedly into £=(0,), one can easily conclude from the Dunford-Pettis Theorem,

d. [Tre67], that each e- tH is an integral operator whose kernel k t (-, .) satisfies

This result is sometimes called Korotkov's Theorem [Kor65] although the DunfordPettis Theorem is much older. It is well known, cf. [Sim82], that ultracontractivity holds in great generality for a large class of Schrodinger operators, in particular it holds for the Laplacian.

The idea for estimating the perturbed eigenvalues is the following: If .A is an eigenvalue of H, then e- t )" is an eigenvalue of e- tH . Hence it might be easier to treat the domain perturbation of the bounded integral operator e-tH than the perturbation of H itself while also giving estimates for .A. But this does not work since the perturbed semigroup is not the semigroup which corresponds to the perturbed operator, i.e., with our notation we have

(3.6)

In the case of the Dirichlet Laplacians H = - ~n, HQ = - ~i\. with open and bounded sets A, D C ]Rd satisfying A cD, cf. Example 3.2, this can be seen by the stochastic interpretation of the semigroup via the Hunt process which corresponds to Brownian motion in ]Rd, see Appendix C. Given any subset A of ]Rd, define the first hitting time of A by

TA :=inf{t>O:Xt EA}.

For x E A we have

(e- tH9xi\.) (x) (e t .6. A Xi\.) (x)

lEx{Xi\.(Xt ) : TiV > t} IP'x{Ti\.c > t,Xt E A}

IP'x{T/I" > t,Xt E A,Tnc > t}.

Our assumption A cD implies that IP'x{T/lc S t,Xt E A,Tnc > t} > O. Hence the calculation continues:

(e- tH9xi\.) (x) < IP'x{Ti\.c > t,Xt E A,TW > t} +lP'x{Ti\.c S t,Xt E A,Tnc > t}

IP'x{T~lc > t,Xt E A}

lEx{X/I(Xt ) : Tnc > t} (e t .6."xi\.) (x).

Since x E A, we end up with

(e- tH9 Xi\.) (x) < Xi\.(x) (et.6."X/I) (x) = ((e-tH)Q Xi\.) (x).

In particular we get (3.6). Therefore we will not discuss this idea further.

296 Andre Noll

Finally we prove a generalization of Dynkin's formula, which has been published in [Nol97b]. To this aim we first take a closer look at Dynkin's formula from a functional analytic point of view. Let (£, F) be a regular Dirichlet form in L2(X, m) with corresponding self-adjoint operator H. See Appendix C.1 for a brief introduction to regular Dirichlet forms. Further let Y be an open subset of X such that its complement K = X \ Y is compact. By Theorem C.20 (a) the restriction of £ to the space

F K := {u E F : u = 0 q.e. on K}

corresponds to a unique self-adjoint and non-negative operator HY in L2(y, m). Dynkin's formula (see Theorem C.23) entails that

(H + 1)-1 f = (HY + l)- lUly) + Ul f-lCff+l)-' f, (3.7)

valid for all non-negative f E L2(X, m). Here f-lkH+ 1)-' f is the unique Radon measure of finite energy integral which is associated to K and (H + 1)-1 f in the

(H+l)-'f sense of Theorem C.23 (a) and Ul f-lK denotes its I-potential. In view of Theorem C.20 (b) we have

U ,,cH+l)-' f = P -L (H + I)-If l""'K FK '

hence equality (3.7) becomes

(H + 1)-1 f = (H Y + I)-lUI ) + PF-L (H + 1)-1 f. Y K

This equation can be generalized to our Hilbert space setting of domain perturbations.

Theorem 3.5. Let H be an arbitrary self-adjoint and non-negative operator in H with corresponding form (£, F). Further let 9 be any closed subspace of (F, £1 (', .)) and HQ its associated domain perturbation in the sense of Definition 3.1. Then, for any f E H the following assertions hold true.

(a) Let HQ = 9 be the Hilbert space in which HQ acts and denote by rrQ E B(H) the orthogonal projection onto HG. Then

(H + 1)-1 f = (HQ + 1)-1 rrQ f + PQ-L (H + 1)-1 f,

(b) £1 ((HQ + l)-lf,PQ-L(H + I)-If) = O.

Proof. (a) Put u := PQ(H + 1)-1 f and let v E 9 be arbitrary. Then

£1 (u, v) = £1 (Pg(H + 1)-1 f, v) = £1 ((H + 1)-1 f, v) = (1, vI = (rrQ f, vI·

This implies u E dom(HQ) and (HQ + l)u = rrQ f, i.e.,

(H + 1)-1 f PQ(H + 1)-1 f + PQ-L (H + 1)-1 f

= u+PQ-L(H+1)-lf

(HQ + l)-lrrQ f + PQ-L (H + 1)-1 f. Part (b) is clear since (HQ + 1)-1 f E 9 and PQ-L (H + 1)-1 f E 91... o


3.3. Defining domain perturbations by means of the associated stochastic process

Only for the sake of completeness we mention another method of defining domain perturbations which works in the context of regular Dirichlet forms. This construction uses the fact that there is a one-to-one correspondence between regular Dirichlet forms and Hunt processes, see Theorem C.18. Hence one may use the process to define domain perturbations. The idea is to define a new process by "killing" the sample paths when leaving the complement of the set K on which the domain perturbation takes place. Therefore this process is sometimes called the process killed on K.

To be precise, assume that ([;, F) is a regular Dirichlet form in L2 (X, m) with associated self-adjoint operator H, see Appendix C.l. Let

M = (0, M, X t , (lP'x)xEXLJ

be the corresponding m-symmetric Hunt process on X. Given a closed subset K of X, the operation of killing the sample paths upon leaving Y := X \ K is defined as follows. Let YL:, be the one-point compactification of Y. When Y is already compact, D. is adjoint as an isolated point. Put

Xr(w):= {~t,(w), if 0 ::; t::; inf{t ~ 0: X t E K}, LC. otherwise.

The quadruple MY := (O,M,Xi, (lP'x)xEY6 ) is called the part of M on the subset Y. It turns out that this construction again yields a Hunt process whose transition function can be given explicitly.

Theorem 3.6. MY is a Hunt process on Y with transition function

pi (x, E) = IP'x{Xt E E, CfK > t}, E E 8(YL:,)

where CfK = inf{t > 0: X t E K}

is the first hitting time of K.

For a proof we refer to [FOT94]' Theorem A.2.10. Denote by ([;Y, F Y ) the regular Dirichlet form of MY and let H Y be the

self-adjoint operator in L2(y, m) which corresponds to ([;Y, F Y ) in the sense of Theorem 2.2, see also the discussion after Theorem C.17 for details. Hence the operation of killing the sample paths upon leaving Y is another method of defining domain perturbations. It is therefore a natural question to ask for an analytic description of H Y . The following theorem gives the answer.

Theorem 3.7. In the situation just described let

FK := {u E F: it = 0 q.e. on K}.

Then H Y = H:FK , i.e., the self-adjoint operator H Y defined by killing the sample paths upon leaving Y coincides with the domain perturbation H:F K of H with respect to the subspace F K, see Definition 3.l.

The proof is carried out in [FOT94], Theorem 4.4.3.

298 Andre Noll

3.4. Preliminary remarks on the spectrum of the perturbed operator

Let HQ be the domain perturbation with respect to the subspace Q which was introduced in Definition 3.1. Even without further specifying the unperturbed operator H one can obtain some information about the spectrum of HQ. The minimax principle (see Appendix A) gives the following information on the discreteness of the spectrum (recall that the spectrum is said to be purely discrete if it consists only of isolated eigenvalues of finite mUltiplicity): HQ has purely discrete spectrum if and only if the sequence

/-Ln:= inf{suP { If~~~ : U E.[\ {O}}:,[ c Q,dim.[ = n}, n'2 1 (3.8)

tends to infinity as n ----+ 00. Clearly this happens if H has purely discrete spectrum. From equation (3.8) we also conclude that /-Ln '2 .An, where

.An := inf { sup {~~~~ : U E .[ \ {O} } : .[ c F, dim.[ = n} , n '2 1

is the corresponding sequence for H. This means that the eigenvalues of H (if there are eigenvalues at all) can only be shifted to the right by passing from H to HQ. It is therefore a natural question to ask for the magnitude of this shift in terms of the subspace Q. If 1{ = L2(X, A, m) and Q is such that 1{Q = £2(y, A, m) with some measurable subset Y of X, this leads to the question of giving estimates on the shift of eigenvalues by geometric properties of Y. In the special case of the Dirichlet Laplacians H = -~r! and HQ = -~A, see Example 3.2, we retain the physical problem of measuring the improvement in cooling efficiency resulting from the cooler n \ A which served in Section 1.4 as a motivation for considering this kind of problems.

As we already noted in Section 1.4 it is the capacity which plays the decisive role in estimating the perturbed eigenvalues of the Dirichlet Laplacian and we are heading towards a generalization of the results stated there to our general type of domain perturbations. To this aim we of course need a notation of capacity appropriate to this level of generality.

4. Several approaches to capacity

In this section we introduce several notions of capacity and compare them with each other. As a motivation the first subsection gives a rather heuristic definition of capacity which is motivated by an electrostatic consideration. The following Section 4.2 treats the capacity of a regular Dirichlet form. It is shown that one may look at this capacity as an orthogonal projection in the associated Dirichlet space. This observation is used in Section 4.3 as the starting point for defining the capacity of a subspace in a general Hilbert space setting. We also show that this capacity may be expressed as an infimum in the case of £2-spaces. As a consequence we prove that the general capacity coincides with the capacity of a regular Dirichlet form in this special case.


Finally in Section 4.4 we give a brief overview on certain other capacities, but we restrict ourselves to the capacity of a non-regular, non-symmetric Dirichlet form and to the higher-order capacity in the sense of Maz'ja.

The following table gives a short overview of the different capacities and their associated operators which are treated in this thesis.

I Section I Hilbert space I Operators

4.1 L2(JRa) -b. 4.2 real £Z(X, m), X a locally H which corresponds to a

compact metric space regular symmetric Dirichlet form

4.3 (1-{, (-, .)) arbitrary, real or H self-adjoint and complex semi-bounded

4.4.1 real L"L(X, m), X a Haus- H which corresponds to a dorff space (non-regular, non-symmetric)

Dirichlet form 4.4.2 complex L"L(~), ~ c JRa open H an elliptic differential

operator of arbitrary order

4.1. Definition of the electrostatic capacity

Let d ~ 3 and consider a compact and connected subset K of JRd together with an electric charge distribution f.1 on K. This means that f.1 is a Radon measure supported on K with finite total charge Q := f.1(K). Then f.1 induces an electrostatic potential UI" and a force F = -\lUI". According to the laws of physics, see e.g., [Jac62] Section 1, UI" satisfies Poisson's equation

b.UI" = f.1

which has to be understood in the sense of distributions. Therefore UI" = f.1 * Ed, where Ed is any fundamental solution of the Laplacian. One usually takes

Ed(X) = cdlxI2-d.

Here Cd is a constant whose exact value is given by

Wd r (~) Cd = 2 - d = 2(2 - d)Jrd / 2 '

and Wd is the surface area of the unit ball in JRd. See [PS78], Section 3.1 or [Joh71], p. 99 for further information. The total energy E(f.1) stored in the system is given by

E(f.1) = r I\lUI"12dx. lJRd

If I\lU1"12 does not belong to L1(JRd) the convention E(f.1) = 00 applies. Let s be any real number. By elliptic regularity, see Appendix B or [Rud73], we have UI" E H?o~S(JRd), whenever f.1 E HtoJJRd). If, in particular, f.1 = pdx with some bounded function p, supported in K, we have UI" E H?oc(JRd). Hence I\lU1"12 E Ltoc(JRd).

300 Andre Noll

Moreover it is easy to see that I'VUJ.LI decays like Ixl 1- d as x tends to infinity which implies l'VUJ.L1 2 E L1(JRd) since d 2: 3.

If K is a conductor it is clear for physical reasons that the charge I-" will distribute itself freely on K to become the equilibrium measure VK,Q which will only depend on the shape of K and on the total charge Q but is independent of 1-". Of course we have vK,Q(K) = Q. The equilibrium measure VK.Q determines the equilibrium potential eK,Q := UVK.Q = VK,Q *Ed. If the system is in its equilibrium state a unit test charge at x E K does not experience any force, i.e" 'VeK,Q = 0 on K. Therefore eK,Q must be constant on K. Denote this constant value by a(K, Q) and define the (zero-order-) capacity of K by

1 CaPo(K):= ( ) . a K,l

We use the index 0 to distinguish this capacity from the more general capacities we shall define later on. If Q = CaPo(K) and x E K, then

eK,Q(x) = a(K, CaPo(K)) = Capo(K)a(K, 1) = 1.

Thus Capo(K) is the charge one has to put on K in order to make the equilibrium potential equal to one on K. By convention one normalizes the equilibrium potential in such a way that its total mass equals the capacity, i.e., instead of VK,Q one considers VK := VK,Cap(K) and eK := VK * Ed. Then

E(VK) = r l'VeKl 2 dx = r eKdvK = vK(K) = Capo(K). lIRd lIRd Therefore the capacity of K equals the energy of the system being in its equilibrium state. On the other hand the initial charge distribution I-" will distribute itself in such a way that minimizes the energy stored in the system. Thus

Capo(K) =inf{E(J-l) : UJ.L 2: 1 on K} = inf {ld l'VUJ.L1 2 dx : UJ.L 2: 1 on K},

and one can prove that in the last expression the potentials UJ.L may be replaced by smooth functions. Thus

Capo(K)=inf {ld 1'V¢1 2dx : ¢ E C~(JRd), ¢ 2: 1 on K}. (4.1)

In fact the unique minimizer of this infimum turns out to be UVK.

4.2. The capacity of a regular Dirichlet form In this section we recall the definition of the capacity associated to a regular Dirichlet form. With the aid of the corresponding Hunt process we show that one may look at the capacity as an orthogonal projection in the form domain. This observation serves as a motivation for defining the capacity in general Hilbert spaces which is done in Section 4.3.

Let X be a locally compact separable metric space and let m be a strictly positive Radon measure on X. Fix a self-adjoint and semi-bounded operator H in L2(X, m). By Theorem 2.2 there is a unique quadratic form (E, F) that corresponds

to H. Throughout this subsection we assume that (E, F) is a regular Dirichlet form in L2(X, m). A comprehensive treatment of the theory of Dirichlet forms can be found in [Fuk80]' [FOT94]' [BH91] or [MR92]. The most important results from the spectral theoretic point of view are summarized in Appendix C.1.

The (first-order) capacity of an open subset U of X is defined by

Cap(U) := inf{El [u] : u E F, u ~ 1 m-a.e. on U}, (4.2)

where El[U] := E(u,u)+ Ilulli2(X.m)' If there is no such u the convention Cap(U) =

00 applies. This definition is extended to arbitrary subsets of X by

Cap(A) := inf{Cap(U) : U ::J A, U open}.

Recall that a statement is said to hold quasi-everywhere (q.e.), if it holds outside a set of zero capacity. Similarly an element u E F is called quasi-continuous if for each c > 0 there is an open set UE C X such that Cap(UE ) < c and the restriction of u to Us is continuous. It is an important consequence of the regularity of (E, F) that each u E F admits a quasi-continuous m-version U, i.e., there is a quasi-continuous function u that coincides with u m-almost everywhere. In view of Theorem C.17 (d) the capacity of an arbitrary subset A of X may be expressed as

Cap(A) = inf{El[u]: u E F,u ~ 1 q.e. on A},

and by Theorem C.1 7 (e) this infimum is attained for some unique element e A E F. The function eA is called the (first-order) equilibrium potential of the set A.

According to Theorem C.18 there is always a unique Hunt process M (D, M, (Xt)t>o, (PX)XEX) which corresponds to (E, F) and H in the sense that

for all f E L 2 (X,m), where lEx denotes the expectation with respect to Px ' See Appendix C for a brief introduction to the theory of Markov processes and Hunt processes. We next show that M can be used to give a stochastic interpretation of the equilibrium potential. The following theorem was published in [DMN97] and its proof is based on ideas taken from [DC].

Theorem 4.1. Let B c X be a Borel set of finite capacity and let

TB := inf{t > 0 : X t E B}

be the first hitting time of B. Then

(4.3)

302 Andre Noll

Proof. We first show that VB(X) := lEx{e-TB : TB < oo} is a I-excessive function, i.e., e-tHvB :::; etvB for each t > O. This is seen from

lEx{VB(Xt )}

lEx{ VB(Xt ) : TB :::; t} + lEx { VB(Xt ) : t < TB < oo}

< IP'x{TB:::; t} +lEx{VB(Xt ): t < TB < oo}

< et lEx{e-TB :TB :::; t} + lEx {lEXt {e- TB : TB < oo}:t < TB < oo}

< e t lEx { e-TB : TB :::; t} + lEx { e-(TB-t) : t < TB < oo}

etvB(x).

According to Theorem C.17 (e) the equilibrium potential eB is also I-excessive. Because of Theorem C.13 (b) it suffices to show

( 4.4)

and VB = 1 q.e. on B. (4.5)

Then we obtain VB E :F as well as EdVBJ :::; EdeB], and we can conclude from (4.5) and Theorem G.17 (e) that eB = VB. That means it remains to prove (4.4) and (4.5).

To get (4.5) suppose that there is some Xo E B such that VB(XO) i= 1. Then IP' Xo {TB > O} > 0, i.e., Xo is an irregular point of B. (4.5) follows since the set of irregular points is a set of zero capacity by Theorem C.25.

For the proof of (4.4) choose an open set U C X and consider the random variables

yt := e-teu(Xt ), (t 2: 0).

We claim that for x E X fixed, (n,M,M t , yt,lP'x) is a supermartingale, see Definition C.6 (a). This follows from the Markov property and the fact that eu is I-excessive by

e-slEx{eu(Xs)IMt} e-slEXt {eu(Xs- t )}

e- s (e-(S-t)H eu ) (Xt )

< e-ses-teu(Xt )

yt,

where s 2: t 2: O. Since this supermartingale is clearly of class (DL), see Definition C.6 (b), we get from Doob's optional sampling theorem (Theorem C.7)

lExfY"ATIMt} :::; ytAT>

valid for any (Mt)-stopping time T. In particular we may choose T = TU, t = 0 and get


By letting s tend to infinity the desired inequality (4.4) follows for open subsets of X. For the general case choose a decreasing sequence (Un)nEN of open subsets of X such that Un :J Band Cap(Un ) -+ Cap(B). Then eUn -+ eB in (F,E1(-, .)) since

£I[eunl + £l[eB]- 2£1(eu", eB)

Cap(Un ) + Cap(B) - 2 f eUn dVB

Cap(Un ) - Cap(B),

where VB is the equilibrium measure of B, see Theorem C.17 (f) and (C.4). Consequently the inequality

VB :S vu" :S eu"

carries over to the limit and yields (4.4), completing the proof. o

We are now heading towards a description of the capacity as an orthogonal projection in the Hilbert space (F, £1 (', .)). Clearly

F B := {u E F : u = 0 q.e. on B} (4.6)

is a closed subspace of (F, £1 (', .)). Therefore we have the orthogonal decomposition

(4.7)

The orthogonal projection P;:.l onto FiJ admits a description via M: By Theorem D

C.20 we have for each U E F,

Using the nice continuity properties of the sample paths of M, see Definition C.3, it is then easy to see that

(4.8)

provided that u = 1 on some neighborhood of B. But the right-hand side of identity (4.8) is exactly the description (4.3) of the equilibrium potential eB. Hence eB = P;:.l u and

R

(4.9)

Equation (4.9) entails that in order to compute the capacity of the set B one can do the following: Consider the space FB as in (4.6), define FiJ := F 8 FB and choose any function u in the form domain which is equal to 1 in some neighborhood of B. Then evaluate the quadratic form £1 at the projection of u to that subspace. From this point of view it is no longer necessary to restrict oneself to regular Dirichlet forms. Hence one can define a more general version of capacity. The next subsection treats this problem in detail.

304 Andre Noll

4.3. Capacity in general Hilbert spaces

Before we come to the main part of the article, let us briefly illustrate the scope of regular Dirichlet forms. The well-known Beurling-Deny formulae, see [FOT94] Theorem 3.2.1, [Fuk80], Section 2.2. or [MR92]' Theorem 11.2.8 state that every regular Dirichlet form (E, F) can uniquely be expressed as the sum of three forms which are called the diffusion part, jump part and killing part respectively. These names come from the stochastic interpretation of the Dirichlet form. For example the paths are continuous if and only if the jump part equals zero. If X = 0 is an open subset of]E.d and F c C~(O), one can even prove that

E(u,v) d

L 1 au av dVij i,j=l n aXi aXj

+ 1 (u(x) - u(y))(v(x) - v(y)) dJ(x, y) + 1 uv dk, (nXn)\6 n

(4.10)

where Vij and k are uniquely determined Radon measures on 0, ,6 = {(x, x) : x E

O} is the diagonal of n x n, and J is a symmetric Radon measure on the product space (0 x 0) \ 6 which is also uniquely determined by (E,F).

This representation shows that the class of operators which can be treated by using regular Dirichlet forms is very restricted. In particular differential operators of order greater than 2 cannot be included.

We now want to generalize the concept of capacity to the general Hilbert space setting. For this we assume as in Section 3.2 that (H, (., .)) is an arbitrary real or complex Hilbert space and H is a self-adjoint operator in H which is semi-bounded from below with spectral bound A := inf cr(H). Define (E, F) as the non-negative closed quadratic form which corresponds to H - A. Using the abbreviations (3.4) the definition of the abstract capacity reads as follows.

Definition 4.2. Let 9 be a closed subspace of (F, E1 (., .)) and let u E :F. The u-capacity of 9 is defined by

where Pg is the orthogonal projection onto 9 in (F, E1 (-, .)).

In view of equation (4.9) this is a quite natural definition and we have the following result.

Proposition 4.3. If H is such that (E, F) is a regular Dirichlet form in L2(X, m), B C X is any Borel set and u E F is such that u = 1 in some neighborhood of B, then

Cap(B) = Capu(F~), where FE and F~ are as in (4.6) and (4.7).


Unlike the zero-order capacity (4.1) the capacity of Definition 4.2 does not have any scaling properties. In Section 7 we shall define a zero-order version in the general Hilbert space setting. In this context is proved a scaling property fOr this capacity in the case of homogeneous differential operators.

Let us summarize some simple facts about the u-capacity which follow immediately from its definition.

Remark 4.4. Let 9 be a closed subspace of (F'£l(-, .)) and let u E F.

(a) 0 ::::; Capu(G) ::::; Edu]. Moreover CaPu(9) = 0 if and only if u E g.1 and Capu(9) = Edu] if and only if u E 9. Notice that "1.." is meant with respect to E1 (-,.) rather than (., .).

(b) Capu (9) + Capu (9.1) = E1 [u]. (c) CaPu(91) ::::; Capu(92) if 91 and 92 are closed subspaces of (F,E1 (-, .)) such

that 91 c 92· (d) If u is a normalized eigenvector of H with eigenvalue A = inf O"(H), then

Capu (9) E [0, 1]. Moreover

Capu(9) = (u, Pgu)

in this case. (e) Cap",u(9) = laI2Capu(9) for each scalar a.

Next we examine what Capu (9) looks like if'H is an L2-space and 9 consists of functions that vanish on a prescribed set. As we shall see in Proposition 4.5 (a), the capacity of Definition 4.2 admits a similar description as in the special case of a regular Dirichlet form.

Let X be a Hausdorff space which is equipped with a strictly positive Borel measure m (i.e., each open set has positive measure) and let A be an arbitrary subset of X. Define FA as the closure of the space

o

FA:= {v E F: v = 0 m-a.e. on some neighborhood of A}, (4.11)

where the closure is taken with respect to the topology induced by the scalar product E1 (·, .). A result similar to Proposition 4.5 (a) below was proved in [No197b].

Proposition 4.5. For any u E F the following equations hold.

(a) Capu(Fi) = inf{Edv]: v E F,v = u m-a.e. on some nhd. of A}, (b) Capu(Fi) = inf{Capu(F&): U:J A,U open}.

Proof. (a) First observe that

inf{Edv] : v E F,v = u m-a.e. on some neighborhood of A} o

= inf{Edu - v] : v EFA

= inf{E1[u - v] : v E FA}. (4.12)

306 Andre Noll

Let PFA be the orthogonal projection onto FA in (F, &1 (', .)). From equation (4.12) we see that

Capu(Ft) = &l[U - PFA u] ~ inf{&l[v]: v E F,v = u m-a.e. on some neighborhood of A}.

To get the converse inequality let v be an arbitrary function in FA. Then, by the Cauchy-Schwarz inequality,

Capu(Ft)2 &1((1 - PFA)U, u? &1((1 - PFA)U, U - V)2

< &1[(1 - PFA)U]&du - v] Capu(Ft)&du - v].

Hence Capu(Ft) :::; &l[U - v]. Since v was arbitrary in FA, this inequality implies together with (4.12)

Capu(Ft) :::; inf{&du - v] : v E FA} = inf{&l[v]: v E F,v = u m-a.e. on some neighborhood of A}.

The proof of (a) is complete. (b) Let U ::) A be an open set. Then clearly Ft; ::) Ft and hence, by

Remark 4.4 (c),

implying

Capu(Ft) :::; inf{Capu(Ft;) : U::) A, U open}.

For the other inequality let e > 0 be arbitrary. From (a) we see that there is some v E F with v = u m-a.e. on some open neighborhood V of A and such that

&dv] :::; Capu(Ft) + c.

We also see from (a) that &1 [v] ~ Capu(Fi7} Thus

inf{Capu(FU) : U::) A, U open} :::; Capu(Fi7-) :::; Capu(Ft) + e,

and the proposition is proved. o

We remark that Proposition 4.5 can be used for an alternative proof of Proposition 4.3 which avoids the use of the machinery of Hunt processes. The proof is carried out in [Nol99].

4.4. Other capacities

After having discussed the capacity of a regular Dirichlet form and its generalization to the general Hilbert space setting we now want to give a brief overview on other notions of capacity. First we define the capacity of a non-regular and non-symmetric Dirichlet form. In Section 4.4.2 we treat capacities on ~d that are not induced by a Dirichlet form, but by higher-order differential operators.


4.4.1. THE CAPACITY OF A NON-REGULAR, NON-SYMMETRIC DIRICHLET FORM

Dirichlet forms on L2(X, m) which are not necessarily symmetric are considered in [MR92]. Furthermore the topological assumptions on X are weakened considerably: X is merely assumed to be a Hausdorff space. Also the local compactness is no longer needed.

Definition 4.6. Let F be a dense subspace of L2(X, m). A (not necessarily symmetric) non-negative bilinear form E : F x F ---+ lR. is called a Dirichlet form on L2(X,m) if the following conditions are satisfied.

(i) The symmetric part

- 1 E1(u,v):="2 (El(U,V) +El(V,U))

is a closed form on L2(X, m), (ii) E fulfills the weak sector condition with some continuity constant K > 0, i.e.,

IE1 (u,v)1 ::; K E1 (u,U)1/2E1(v,V)1/2

for each u, v E F, (iii) u E F implies v := u+ 1\ 1 E F and

E(u+v,u-v) :::::0, E(u-v,u+v) :::::0.

It is easy to see that the condition (iii) is equivalent to the Markov property (C.2) in the case of a symmetric form. It is shown in [MR92] that if (E, F) is a Dirichlet form, there is a unique operator H (not necessarily self-adjoint) corresponding to (E, F) in the sense that

E(u, v) = (Hu, v),

valid for v E F and u in the domain of H. It follows from the Hille-Yosida theorem that there is a strongly continuous semigroup (St It>o on L2 (X, m) associated to H and hence to (E,F). The adjoint semigroup (S;)t>o is also strongly continuous and is called the cosemigroup associated to (E, F). Analogously (( H* + a) -1 ) 0:>0

is called the coresolvent of H. The definition of capacity uses the concept of balayaged functions which will

be described briefly in the following proposition.

Proposition 4.7. Let U be an open subset of X and let h : X ---+ lR be an arbitrary function. Assume that

Lh,U := {u E F: u::::: h m-a.e. on U} =1= 0.

Then there is a unique function hu E Lh,U, called the balayaged function of h on U such that E1(hu ,u) ::::: Edhu] for all u E Lh,U. Analogously there is a unique function h'u E Lh,U, called the cobalayaged function of h on U such that E1(u, h'u) ::::: E1[h'u] for all u E Lh,U.

308 Andre Noll

A proof can easily be deduced from Stampaccia's theorem, see [MR92] Theorem 2.6. Using balayaged functions the capacity Caph,g(') can be defined for functions h, 9 such that one of the following two conditions hold.

(i) h,g E F, h is I-excessive (i.e., 5th::; eth) and 9 is l-coexcessive (i.e., 5;g::; etg),

(ii) 9 = hand hu /\ h, h'u /\ h are I-excessive resp. l-coexcessive functions in F for all open U c X with Lh,U -I- 0.

Given h,g with (i) or (ii) define for open U c X,

if Lh,U -I- 0 and Lg,U -I- 0, else

and extend this definition to arbitrary A c X by

Caph,g(A) := inf{Caph,g(U) : U => A, U open}.

(4.13)

It is easy to see that h = 9 = 1 fulfills condition (ii). If (E, F) is a symmetric regular Dirichlet form, then Capl,l (-) coincides with the capacity of Definition C.14.

Now suppose that

h = (H + 1)-1¢, 9 = (H* + 1)-1¢, (4.14)

where ¢ E L2(X, m) with 0 ::; ¢ ::; 1 m-a.e. Then h is I-excessive and 9 is l-coexcessive, so Caph,g(') is properly defined. It is remarkable that the sets of zero capacity do not depend on h, 9 in this case. This leads to the concept of E-exceptional sets: A subset A c X is called E -exceptional if Caph,g (A) = 0 for some/any choice of ¢ in (4.14). Hence quasi-continuous and quasi-everywhere can be defined in the obvious way. Since (E, F) was not assumed to be a regular Dirichlet form, one cannot expect that each u E F admits quasi-continuous mversions, as is true in the case of a regular symmetric Dirichlet form on a locally compact metric space. Hence an analog of parts (d)-(f) of Theorem C.17 cannot be true in general, but if each u E F admits a quasi-continuous m-version u and X fulfills some mild topological assumption, it is indeed true that for arbitrary AcX

Caph,g(A) = E1(hA,g'A),

where hA is the unique element (which turns out to exist) of

Lh,A := {u E F : u :::::: Ii quasi-everywhere on A}

which satisfies

for all u E Lh,A; g'A is defined similarly.

4.4.2. CAPACITIES ON ffi.d In this subsection we suppose that X = ffi.d. There are many different definitions of capacity available concerning more general operators than partial differential operators of order smaller than 2. The various variants are conditioned by the various aspects of applications. For an overview of the different notions of capacity see Schulze, Wildenhain [SW77] IX.7 and IX.8. A comprehensive treatment can be found in the books of Maz'ja [Maz85], Landkof [Lan72] or, more recently, Adams and Hedberg [AH95]. We confine ourselves to the capacity in the sense of Maz'ja.

Definition 4.8. Fix p > 1 and fEN. For n c ffi.d open and Ken compact the (p, f)-capacity of K in n is defined as

( fl) C (K n)._· f{l "'"' IDa"'IPd . ¢ E C~(n),o:::; ¢:::; 1,</J = I} p, t- - ap ,H.- in ~ 'I' X. . hb h d f K . rl on some neig or 00 0

lal=£ (4.15)

From the spectral theoretical viewpoint the case p = 2 is the most important one. The (2, l)-capacity can be used to give necessary and sufficient criteria for the operator (_~)C to have discrete spectrum. It is also possible to give capacitary criteria for the semi-boundedness of Schrodinger operators, see [Maz85], Section 12.5 for details. Let us remark that these considerations are not restricted to powers of the Laplacian but can be generalized to elliptic operators of arbitrary order defined via quadratic forms.

Suppose that p = 2 and f = 1. Then (4.15) looks similar to the Definition C.14 (d) of the zero-order capacity in the case of the regular Dirichlet form which corresponds to the Dirichlet Laplacian in L 2 (n), see Section 1.1. But there are three differences: Firstly in (4.15) only smooth functions with compact support are used whereas in Definition C.14 (d) all functions in the form domain:F = HJ(n) were considered. Secondly in (4.15) the infimum is taken only over those functions that are equal to 1 on K in contrast to Definition C.14 (d) where the functions were allowed to be greater than or equal to 1. Finally in (4.15) the functions must take values between 0 and 1, unlike in Definition C.14 (d). But it turns out that all these differences do not affect the value of the infimum.

Proposition 4.9. Let Ken be compact. Then

(2, l)-Cap(K, n) = CaPo(K),

where Capo (-) is the zero-order capacity of the regular Dirichlet form which corresponds to the Dirichlet Laplacian in U(n), see Definition C.14 (d).

The proof is carried out e.g., in [DMN97].

5. Capacitary estimates for domain perturbations

This section contains the main results of the thesis in hand. Although a major part is already contained in [No197b], [DMN97], [No197a], [NolOO] or [No199], some

310 Andre Noll

results are generalized or the proofs are simplified. Section 5.1 contains two upper bounds for the bottom eigenvalue, the first of which is a generalization of McGillivray's work [McG96], whereas the second eigenvalue estimate works in the general Hilbert space setting of Section 4.3.

Proving lower bounds for the bottom eigenvalue turns out to be a harder problem than proving upper bounds, even in the case of the Laplacian. This is due to the fact that one cannot simply guess a good trial function Uo for the perturbed ground-state and then estimate [[uo]. Nevertheless, in Section 5.2 we shall obtain a lower bound for the shift of the bottom eigenvalue which is of the same magnitude as the upper bounds of Section 5.1. By using a simpler method we obtain another lower bound which generalizes Thirring's inequality.

Assuming that the underlying self-adjoint operator has purely discrete spectrum, we prove in Section 5.3 a capacitary upper bound for the second eigenvalue. It is not possible to obtain capacitary lower bounds for higher eigenvalues by using the classical versions of capacity as is seen from the example of the Dirichlet Laplacian on a square in ]R2.

In Section 5.4 we treat additive perturbations together with domain perturbations and prove an upper bound for the first eigenvalue under the assumption that the additive perturbation is infinitesimally form bounded.

Finally, Section 5.5 contains a capacitary characterization for the spectral bound being unaffected by a given domain perturbation.

5.1. Upper bounds for the bottom eigenvalue

During the last decades many mathematicians studied the problem of analyzing the connection between domain perturbations of a given self-adjoint operator and changes of its spectrum. Among others Rauch and Taylor [Rau75], [Tay76], [Tay79], [RT75a], [RT75b] worked on that subject and pointed out the relevance of this kind of problem for various areas of mathematical physics. The following result due to Taylor [Tay76] shows that the capacity can be used to give quantitative upper and lower bounds for the bottom eigenvalue of the Laplacian.

Theorem 5.1. If 0 C ]Rd is a bounded domain, ), is the lowest eigenvalue for -6. on O\K with Dirichlet boundary conditions on aK, Neumann boundary conditions on a~, K c 0 compact, then there are constants Cl, C2 > 0 such that for small Capa(K),

Cl Capa(K) ::; ), ::; c2Capa(K),

where CapaC) is the classical zero-order capacity in ]Rd, i.e., the capacity of Definition C.14 (d) in the case of the Laplacian in £2(]Rd).

Taylor's result has been generalized in many directions. For example Gesztesy and Zhao proved in [GZ94] that the bottom eigenvalue of certain Schrodinger operators remains unaffected by domain perturbations if and only if the perturbed domain differs from the unperturbed one only by a set of zero capacity. The proof is based on Brownian motion and on the Feynman-Kac formula. One year later


Arendt and Monniaux gave an analytical proof of this result which used a domination argument for semi groups as its main ingredient [AM95]. Their result allowed the potential to vary as well. We refer to the survey article [DMN97] for precise statements and the proofs as well as for further literature on this subject. Let us also remark that similar estimates for the bottom eigenvalue also hold for the Laplace-Beltrami operator on a Riemannian manifold, see [CF78], [CF88], [Cou95], [Oza82], [Oza83]. Another generalization of Taylor's Theorem was given by McGillivray [McG96] who proved the same estimate in the context of regular Dirichlet forms. We next state this result precisely and show that one can improve the estimate by introducing the so-called ground-state transformation.

Let (E, F) be a regular Dirichlet form in L2(X, m) and let H be the associated self-adjoint and non-negative operator, see Appendix C.l. Assume that the spectrum of H is purely discrete. Fix a compact subset K of X and put Y := X \ K. According to Theorem C.20 (a) the restriction of E to the space

F K := {u E F : it = 0 q.e. on K}

defines a regular Dirichlet form in L2(y, m) with associated self-adjoint and nonnegative operator H Y . The spectrum of H Y remains discrete by the minimax principle, see Appendix A. Denote the bottom eigenvalues of Hand H Y by .x. and .x.Y respectively. Recall that a Borel subset E of X is called invariant for the semigroup (e-tH)t>o if f = fXE implies e- tH f = XE(e- tH f) for all t > O. The semigroup (e-tH)t>o is called irreducible if X and f/J are the only invariant sets, modulo null sets. According to Proposition 1.4.3 in [Dav89] irreducibility has the following important consequences for the bottom eigenvalue .x. of H:

(i) .\ is a simple eigenvalue, (ii) the eigenfunction ¢ corresponding to the ground-state energy .x. may be cho

sen such that ¢(x) > 0 m-a.e. on X.

The semigroup is said to be ultracontractive if for each t > 0 there is a constant Ct 2: 0 such that

Ile-tH fIIL=CX.m) :=:; Ct Ilfll£2(X,m)

for all f E L2(X, m). Note that ultracontractivity implies that each eigenfunction of H is bounded which was the main reason for assuming ultracontractivity in Theorem 5.2 below.

Theorem 5.2. (1. McGillivray 1996) Let (E, F) be a regular Dirichlet form in L2 (X, m) such that its associated self-adjoint operator H has purely discrete spectrum. If (e-tH)t>o is irreducible and ultra contractive, then there are constants Cl, C2 E (0,00) such that

for all compact subsets K of X with capacity smaller than C2.

312 Andre Noll

5.1.1. THE GROUND-STATE TRANSFORMATION One drawback of Theorems 5.1 and 5.2 is that these results are only valid for sets of sufficiently small capacity. If in the situation of Theorem 5.2 the obstacle region K comes close to the boundary of X one must expect that the capacity increases so McGillivray's result cannot be applied.

We next show that this problem can be circumvented by introducing the ground-state transformation by which we mean the unitary map

U: L2(X, m) ----> L2(X, ej}m) , f f---+ ¢-l f. Here ¢ is the unique function in the eigenspace ker (H - A) which is normalized and strictly positive m-a.e. Define the self-adjoint operator H' in L2(X, ¢2m) by

dom(H') := {Uu: u E dom(H)}, H'u:= UHU-1u.

Then a(H') = a(H) and the constant functions are eigenfunctions of H' with eigenvalue A. In particular these eigenfunctions are automatically bounded. This observation allows us to drop the ultracontractivity condition in Theorem 5.2, see Theorem 5.4 below. Let (£', F') be the closed form associated to H'. It is easy to see that (£', F') is again a Dirichlet form with

F' = {¢-lU : u E F} and £'(¢-lU, ¢-lV) = £(u, v)

for u, v E F. Denote by Cap'(U) the £'-capacity of some open set U C X. If U is such that ¢(x) ::; 1 m-a.e. on U, then

Cap'(U) inf{£l'(u', u') : u' E F', u' 2': 1 m-a.e. on U}

inf{£l'(¢-lu, ¢-lU) : u E F: u 2': ¢ m-a.e. on U}

inf{£l(u,u) : u E F: u 2': ¢ m-a.e. on U}

< Cap(U).

These calculations carryover to arbitrary A eX. For instance, if X = n is a sufficiently regular domain and H is the Dirichlet Laplacian in L2(n), then ¢(x) ----> 0 as x ----> an. Therefore we can hope that the new capacity Cap'(·) yields a better estimate for the ground-state shift A Y - A. Before we can prove the generalization of Theorem 5.2 we have to guarantee that (£', F') is regular. This turns out to be true if the ground-state ¢ is a continuous function which clearly is not a serious restriction. For example, if H is a (second-order) elliptic differential operator the eigenfunctions are even Coo because of the elliptic regularity theorem, see Appendix B.

The following Lemma is taken from [N 0197 a].

Lemma 5.3. Let (£, F) be a regular Dirichlet form in L2(X, m) possessing the following properties:

(i) The associated self-adjoint operator H has purely discrete spectrum, (ii) the semigroup (e-tH)t>o is irreducible.

(iii) the ground-state ¢ is continuous on X.

Then (£', F') is regular and (e-tH')t>o is irreducible.

Proof. The proof of the irreducibility is straightforward, so let us concentrate on the regularity of (£',F'). We have to show the denseness of Co(X) n F' in F' as well as in (Co (X), II . II CXl). The first statement is again straightforward, so we will only prove that Co(X) n F' is dense in Co(X) with respect to uniform norm. Let g E Co(X) arbitrary. Because of the continuity of rp we conclude h := rpg E Co(X), and the regularity of £ guarantees the existence of a sequence hn E F n Co(X) such that

hn ----+ h in Co(X). (5.1 )

Urysohn's theorem (d. [Rud66], Theorem 2.1.2) gives us some TJ E Co(X) with

TJ == 1 on supp(h). (5.2)

Now we use again regularity of £ to find for arbitrary E > 0 a function TJ' E

Co(X) n F satisfying

Put

gn := TJ'rp- 1 hn .

The proof is completed if we show

and

gn ----+ g in Co(X).

Proof of (5.4): Obviously we have gn E Co(X). Because of

F' = {rp -1 U : u E F}

(5.3)

(5.4)

(5.5)

it suffices to show that TJ' hn E F. This is a consequence of TJ', hn E F n LCXl (X, m) and Theorem C.13 (a). It remains to prove (5.5), but this is easy:

Ilgn - gllCXl IITJ'rp-1hn - rp-1hll oo

IITJ'rp- 1hn - TJrp-1hlioo

< liTJ'rp-1hn - TJ'rp- 1 hll oo + IITJ'rp- 1h - TJ4>- l hll oo

< IITJ'rp-11Ioollhn - hll oo + Ilrp-1hllooIITJ' - TJlloo

< 2Ellrp-1hll oo

if n is sufficiently large because of (5.3) and (5.1) (note that TJ', h E Co (X) implying Ilrp-1hll= < 00). 0

With the aid of Lemma 5.3 we can prove a generalization of Theorem 5.2. Theorem 5.4 below is a slight improvement of Theorem 6 in [Nol97a], where a similar estimate has been proved with larger constant co.

314 Andre Noll

Theorem 5.4. Under the hypothesis of Lemma 5.3 the following is true: For each o :::; 5 < ~(1 +..\.) there is some constant Co > 0 such that

..\.Y ~..\. :::; coCap'(K),

whenever K is a compact subset K of X whose capacity is smaller than 5. The t t OO lO °tl b .~ 2(H'\) cons an Co zs gwen exp zcz y y Co .~ 1+,\-20'

Proof. From Lemma 5.3 we conclude that (£', F') is again a regular Dirichlet form. Let (n',M',M~,X~,IF'~) be the Hunt process associated to (£',F') and define the set S6 of measures of finite energy integral and their associated 1-potentials U{f.1 in the obvious way. Further let v := f.1k E S6 be the unique measure corresponding to K and 1 in the sense of Theorem C.23 (a). Because of

and

veX) = [ldV = £~ (1, U{v) < 00

we see from Theorem C.24 that

veX) = v(K) :::; Cap'(K).

From Dynkin's formula, see Theorem C.23 (b), we obtain

U{v U{f.1~/+l)-l(HI+l)l

(H' + l)-l(H' + 1)1 ~ (H'Y + l)-l((H' + l)l)ly

1 ~ (1 + ..\.)(H'Y + 1)- l ll y '

i.e.,

and

[~((H'Y + 1)- l ll y ' (1 + ..\.)-lU{v) = O.

Putting everything together we get

<

<

<

<

11(1 + A)-11- (1 + A)-lU{vII12(x,¢2 rn )

(1 + A)-l (1 + A)-2 - 2(1 + A)-2(1, U{v) P(X,¢2m)

I+A 1 - 2(1 + A)-lE{ (1, U{v)

(1 + A)2 1 + A - 2v(K)

(1 + A)2 1 + A - 2Cap'(K)

1 \ 2(1 + A)Cap'(K) + /\ + '( ) 1 + A - 2Cap K

1 + A + c8Cap'(K),

finishing the proof of the theorem, o

5.1.2. AN UPPER BOUND IN THE GENERAL HILBERT SPACE SETTING The results of Section 5.Ll are restricted to operators which correspond to regular Dirichlet forms. It is a consequence of the Beurling--Deny formulae (4.10) that the scope of regular Dirichlet forms in L 2 (n), 0, c IR.d an open set, is limited to (pseudo-) differential operators of order at most 2. In particular the clamped plate cannot be handled as it may be identified with the biharmonic operator ~2 subject to Dirichlet boundary conditions. Another class of operators which are not included are semi-bounded operators H 2': A having discrete spectrum in some interval [A, /1) and essential spectrum in [/1, 00). Examples of such kind of operators are oneand many-particle Schrodinger operators in quantum mechanics. In particular the Schrodinger operator of the hydrogen atom is of this kind, see [BS91], [Wei95a], [Wei95b].

We shall overcome these restrictions by proving abstract eigenvalue estimates which use the capacity of Definition 4.2. To this aim we assume from now on unless otherwise stated, that He;; is the domain perturbation of some self-adjoint and semi-bounded operator H with respect to the subspace g. This was explained in detail in Section 3.2. In what follows we use the notation of this section without further notification. Our first result gives an upper bound for the spectral bound Ae;; = inf cr(He;;) of He;; in terms of the </in-capacities of g..L where the </in's are normalized eigenfunctions of H.

Theorem 5.5. Let {An} be the (finite OT infinite) set of eigenvalues for H with corresponding normalized eigenfunctions </in. Then

316 Andre Noll

Proof. Let P be the orthogonal projection onto g..L, put "'n := Capcpn (9..L) and J-Ln := An - A. Then

(¢n, P¢n) = J-Ln ~ 1 £1(¢n, P¢n) = J-Ln ~ 1 £I[P¢nl = J-tn"': 1·

Moreover, by the Cauchy-Schwarz inequality

IIP¢nll ~ I(P¢n,¢n)1 = J-Ln"': 1·

Hence

1 + AQ - A inf { ~~[I~~ : u E 9 \ {O} }

< £1[(1 - P)¢nl 11(1 - P)¢nI1 2

1 + J-tn - "'n 1- 2Re(¢n, P¢n) + IIP¢nI1 2

< 1 + J-tn - "'n 1 2~ K~

- Iln+1 + (Iln+ 1)2

(1 + J-tn)2 1 + J-tn - "'n 1 "'n + J-tn"'n +J-tn+--,----.:..:-'---'-'-....:..:....

1 + J-tn - "'n "'n 1 + An - A + I<n

1 - HAn A

and the desired estimate follows.

As an immediate consequence we note the following corollary.

Corollary 5.6.

o

(a) Suppose that A is an eigenvalue of H with normalized eigenfunction ¢ ~ g..L. Then

AQ _ A < Capcp(9..L) . - 1 - Capcp(9..L)

(5.6)

(b) Fix 0 E [0,1). For each closed subspace 9 of (F'£1C,·)) with Capcp(9..L)::; 0 we have

Q 1 ..L A - A ::; 1 _ oCap cp (9 ).

Remark 5.7. (a) Note that we did not assume that A is a simple eigenvalue of H. If A is degen

erate we may apply Corollary 5.6 (a) for any normalized ¢ in the eigenspace ker (H - A) and obtain

Q • { Capcp(9..L) } A - A::; mf 1- Capcp(9..L) : ¢ E ker(H - A), II¢II = 1 .

(b) The condition ¢ t/:- gJ.. is necessary to rule out the possibility Capq,(QJ..) = 1 (see Remark 4.4) because the denominator of the right-hand side in (5.6) vanishes if ¢ E gJ... Intuitively speaking ¢ E gJ.. means that ¢ was "completely thrown away" by restricting E to the subspace g. From this point of view it becomes clear that (5.6) will only be a good estimate if Capq,(QJ..) is sufficiently small, see also Example 5.2. Roughly speaking, Capq,(QJ..) measures how much of ¢ will be lost if the form E is restricted to the smaller subspace g, and Capq,(QJ..)/(l- Capq,(QJ..)) gives a bound for the energy of the projected function Pg¢.

The following simple example shows that ¢ E gJ.. might happen if 1-{ =

£ 2 (X, m) with a non-connected set X.

Example. Let (E, HJ ((0, 1f) U (4,5))) be the quadratic form of the Dirichlet Laplacian on £2((0, 1f) U (4,5)). The ground-state of H is given by

¢(x) = {sinx, x E (0,1f), 0, else,

and we see that ¢ E gJ.. if 9 c HJ(4,5).

We next show that in the case of a Dirichlet Schrodinger operator on a domain, i.e., an open and connected subset oflRd , we always have ¢ t/:- gJ... Let us first describe the definition of a Schrodinger operator via sesquilinear forms. Let Sl be an open subset of lRd and let q E £Foc(Sl) be a real-valued function such that multiplication by q is form-bounded with respect to the form

( u, v) /--t r Vu V v dx, u, v E F : = H 6 ( Sl ) . in

It is well known (see e.g., [RS75], Section X.2) that if the relative bound is less than 1, the form

(u,v) /--t r VuVvdx + r quvdx, u,v E F, in in

is closed and semi-bounded from below. Denote the corresponding self-adjoint operator by H and let A. : = inf 0"( H) be its spectral bound. Let (E, F) be the nonnegative form associated to H - A.. For the definition of the domain perturbation let A be an open subset of Sl. In order to avoid trivialities we assume A =1= Sl and A =1= 0. Put 9 := HJ(A). Then 9 c F and hence (E, g) defines a self-adjoint operator Hg in 1-{g := £2(A).

Theorem 5.8. In the situation just described suppose that Sl is a domain and that A. is an eigenvalue of H with normalized eigenfunction ¢. Then Capq,(QJ..) < l.

Proof. Connectedness of Sl implies that the semigroup (e-tH)t>o is irreducible. This together with the positivity preserving property of (e-tH)t>o implies that ¢ is strictly positive almost everywhere on Sl (for details concerning this type of

318 Andre Noll

argumentation see e.g., [Dav89] or [Goe77]). Then for any u E 9 \ {O} with u 2: 0 m-a.e.

£l(CP,U) = (cp,u) = f cpudx > 0,

hence cp t/:. g..L, i.e. Capq,(Q..L) < 1. D

Corollary 5.6 (a) can be used to prove a capacitary criterion for the perturbed operator to have a bottom eigenvalue.

Corollary 5.9. If the spectral bound>" is an isolated eigenvalue of H of finite multiplicity and some normalized cp in the eigenspace ker (H - >..) satisfies

Capq,(Q..L) >.. + C (Q..L) < inf(a(H) \ {A}), 1 - apq,

then >..9 is an isolated eigenvalue of H9 of finite multiplicity.

Proof. Let k be the dimension of the eigenspace ker (H - >..). By Corollary 5.6 (a) and the minimax principle, see Appendix A, we have

Capq,(Q..L) >.. + ( ..L) 1- Capq, 9 <

< inf(a(H) \ {>..})

inf {sup {(£ :u~~[u] : u E L \ {O} } : L c F, dim L = k + 1 }

< inf {sup { (£ ~~~[u] : u E L \ {O} } : Leg, dim L = k + I} . Therefore >..9 is not the least upper bound of the sequence

(inf {sup {(£ :u~~[u] : u E L \ {O}} : Leg, dimL = n}) nEN'

and the assertion follows again by the minimax principle. D

5.2. Lower bounds for the bottom eigenvalue

McGillivray proved in the context of regular Dirichlet forms that the shift of the spectral bound can be estimated from below by some constant times the capacity [McG96], but his result needs strong additional assumptions and does not carry over to more general operators. As we shall see in this section it is possible to obtain capacitary lower bounds for the shift >..9 - >.. similar to those of Corollary 5.6 if >.. is a simple and isolated eigenvalue of H.

Theorem 5.10. Suppose that>.. is a simple eigenvalue of H with normalized eigenfunction cp. Let J-l E [0,00) be the spectral gap, i. e.,

J-l := inf{a(H) \ {>..}} - >... (5.7)


Then

Proof. The second inequality is obvious, so let us prove the first one. Let lC be the eigenspace of Hand A. Denote by PK and lC.l. the orthogonal projection onto lC and the orthogonal complement of lC respectively, where orthogonality is meant with respect to the scalar product E1 (.,.) on F. Put PK.l := 1- PK. Then for any UEF,

Capu(lC.l.) = EdPK.lU] 2: (1+fLlIIPK.luIl2.

Observe that PKU and PK.l u are not only orthogonal with respect to £1 (.,.) but are also orthogonal with respect to (., .). This gives us

Capu(lC.l.) > (J-l + 1)(lluI1 2 - IIPKuI1 2 )

= (J-l + 1)(lluI1 2 - EdPKU])

= (J-l + 1)(lluI1 2 - Capu(lC))· (5.8)

Let Pg be the orthogonal projection onto Q. Since K is spanned by ¢ and £d¢] = 1 we obtain for u E Q

Capu(lC) = EdPKU] = EdE1(¢'U)¢] = IE1(¢,uW = IE1 (¢,Pgu)1 2

= IEl(Pg¢, uW ::; El[P9¢]Edu] = Edu]Cap¢(Q). (5.9)

Now let u E Q with Ilull = 1. Then, by Remark 4.4 (b), (5.8) and (5.9)

(1 + J-lCap¢ (Q) )£1 [u] J-lCap¢(Q)E1 [u] + Capu (lC) + Capu (lC.l.)

Consequently

which completes the proof.

> J-lCapu(lC) + Capu(K) + (J-l + 1)(1 - Capu(lC))

J-l + 1.

>

inf{E1[u] : u E Q, Ilull = I} J-l+1

1 + J-lCap¢(Q)

1 + _J-l_-_J-l_C.,-a_p-,-¢..,...(Q=-c-) 1 + J-lCapq,(Q)

J-lCapq, (Q.l.)

1 + 1 + J-l(1- Capq,(Q.l.))'

o

From Corollary 5.6 (a) and Theorem 5.10 we immediately obtain the following characterization for the shift Ag - A to be positive.

Corollary 5.11. Suppose that A is a simple and isolated eigenvalue of H with eigenfunction ¢. Then Ag > A if and only if Capq,(Q.l.) > O.

320 Andre Noll

In practice it seems to be difficult to compute the capacities involved in Theorems 5.5 and 5.10 since both 9 and g~ are typically infinite-dimensional, e.g., if H is an operator in £2(JRd) and Hg arises from H by imposing boundary conditions on a set with non-empty interior. It is possible, however, to prove another lower bound for the shift of the spectral bound which involves the u-capacity of the eigenspace K for u E 9 rather than the ¢-capacity of 9 for ¢ E K as in Theorems 5.5 and 5.10. Since K happens to be one-dimensional in many interesting cases, e.g., if H is the Dirichlet Laplacian on some bounded domain in JRd, it is possible to compute the u-capacity of K explicitly in these cases.

Theorem 5.12. Suppose that A is an eigenvalue of H. Denote the corresponding eigenspace by K and let

5:= sup{CaPu(K) : u E g, Ilull = I}.

Then Ag - A 2': fL(l - 5), where fL is as in (5.7) the spectral gap.

Proof. Let PK and K~ be the orthogonal projection onto K and the orthogonal complement of K respectively and put PK.l := 1 - PK. Then for u E 9 such that Ilull = 1:

£I[u] £1[PKU] + £1[PK.l u] = IIPKul1 2 + £I[PK.l u]

> IIPKul1 2 + (fL + 1)IIPK.luI1 2 = 1 + fL11PK.lu11 2

1 + fL(1 - IIPKuI1 2 ) = 1 + fL(l - £I[PKU])

1 + fL(1- CaPu(K))

> 1 + fL(1 - 5),

and the theorem follows by

).p - A + 1 = inf{£l[u] : u E g, Ilull = I}.

o

As an application of Theorem 5.12 we will prove a generalization of Thirring's inequality which reads as follows: Let 0, A C JRd be bounded domains with A c O. Denote by (Aj (0) k"'l and (Aj (A) )j~l the eigenvalues of the Dirichlet Laplacian in £2(0) and in £2(A) respectively. Then

Al (A) 2': Al (0) r 1¢1 2 dx + A2(0) r 1¢1 2 dx. JA JO\A A proof of this result can be found in [Szn98]. For the generalization of Thirring's inequality we use the previous notation and assume additionally that (X, m) is a measure space, 1i = £2(X, m) and 9 is a subspace of:F such that Hg = £2(y, m) with some measurable set Y c X.


Corollary 5.13. Suppose that ,\ is a simple eigenvalue of H with normalized eigenfunction ¢. Then

,\9 2:'\ r 1¢1 2 dm + (,\ + /-L) f 1¢1 2 dm, }Y }X\Y

where /-L is as in (5.7) the spectral gap.

Proof. Since ,\ is simple the orthogonal projection PK onto the eigenspace K. =

ker (H - ,\) is given by

FKU = El(¢'U)¢ = (¢,u)¢, u E:F.

Hence if u E 9 such that Ilull = 1:

Capu(K.) = EdFKU] = I(¢, uWE1 [¢] = Ii ¢udml2 ~ i 1¢1 2dm,

and we see that

8:= sup{Capu(K.) : u E g, Ilull = I} ~ i 1¢1 2 dm.

Thus by Theorem 5.12

,\9 > /-L(1 - 8) + ,\ 2: /-L f 1¢1 2 dm + ,\ }x\Y

(/-L +,\) r 1¢1 2dm +.x. r 1¢1 2 dm. }X\Y }y

Note that we did not assume that .x. + /-L is an eigenvalue of H.

o

Remark 5.14. Let us look at Corollary 5.13 in the special case of domain perturbations on small balls in JRd. More precisely, assume that the measure space X is some open subset D of JRd, dm = dx is the Lebesgue measure and Y = D \ Br(xo), where Br(xo) is the closed ball of radius r centered at Xo. If ¢ is continuous at Xo and ¢(xo) # 0 we have by Corollary 5.13

,\9 -.x. 2: ,\ J 1¢1 2dx + (,\ + /-L) f 1¢1 2 dx - ,\ = /-L f . 1¢1 2 dx 2: crd

D\Br(xo) }Br(xo) }Br(xo)

for some constant c > 0 and sufficiently small r.

Example. Let D be an open and bounded subset of JRd and let

H = L (-l) Ct D a(a CtI,D{3u), aa{3 = a{3a E C lal,I{3I~rn

be an elliptic constant coefficient operator of order 2m, defined as the closure of the form

(U, v) f--+ L 1 aa{3 D a7jjD{3 v dx, lal,131~m n

322 Andre Noll

initially defined on C;:x'(O), the space of smooth functions with compact support. Then the spectrum of H is discrete by Theorem 14.6 of [Agm65] and all eigenfunctions are analytic (see e.g., [Joh49], [Hor83a], Section 8 and [Hor83b]' Section 11). Assume that the bottom eigenvalue A is simple which happens to be true e.g., in the case of second-order differential operators on a bounded domain. Fix a point Xo E 0 such that the first eigenfunction ¢ of H does not vanish in Xo. Define the self-adjoint operator Hr,xo in L2(0 \ BT(XO)) in the same way as H, but with initial form domain C;:x'(O \ BT(XO))' Denote its bottom eigenvalue by AT,xo' Then Remark 5.14 gives us

AT •XO - A 2:: crd

for some constant c > ° and sufficiently small r. This estimate is far from optimal since we shall see in Section 6, Theorem 6.2, that for d > 2m one even has

Ax,xu - A 2:: const rd- 2m .

The proof of this result is based on Theorem 5.10.

Let us finally illustrate the upper and lower bounds by an almost trivial example. Further examples are given in Section 6.

Example. Let :F = 1i = ffi.2, H = (~ ~) with some J.t > ° and let 9 be the

linear span of e", := (cos a, sin a) where a E [0, n). Since [[e",l = J-L sin2 a we have H9 = J.t sin2 ex and A9 = J-L sin2 a. Now let us compute the ¢-capacity of 9J.. where ¢ = (1,0) is the ground-state of H. Since [de",P/2 = (1 +J.tsin2a)1/2 =: c'" we get

which gives us

Hence, by Theorem 5.6

and by Theorem 5.lO

J.t(1 + J.t) sin2 ex 1

1 + J.t sin2 ex 1 + J.t l;~:~r~ '" and we see that the upper bound is a good estimate only if Cap¢(9J..) is sufficiently small, whereas the lower bound always gives the correct value.

5.3. Higher eigenvalues

There is an enormous literature on estimates for eigenvalues of the Laplacian or other special operators as well as relations between these eigenvalues such as the ratio between the first and the second one. We do not try to list these results but remark instead that there are only few results which can treat general classes of operators. One of those is the previously mentioned work of McGillivray [McG96] who proved in the context of regular Dirichlet forms that under an ultracontractivity condition the shift of the higher eigenvalues may be estimated from above by some constant times the capacity of the set on which the domain perturbation takes place. We shall obtain a similar estimate for the second eigenvalue in our general Hilbert space setting.

To keep things simple we assume from now on that H has purely discrete spectrum, i.e., there is an orthonormal basis (¢j) j EN of 1{ and a non-decreasing sequence (Aj )jEN of real numbers such that

and

Hu = L Aj(¢j, u)¢j jEN

for all u E dom(H). Since we are interested mainly in the shift of the eigenvalues we may assume

without loss of generality that the bottom eigenvalue Al is equal to zero. Define £, F, Q, 1{9 ,H9 as in the previous sections. It is a consequence of the minimax principle that the spectrum of H9 is again discrete and that each eigenvalue, not only the lowest, gets shifted to the right. Hence there is an orthonormal basis (1/JjLEN of1{9 and a sequence (A;)jEN satisfying A;;::: Aj such that we have the following representation.

for all

H9 V = L A;(1/Jj,v)?jJj JEN

v E dom(H9) = {w E Q: (A;(1/Jj,W))jEN E fl2}.

Now the main result on the shift of the second eigenvalue reads as follows:

Theorem 5.15. Suppose that

.l (1 + A2 - Capq,,(Q.l))2 Capq,l (Q ) < A2 (1 + A2)4 + (1 + A2 - Capq,2 (Q.l))2 A2 . (5.10)

Then

A~ - A2

(1 - Capq,l (Q.l))(1 + A2)2CaP¢2(Q.l) + (1 + A2)4Capq,1 (Q.l)

S; A2 (1 _ CaP¢1 (Q.l) )(1 + A2 - Capq,2 (Q.l ))2 A2 - CaP¢l (Q.l )(1 + A2)4 .

324 Andre Noll

As we shall see in Example 6.1, inequality (5.10) is fulfilled in great generality for elliptic differential operators if the domain perturbation takes place on a sufficiently small ball. In this context we prove an inequality which links the two capacities CaP4>l (Q.1) and CaP4>2 (g.1) by

CaP4>l (Q.1) :s: cCaP4>2 (g.1) (5.11)

for some constant c > 0 depending on the eigenfunctions (h and <P2 but independent of the class of g's in question. If (5.11) holds we can prove that the shift of the second eigenvalue can at most grow linearly with respect to the capacity CaP4>2 (g.1). Let us first state and prove this corollary before turning to the proof of Theorem 5.15.

Corollary 5.16. Let Al be a simple eigenvalue of H. Then, for each c > 0 there are constants Cl, C > 0 such that for all closed subspaces 9 of:F with CaP4>2 (Q.1) < c and satisfying inequality (5.11) we have

A~ - A2 :s: Cl CaP4>2 (g.1).

The constant Cl can be given explicitly:

Cl = 2(1 + A2)2(1 + 2c(1 + A2)2).

Proof. Since Al is simple we have A2 > O. In view of inequality (5.11) there is a constant 1'0 > 0 such that the hypothesis of Theorem 5.15 is satisfied whenever CaP4>2 (Q.1) :s: 1'0. Similarly we can achieve that

Ca (g.1) < A2 (1 + A2 - CaP4>2 (g.1))2 P4>2 - 4c (1 + A2)4

whenever CaP4>2 (Q.1) :s: 1'1 for some other constant 1'1 > O. Put

. 1 1':= mm{1'0,I'I,A2, -} > O.

2c

Hence, if CaP4>2 (Q.1) :s: 1', Theorem 5.15 gives us

A~ - A2

(1 + A2?(1 - CaP4>l (Q.1))Cap4>2(Q.1) + (1 + A2)4CaP4>1 (Q.1)

Since CaP4>2 (g.1) :s: 1c we have CaP4>l (Q.1) :s: 1/2. Hence

CaP4>l (Q.1) .1 1 - CaP4>l (Q.1) :s: 2CaP4>1 (Q ).

(5.12)

From this inequality and (5.11) we conclude

<

(1 + .:\2 - Cap</>2 (Q-L))2 - 2 Cap</> 1 (Q-L)(1 + .:\2)4/.:\2

(1 + .:\2)2Cap</>2 (Q-L) + 2c(1 + .:\2)4 CaP</>2 (Q-L)

Because of inequality (5.12) the calculation continues and we obtain

< 2 (1 + .:\2)2 + 2c(1 + .:\2)4 Ca (Q-L) (1 + .:\2 - CaP</>J~F ))2 P</>2

< 2(1 + .:\2)2(1 + 2c(1 + .:\2) 2) CaP</>2 (Q-L),

o

Before we can prove Theorem 5.15 we have to prepare some lemmas, the first of which already gives an upper bound for the second eigenvalue but involves the scalar product I(PO¢2, 'lj!1)1 which shall be estimated later on.

Lemma 5.17. Let Po be the orthogonal projection onto Q in the Hilbert space (F, £1(', .)). Suppose that

I(Po rh ni')1 1 +.:\2 - Cap</>2(Q-L) 0'1'2,'1'1 < 1+.:\2 .

Then

Proof. Put

By multiplying ¢2 with some constant of modulus 1 we can achieve that 0: is real without changing the value of Cap</>2 (Q-L). Since

we get

. {£l[ul } mf Ilu11 2 : u E Q \ {O}, (u, 'lj!1) = 0

£1 [PO¢2 + o:'lj!ll IIPO¢2 + 0:'lj!111 2 .

<

326 Andre Noll

We now estimate numerator and denominator separately. Let Pg~ = 1 - Pg be the projection onto Q~. The numerator can be calculated exactly:

EdPg¢2 + O:~ll = E1 [Pg(¢2 + O:~l)l = E1[¢2 + O:~ll- El[Pg~(¢2 + o:~I)l = Ed¢2 + O:~ll- El[P9~¢2l

= A2 + 1 + 20:Re(E1(¢2, ~1)) + 0:2El[~d - Cap¢2(9~)

= A2 + 1 + 20:(Af + 1)Re((pg¢2'~I)) + 0:2(Af + 1) - Cap¢2(Q~)

= A2 + 1 - Cap</>2 (Q~) - 0:2(Af + 1).

For the denominator we have the following estimate:

IIPg¢2 + 0:~1112 = IIPg¢211 2 + 20:Re( (P9¢2, ~1)) + 0:2

= 11(1 - Pg~)¢2112 - 0:2

= 1- 2Re((¢2,Pg~¢2)) + IIPg~¢2112 - 0:2

2: 1- 2Re((¢2,Pg~¢2)) + 1(¢2,P9~¢2W - 0:2

2 1 2 2 = 1 - 1 + A2 Re(E1(¢2, Pg~¢2)) + (1 + A2)21E1 (¢2, Pg~¢2)1 - 0:

= (1- Cap¢2(Q~))2 _ 0:2 1 + A2

(1 + A2 - Cap¢2(Q~))2 - 0:2(1 + A2)2 (1+A2)2

and the assertion follows immediately. o In view of this lemma it is clear that an upper bound for I(P9 ¢2, ~1)1 is useful.

The next lemma reduces the problem of finding upper bounds for 1 (P9¢2, ~1> 1 to finding lower bounds for 1 (¢1 , ~l) I· Lemma 5.18. The following inequality holds.

I(P9¢2'~1)12 ::; (A2 + 1)2(1-1(¢I'~IW)· Proof. First we get rid of the projection Pg by observing that

1 (P9¢2, ~1) >..f + 1 E1 (Pg¢2, ~1)

1 >..f + 1 El(¢2,~I)

>"2 + 1 >"T + 1 (¢2'~1> >"2 + 1 >..f + 1 (¢2, ~1 - (¢1, ~1)¢1)'

Then the Cauchy-Schwarz inequality gives us

Finally we give a lower bound for 1 (¢1, 'lh) I.

Lemma 5.19. If Al is simple, i.e., if A2 > 0, the following inequality holds.

1(¢I,'lhW 2 A2 ~2 Af.

Proof. This is seen by the following calculation.

£1 ['lj;I]

E, [~(¢" ,p')¢j 1 00

j=1 00

j=2 00

j=2

1(¢I,'lj;IW + (1 + A2)(1-1(¢I,'lj;1)12) 1 + A2 - A21 (¢1, 'lj;1) 12.

o

o

Since AT can be estimated from above in terms of the ¢1-capacity of g.l.., we are now in the position to prove a capacitary upper bound for I(Pg¢2,'lj;I)1 which involves only A2 and Cap,Pl (Q.l..).

Lemma 5.20. If Al is simple and Cap</>l (Q.l..) < 1, then

I(P. '" .1. )12 < (A2+1)2 Cap</>l(Q.l..) g'l'2, '1'1 - A2 1 _ Cap</>l (Q.l..) .

328 Andre Noll

Proof. In view of Lemma 5.18 and Lemma 5.19 we have

(A2 + 1)2(1-1(</Jl,1h)1 2)

< (A2 + 1)2 (1 - A2 ~2 Af)

(A2 + 1)2 A9 A2 1

(A2+1)2 Cap'h(Q.1.) A2 1 - CaP¢l (Q.1. ) ,

<

where in the last inequality we used the capacitary upper bound of Corollary 5.6 (a) for the first eigenvalue. 0

Putting together what we have achieved so far, we can now easily prove Theorem 5.15.

Proof. (of Theorem 5.15) Put 0::= -(P9cP2,'ljJl). Observe that (5.10) implies that Al is simple and that

CaP¢l (Q.1.) < 1. Hence Lemma 5.20 and inequality (5.10) yield

2 (A2 + I? CaP¢l (Q.1.) (1 + A2 - CaP¢2 (Q.1.)) 2 0: < <

A2 1- CaP¢l (Q.1.) 1 + A2 (5.13)

Therefore the hypothesis of Lemma 5.17 is verified and we get

A~ - A2 = 1 + A~ - (1 + A2)

1 + A2 - CaP¢2(Q.1.) - 0:2(1 + Af) 2 < (1 + A2 - Cap¢2(Q.1.))2 _ 0:2(1 + A2)2 (1 + A2) - (1 + A2)

(1 + A2)(1 + A2 - CaP¢2 (Q.1.))Cap¢2 (Q.1.) + 0:2(1 + A2)2(A2 - Af)

(1 + A2 - CaP¢2 (Q.1.))2 - 0:2 (1 + A2)2

(1 + A2)2CaP¢2 (Q.1.) + 0:2(1 + A2)2 A2 < .1. . - (1 + A2 - Cap¢J9 ))2 - 0:2(1 + A2)2

Now the first inequality of (5.13) gives us

(1 + A2)2CaP¢2 (Q.1.) + (1 + A2)4 l~~Ph (~;~) A9 _ A < aPh

2 2 - (1 + A _ C (9.1.))2 _ (1+A2)4CaP¢1 (9-1.) 2 aP¢2 A2(1-CaP¢1 (9-1.))

(1 + A2)2(1 - CaP¢l (Q.1.) )CaP¢2 (Q.1.) + (1 + A2)4CaP¢1 (Q.1.)

A2(1 + A2 - CaP¢2 (9.1.) )2(1 - CaP¢l (Q.1.)) - (1 + A2)4CaP¢1 (Q.1.) A2,

and Theorem 5.15 is proved.

Applications of this theorem are given in Section 6.

o

Remark 5.21. Let H be a self-adjoint operator which corresponds to a regular Dirichlet form in L2(X, m). If Y is an open subset of X such that K := X \ Y is compact and H Y arises from H by imposing Dirichlet boundary conditions on K, one can not prove a capacitary lower bound of the form

AJ - Aj ~ const Cap(K) (5.14)

for the second eigenvalue or for higher eigenvalues (here Cap(·) denotes the capacity of the regular Dirichlet form associated to H). This is seen from the following example (cf. Figure 3 below). Let n = (0,7f)2 C ~2, H = -.6. with Dirichlet

7f

7r

2"

° 7r

2" 7f

FIGURE 3. A domain with A2 = Ar

boundary conditions on an and let H Y be the Dirichlet Laplacian in L2(fl \ K), where

K = {(x,y) En: x::; 7f/2,y = 7f/2}.

One can easily prove that the second eigenvalues A2 and Ar of Hand H Y coincide although the capacity of K is positive, see [GZ94] or [DMN97].

The crucial point with this example is that the second eigenfunction (P2 of H given by

(/J2(x, y) = sin(x) sin(2y)

already belongs to the form domain 9 = HJ(n \ K) of H Y . Let us ask what this means for the general capacity of Definition 4.2: Since (P2 E 9 the projection of (P2 onto g~ vanishes, i.e., we have Cap</>2 (Q~) = 0, unlike as in the classical case. Hence the inequality Ar - A2 ~ const Cap</>2 (Q~) indeed holds for this particular example but we cannot prove it in general.

330 Andre Noll

5.4. Additive perturbations

It is also possible to include additive perturbations together with the domain perturbation treated in the previous sections. In what follows we ask for an estimate for the spectral bound of operators HQ, v of the form

where HQ is as in Definition 3.1 the domain perturbation with respect to the closed

subspace g of (oF, £1 (-, .)) and V is a suitable perturbation in HQ = g1{. Recall that a symmetric operator V in HQ is called infinitesimally form

bounded with respect to HQ if the domain of V contains the domain of HQ and for each c > 0 there is some c~ 2: 0 such that for all u E dom(HQ) the following inequality holds:

Observe that this is the case if and only if for each c > 0 there is some Co 2: 0 such that

(5.15)

The latter version is more convenient for our purpose. We shall always assume that Co is chosen as small as possible. Then c f---7 Co is a monotonically decreasing function on (0,00). It is well known, see Theorem VL1.33 of [Kat80], that if V is infinitesimally form bounded with respect to HQ, then the form

(u, v) f---7 (HQ u , vj + (Vu, vj, u, v E dom(HQ)

is closable and semi-bounded from below (actually it would be enough to assume form boundedness with some bound strictly less than 1, see [Kat80], Section VI.6). Hence its closure £ + A + V determines a unique self-adjoint operator HQ,v in 1{Q. Moreover the form domain of HQ,v equals g and one has

for all u E g. In the next theorem we prove an upper bound for the spectral bound AQ,v = inf a(HQ,v) of HQ,v.

Theorem 5.22. Let A be an eigenvalue of H with normalized eigenfunction ¢ tf- g and let V be a symmetric operator in 1{Q which is infinitesimally form bounded with respect to HQ. Then

where Co is the constant in inequality (5.15).

Proof. Let PQl- be the orthogonal projection (in (F'£l("')) onto g.L. Then, by (5.15)

1 + AQ,v - A . {(£1 + V)[u] } wf IIul1 2 :UEg\{O}

< (£1 + V)[(1- PQl-)<t>]

II (1 - PQl- )¢11 2

< £1 [(1 - PQl- )<t>]

(1 + E) II (1 _ P Q l- ) <t> 112 + Co'

From the proof of Theorem 5.5 we see that

and the result follows. D

If V is a symmetric operator in H which is infinitesimally form bounded with respect to H we can define HQY for any closed subspace of 9 by restricting the form [; + A + V to g. In this case we can say more.

Corollary 5.23. Fix 0 < 01 < 02 < 1. Let A be an eigenvalue of H with normalized eigenfunction ¢ and let V be a symmetric operator in H which is infinitesimally form bounded with respect to H. Then for every closed subspace 9 of F with Capq\ (g.L) E [01,02] the following inequality holds.

2 AQ,v - A:S C,s, + --s:-Capq\(9.L),

1 - U2

where c,s, is the constant in inequality (5.15) with c = 01.

Proof. Since Capq\(9.L) :S 02 < 1 the hypothesis of Theorem 5.22 on ¢ is fulfilled. Thus Theorem 5.22 gives us

D

For applications of this result, see Section 6.2.

332 Andre Noll

5.5. Operators with spectral bound of arbitrary type

In Section 5 the spectral bound ,\ := inf O"(H) was always supposed to be an eigenvalue. We proved capacitary estimates on the magnitude of the shift of the spectral bound when introducing a domain perturbation. These estimates measured the shift in terms of the capacity with respect to a normalized eigenfunction to ,\. In this section we no longer assume that ,\ is an eigenvalue. Theorem 5.24 below gives a capacitary characterization for the spectral bound being unaffected under a domain perturbation.

Assume as in Section 3.2 that (1i, (', .)) is an arbitrary real or complex Hilbert space and H is a self-adjoint operator in 1i which is semi-bounded from below with spectral bound'\:= infa(H). Define (E,F) as the non-negative closed quadratic form which corresponds to H -'\. Given a closed subspace 9 of (F,E1 (-, .)), let HQ be the domain perturbation of H with respect to 9 in the sense of Definition 3.1 and put ,\Q := inf a(HQ). The next theorem states that there is no shift of the spectral bound if and only if one can find normalized ¢ E F of arbitrarily small energy E[¢] such that also the capacities Cap¢(Q-.l) remain arbitrary small.

Theorem 5.24. The following assertions are equivalent:

(i) ,\Q = ,\. (ii) For each E > 0 there is a ¢c E F such that II¢c II

CaP¢E (Q-.l) < E.

1, E[¢c] < E and

Proof. The proof of (i) ~ (ii) is easy. Since

inf{£[u] : u E 9 : Ilull = I} = inf a(HQ) - ,\ = 0

we can find for each E > 0 some ¢c E 9 with E[¢c:] < E and 1I¢c: II = 1. But cPc E 9 implies CaP¢E (Q-.l) = 0, see Remark 4.4 (a). For the other implication let PQ "- be the orthogonal projection onto g-.l. Then

which gives us

IIPQ"-¢cI1 2 :S Cap¢£(Q-.l) < E

<

<

<

<

E1 [(1 - PQ "- )¢c] 11(1- PQ"-)¢c:)11 2

E1 [¢c] 1 - 2Re(¢c, PQ"-¢E) + IIPQ"-¢c:11 2

l+E 1- 21(¢c:,Pg "-¢c:)1

l+E 1 - 21IPQ"-¢EII

l+E 1- 2y1i'

The result follows by letting E tend to zero. D

Example. Let H = - d~2 in 1-{ = £2(IR). The corresponding form is then given by

£(u,v) = l u'v'dx, (u,v E F= H1(IR)).

Obviously the spectrum of H is purely absolutely continuous and is equal to the half-axis [0,00). Let 9 = HJ((O, 00)). Then H9 is the Dirichlet Laplacian in 1-{9 =

£2((0,00)). To derive from Theorem 5.24 that >..9 = ° (which is of course obvious for other reasons too) it is sufficient to consider for instance the sequence cPn (x) = CnX[O,n] (x) sin(11'x/n) , where Cn is such that IlcPnll = 1.

6. Applications to differential operators of arbitrary order

The general results of Section 5 are applied to the special case of differential operators on open subsets of IRd. We shall treat operators of arbitrary even order 2m with minimal assumptions on the coefficients. In Section 6.1 we show that the eigenvalue estimates of Section 5 yield estimates of the form

clrd - 2m ::; >..9 _ >.. ::; C2rd-2m

and >..9 _ >.. < C r d - 2m

2 2 _ 3

if d > 2m and the domain perturbation takes place on a ball of sufficiently small radius r. The results of Section 5.4 are applied in Section 6.2 where we look at perturbations consisting of the above domain perturbation together with an additive perturbation belonging to the Stummel class of suitable order.

For further relevant literature which is not restricted to second-order operators we refer to [Maz85], [Wei84] and [Fre82].

6.1. Capacitary eigenvalue estimates for domain perturbations

Let 0 C IRd be an open set. We want to treat operators in £2(0) which are formally given by

(Hu)(x) = L (-1)la IDa(aae(x)Di3u (x)) lal,Ii3I:Sm

(6.1)

with coefficient functions aai3 = ai3a E £toc(O). To give a rigorous definition of H we consider the quadratic form

a(u,v):= L 1 aai3(x)D Q uDi3v dx, lal,Ii3I:Sm

(6.2)

initially defined on C;:,,",(O), the space of smooth functions with compact support. In order to obtain a self-adjoint operator we need to know that a is closable. The requirement of closability is an implicit assumption on the coefficients aai3 and there are many criteria known on the coefficients aai3 implying closability of a, see e.g., Davies [Dav95], [Dav97] or Agmon [Agm65], Section 7. We will always

334 Andre Noll

assume that for some constants b E 1R, c > 0, and all u E C.;"'(O) the following inequality holds.

c-1 11ull;" S a( u, u) + bllul1 2 s cllull;", where II· 11m is the Sobolev norm

Iluli m := ( L IIDaUI2dX) 1/2 lalSm n

(6.3)

Observe that a quadratic form satisfying inequality (6.3) is always closable and that (6.3) is still valid for u E H[f'(O) , the Sobolev space of order m, because H[f'(O) is by definition the completion of C.;"'(O) with respect to the norm II· 11m. This also makes clear that the closure 0; of a has domain F = HO'(O). Let H be the self-adjoint operator corresponding to (0;, F), put A = inf 0"( H) and let £, := 0; - A be the non-negative form associated to H - A.

Although there is in general no explicit description of the domain dom(H) of H we call H an operator with Dirichlet boundary conditions since H is the Friedrichs extension of the operator (6.1), initially defined on C.;"'(O), provided that the coefficients aQ (3 are sufficiently smooth.

We now want to investigate the u-capacity (see Definition 4.2) of subspaces of F consisting of functions that vanish on some set A c n. Therefore we define FA as in (4.11). If A = Br(xo) := {x EO: Ix - xol S r} we write Fr,xo for short. Lemma 6.1 below compares the u-capacity of FA with the (m,2)-capacity Capm,2(A) treated in the book of Adams and Hedberg [AH95]. Recall that this capacity is defined by

Capm,2(A) := inf{llull;" : u E C.;"'(lRd), u 2:: 1 on some nhd. of A},

see Section 2.2 of [AH95] for details.

Lemma 6.1. Let aa(3 = a(3a E Lfoc(O) such that inequality (6.3) holds and let Uo E H[f'(O).

(a) For each c > 0 there is a constant Cc > 0 such that for all Xo E 0 and all r E (0,1) with Br(Hc) (xo) C 0 the following inequality holds.

Capuo(F/,"xo) S ccr-2mlluoll~o(Br(1+e)(xo))' (b) If U is an open subset of 0 satisfying

essinfRe(eilluo(x)) > 0 xEU

for some () E [0, 27l"), then there is some constant Cu > 0 such that for all AcU,

Capuo (FX) 2:: Cu Capm,2(A).

Proof. (a) Choose 'IjJ E C.;"'(lRd) such that 'IjJ(x) = 1 if Ixl < 1 and 'IjJ(x) = 0 if Ixl 2:: 1 + c. Put

Then wruo E Htr'(rl), wruo = Uo on Br(xo) and supp(1.}!,,) c Br(HE)(XO), In view of Proposition 4,5 (a) and inequality (6,3) we have

Capuo (F,:\o) :S EdWruo] :S cllwruoll~ = cIIWruoll~o(Br(l+C)(XO»'

In what follows the symbol "const" stands for a generic constant, possibly changing its value from line to line, which is independent of r. By Leibniz' formula and the Cauchy-Schwarz inequality we obtain for 10'1 :S m,

IDaWruol2 < const L 1 (DfJ w,,) (Da-fJ uo ) 12 fJ<a

< const L r- 21fJ1 IDa -fJuo l2

< const r- 2m L IDa -fJ uo l2 ,

fJ~a

Hence the result follows by

(b) Put

< c L 1 IDaWruol2 dx lal~m B r (1+c)(XO)

< constr-2m L 1 L IDa -fJuo l2 dx

lal~m B r (l+E)(XO) fJ~a

< constr-2mlluoll~m(B (x »' o r(1+E) 0

, := essinfRe(eiBuo(x)). xEU

Then by Proposition 4.5 (a) and inequality (6.3)

Capuo(Fi)

= inf{Edu]: u E H~n(rl),u = Uo m-a.e. on some nhd. of A}

= ,2 inf{Edu] : u E C;:x'(rl), u = eiB,-luo m-a.e. on some nhd. of A}

;::: c-l,2inf{llull~,u E C;-:O(rl),u = eiB,-luo m-a.e. on some nhd. of A}

;::: C- 1,2 inf{llull~, u E C;-:O(rl), u ;::: 1 m-a.e. on some nhd, of A}.

This proves the desired estimate with Cu = C- 1,2, o

Next we combine the inequalities just proved with the capacitary eigenvalue estimates that have been proved in Section 5. In the following theorems Hr,xo denotes the self-adjoint operator in L2(rl \ Br(xo)) which is associated to (0;, F",xo) and Ar,xo := inf (J'(Hr .xo ) its spectral bound.

Theorem 6.2. Let d > 2m and let aafJ = afJa E Lfoc(fl) such that inequality (6.3) holds. Moreover suppose that A = inf (J'(H) is a simple and isolated eigenvalue with

336 Andre Noll

eigenfunction ¢ tI- :F~x . If Xo E n is such that the conditions of Lemma 6.1 hold • 0

with uo = ¢, then there are constants Cl, C2 > 0 such that for small r > 0,

c rd - 2m < A - A < C r d - 2m . 1 _ r,xo _ 2 (6.4)

Proof. The proof follows easily from Corollary 5.6 (b), Theorem 5.10 and Lemma 6.1. D

Remark 6.3. Inequality (6.4) is already contained in [Maz85], Chapter 10 in the case of operators being defined on a sqare in IRd. See also the original work [Maz73].

If the coefficients aa{3 are (real- ) analytic and n is a domain, we can prove the same lower bound even if ¢(xo) = O.

Theorem 6.4. Let d > 2m and let aa{3 = a{3a be (real-Janalytic functions on a domain n such that H is elliptic (i.e., Llal=m L{3+{3'=a a{3{3'(x)~a =f. 0 for all (x,~) E n x IRd \ {O} J and inequality (6.3) holds. If A is a simple and isolated eigenvalue of H, then for any Xo E n there is a constant c > 0 such that for small r>O

\ \ > d-2m Ar,xo - /\ _ cr .

Proof. It is known (see e.g., [Joh49], [Hor83a], Section 8 and [Hor83b], Section 11) that the eigenfunction ¢ of H and A is analytic, i.e., for each unit vector v E IRd the mapping

Iv: (-r,r) ----> <C, t f-+ ¢(xo +tv)

can be extended to an analytic function (here r > 0 is such that {x E IRd : lx-xol < r en). It follows that the case ¢(xo) =f. 0 is already covered by Theorem 6.2, so let us assume that ¢(xo) = O. Since ¢ oj:. 0 and n is a domain there is some unit vector v such that Iv oj:. O. Hence the set of zeros for Iv must be discrete. By decreasing r we may assume that t = 0 is the only zero of Iv in (-r, r). Put Yo := Xo + ~v. Then ¢(yo) =f. 0 and B~ (yo) C Br(xo) C n. By Theorem 6.2 there is some c' > 0 such that

( r)d-2m Ar,xa - A 2: A~ ,Yo - A 2: c' "2 = crd - 2m ,

with c = 22m - d c'. This completes the proof. D

Next we treat the second eigenvalue.

Example. Let Hand Hr,xo be the operators defined in Example 5.2. Suppose in addition that d > 2m and that the second eigenfunction ¢2 of H does not vanish in Xo. Then there is a constant c > 0 such that for small enough r,

\ \ < d-2m A2,r,xQ - /\2 _ cr ,

where A2 and A2,r,xo are the second eigenvalues of Hand Hr,xo respectively.

Proof. This follows immediately from Corollary 5.16 and Lemma 6.1. D

6.2. Stummel class perturbations

Let Ha be a symmetric constant-coefficient partial differential operator in L2(I1~d) of order 2m, i.e.,

fY'u Ha u = 2:= (-i)IO'laO' oXO" u E C~(JRd), ac> E JR.

10'1::;2rn

Assume that H o is elliptic, i.e.,

2:= aa~o. =I 0 for all ~ E JRd \ {O}. lo.l=2rn

Then H o is essentially self-adjoint and the unique self-adjoint extension of H a, which we again denote by Ho, has domain H 2rn (JRd). Now suppose that 0 is an open and bounded subset of JRd and that for lal < 2m we are given measurable functions qo. : JRd ---> JR with supp(qo.) C O. Moreover assume that each qa belongs to the Stummel class of order Pa with Pa satisfying

Pc> < 4m - 2a. (6.5)

Recall that this means that the function

{ J Iqo.(y)12Ix - ylPn- d dy,

Ix-YI<l x f-7 J- Iqo.(y)12dy,

Ix-yl::;l

if d > Pc"

if d :s; Pa

remains bounded. It is known, see [Wei76] Theorems 10.17 and 10.lS, that then the operator

(J"'u Vu:= 2:= (-i)IO'lqo.oxa

lal<2rn

maps H2rn(JRd) into L 2(JRd) and that V is infinitesimally small with respect to Ha, i.e., for each c > 0 there is some Co 2' 0 such that

IIVul1 2 :s; cilHaul12 + cE Ilul1 2

for all u E H 2rn (JRd). In particular V is infinitesimally form bounded with respect to Ha since this is implied by infinitesimal smallness, see [KatSO], Theorem V1.1.3S.

In order to obtain discreteness of the spectrum we localize to O. Let H be the Friedrichs extension in L2(0) of the restriction of Ho to C;:x'(O) and put ,\ = inf u(H). Then the quadratic form E which corresponds to H -,\ has domain:F = Htl'(O) and the restriction of V to C;:x'(O) remains infinitesimally form bounded with respect to H. Hence for each c > 0 there is a constant CE 2' 0 such that

I(Vu, u)1 :s; cEdu] + cE llul1 2 .

Since 0 is bounded the spectrum of H is discrete, see [Agm65], Theorem 14.6. Let cP be a normalized function in the eigenspace ker (H - ,\). As an immediate consequence of Corollary 5.23 we obtain the following result.

338 Andre Noll

Theorem 6.5. Fix 0 < 51 :S: 52 < 1. For any closed subspace Q of the form domain Ho(n) with Capq\(Ql.) E [51 ,52 ] we have

2 .xQ'v - .x :S: c,5, + --;:-Capq\(Ql.),

1 - U2

where .xQ'v is the spectral bound of HQ,V defined as the self-adjoint operator which corresponds to the form

(u, v) f--7 feu, v) + 2: (qau, v), u, v E Q. lal <2m

Example. Let d E {5, 6, 7}, 0, = (0, l)d, H = ~2 in L 2 (n), defined as the closure of the quadratic form

U f--7 L l~uI2dx, U E c;:,,(n).

Further let V E Lfoc.uniJ(IRd) with supp(V) c n. Define

A := 0, \ {x En: Ix - (1/2, ... , 1/2) I :S: r}

and let HA,v := ~2 + V in L2(A). Then there are constants C1,C2 2: 0 such that for small enough r,

.x A,V - .x :S: C1 + C2rd-4,

where .xA,V = inf a(H) is the spectral bound of HA'v.

Proof. Since V E Lfoc.uniJ (lRd), V belongs to the Stummel class of order d. Because d < 8 the hypothesis (6.5) is fulfilled. Hence

2 .xA,V -.x:S: c + 1 _ 5 Capq\(H;5 (A)l.)

with some c 2: O. Moreover d > 4 implies Capq\(Ho(A)l.) :S: c'rd -4, see Lemma 6.1 and Theorem 6.2. 0

7. Scaling methods

This final section is dedicated to scaling properties of the zero-order capacities. In Section 7.1 we derive the scaling property

CaPo(rA) = r d - 2 Capo(A)

for the classical zero-order capacity in lRd , d 2: 3. After introducing a zero-order version of the general capacity of Section 4.3 we prove in Section 7.2 zero-order counterparts of some estimates of Section 5. In Section 7.3 we specialize to differential operators with homogeneous symbol and prove an appropriate scaling property for the zero-order capacity of Section 7.2 for this class of operators. After introducing a general method for optimizing eigenvalue estimates by scaling we finally show that the results of Section 7.3 cannot be improved by this method.

7.1. The scaling property of the classical zero-order capacity in ]Rd

Let U be an open subset of]Rd where d 2: 3. The classical (zero-order) capacity in ]Rd can be defined by

C (U) .= { inf{f]Rd lV'ul 2dx : u E C~(]Rd), U 2: 1 on U}, apo . 00, if there is no such u. (7.1)

The capacity can be extended to arbitrary subsets of]Rd by

CaPo(A) := inf{Capo(U) : U :J A, U open}.

See Section 4.1 for a physical motivation of this definition in the context of electrostatics. Let £ be the quadratic form of the Laplacian in £2(]Rd) and let F := HI (]Rd) be its form domain. Then

£[u] = { lV'uI 2 dx, u E :F. i]Rd Therefore we may write (7.1) as

C (U) = { inf{£[u]: u E F,u 2: 1 a.e. on U}, apo 00, if there is no such u.

This formula makes plain the name "zero-order capacity" since £ is used in the definition rather than £1 as in Definition 4.2. In this context we also call the capacity of Definition 4.2 the "first-order capacity" .

Clearly the symbol d

LI~jI2 j=1

of the Laplacian is a homogeneous function of degree 2. This allows us to compute the zero-order capacity of a ball in ]Rd. For simplicity we consider only the case d 2: 3. Because of the translation invariance of the Laplacian we may restrict ourselves to balls which are centered at the origin. Let

Br := {x E ]Rd : Ixl < r}

be the open ball of radius r and let '¢r(x) := r- 1x, x E ]Rd. Then

CaPo(Br) inf {ld lV'ul 2dx : u E C~(]Rd), U 2: 1 on Br}

inf {ld 1V'(u 0 '¢r)12 dx : u E C~(]Rd), U 2: 1 on Bl }

r- 2 inf {ld 1(V'u)1 2 0 '¢rdx : u E C~(]Rd), U 2: 1 on Bl }

r d- 2 inf { ( 1(V'u)1 2 0 '¢r IdetD'¢rl dx: u E C~(]Rd), } i]Rd U 2: 1 on Bl

r d - 2Capo(BI). (7.2)

340 Andre Noll

Therefore it is enough to know the capacity of the unit ball which is given by

4nd / 2 CaPo(Bd = d .

f("2- 1) (7.3)

This formula can be proved stochastically with the aid of Brownian motion (see [PS78], Section 3.1) or analytically as it is possible to explicitly write down the zero-order equilibrium potential eB" i.e., the unique minimizer of the infimum in (7.1). It is given by

eB, (x) = { ~XI2-d if x E B 1 ,

if x E JRd \ B 1 .

For Ixl > 1 we have l'VeB l l2 = (2 - d)2IxI 2- 2d . Therefore we get

CaPo(Bd = r l'VeB l l2dx = (2 - d) 2Wd rOC) r2-2drd-1dr = (d - 2)Wd' J~d J1

where 2nd / 2

wd := f(d/2)

is the surface area of the unit ball in JRd. Now equation (7.3) follows immediately from the functional equation of the gamma function.

With the same method one proves that the scaling property (7.2) of the zeroorder capacity holds not only for balls but is true for any subset A of JRd, i.e., we have

(7.4)

where rA := {rx : x E A}

denotes the set A, scaled by r.

7.2. The zero-order capacity in the general Hilbert space setting If H is the Laplacian in 1t := £2(JRd) there is no scaling property for the first-order capacity of Definition 4.2. This is due to the fact that in this case A = inf CT(H) = 0, i.e., the form £1 (', .) corresponds to - b. + 1 whose symbol is

d

LI~jI2+1 j=l

and the constant term destroys the homogeneity. Therefore we introduce a zeroorder capacity in the general Hilbert space setting which can be defined in a straightforward way if H is a strictly positive operator. As we shall see in Theorem 7.4, it is possible to prove zero-order counterparts of the results of Section 5. This leads to bounds which behave appropriately under scaling. Since the scaling property (7.4) can be used to optimize estimates for eigenvalues, this method is of general interest.

Remark 7.1. There are also good reasons to prefer the first-order capacity: Firstly the operator associated to £1 is, per definition, H - ). + 1, where). = inf cr(H) is the spectral bound of H. If ). is an eigenvalue of H with normalized eigenfunction ¢ this implies

£1 [¢] = {(H - ). + 1)¢, ¢) = 1 and 0:::; Cap¢(9) :::; 1,

for any closed subspace 9 of (F'£l(', .)). Therefore our notation simplifies most of the formulas and makes the proofs more transparent and easier to read.

Moreover one can only hope for a homogeneous capacity in the case of nonnegative differential operators with homogeneous symbol. The theory presented in Section 5 is much more general as it allows arbitrary self-adjoint and semibounded operators in general Hilbert spaces, i.e., the underlying Hilbert space need not necessarily be an L2-space. In this setting there is no meaning of scaling anyway.

The definition of the abstract zero-order capacity reads as follows.

Definition 7.2. Let H be a self-adjoint operator in H with positive spectral bound A. Let 9 be a closed subspace of (dom(H1/2), {Hl/2., H 1 / 2.)). The zero-order capacity of 9 with respect to u is defined by

Cap~(9) := {H1/2 Pgu, H 1/ 2 Pgu) = IIH1/ 2 Pgu11 2,

where Pg is the orthogonal projection onto the space 9 in the Hilbert space (dom(H1/2), {H 1/ 2., H 1/2.)).

Remark 7.3.

(a) Clearly {H 1/ 2 ., H 1/ 2 .) and £1(',') (see Definition 4.2) are equivalent scalar products on F = dom(H1/2). Hence the classes of perturbations one can handle are the same.

(b) If ). = 1, then the zero-order capacity of Definition 7.2 and the first-order capacity of Definition 4.2 coincide.

It is not difficult to prove zero-order versions of Corollary 5.6 (a) and Theorem 5.10.

Theorem 7.4. Let). be an eigenvalue of H with normalized eigenfunction ¢.

(a) If ¢ tf. g, then

).g _ ). < ACapg(9..l) . - A - Capg(~j..l)

(7.5)

(b) Suppose that A is a simple eigenvalue of H. Let /1 := inf{O"(H) \ {A}} - A be the spectral gap. Then

(7.6)

342 Andre Noll

Proof. (a) Let Pg "- be the orthogonal projection onto g~ in the Hilbert space (dom(Hl/2), (H 1/ 2., H 1/ 2 .)). Then

. { IIH 1/ 2u11 2 } )... 9 mf II u 112 : u E 9

<

<

IIH1/2(1 ~ Pg"-)¢11 2

II (1 ~ Pg "- )¢112

IIH1/ 2¢11 2 ~ IIHI/2 Pg "- ¢11 2

1 ~ 2Re(¢,Pg"-¢) + IIPg"-¢11 2

)... ~ Cap~(Q~)

1 ~ ~Re(H¢,Pg"-¢) + IIPg"-¢11 2

)... ~ Cap~(Q~)

1 ~ ~Re(Hl/2¢,Hl/2Pg"-¢) + I(Pg"-¢,¢)12

)... ~ Cap~(Q~)

1 ~ ~Cap~(Q~) + ;2 Cap~(Q~)2 )...

1 ~ Cap~(Q"-) ,\

)"'Cap°(Q~) )... + q, . )... ~ Cap~(Q~)

(b) Let K be the one-dimensional eigenspace of Hand ).... Denote by PK and K ~ the orthogonal projection onto K and the orthogonal complement of K respectively, where orthogonality is meant with respect to the scalar product (H 1/ 2 ., H 1 / 2 .) on dom(H 1/ 2). Put PK"- := 1 ~ PK. Since K is spanned by ¢ and IIH1/ 2¢11 = v:x we get for any u E F,

Hence

Consequently

and

(¢, u) (PK "- U, ¢) 1 >.. (¢, u) (PK "- U, H ¢)

~(¢, u)(H 1/ 2 PK"- U, Hl/2¢).

O.


Putting all these facts together we obtain for u E 9 with Ilull = 1,

(1 + ~Cap~(g)) IIH1/2u11 2

= Cap~(K) + Cap~(K.1) + :21I H1/ 2 P,,¢11 21IH1/ 2uI1 2

:2: Cap~(K) + (,X + p)IIPK-LuI1 2 + ~ I(H1/2p,,¢,H1/2u)12

= IIH1/ 2 PK ul1 2 + (,X + p)(l - IIPKUI1 2) + :21 (H1/2¢, H1/2U) 12

= 'x1(u, ¢)1 2 + (,X + p)(l -I(u, ¢W) + pl(u, ¢W

= ,X + p.

This inequality gives us the following estimate for the perturbed bottom eigenvalue:

,X" inf{IIH1/2uI1 2 : u E 9, Ilull = I}

The proof is complete.

Remark 7.5.

> ,X2(,X+p)

,X2 + pCap~(~1)

,X - Capg(Q) ,X + ,Xp-----'---rc

,X2 + pCapg(9)

Capo (Q.1) ,X+,Xp ¢ .

,X2 + ,Xp - pCapg(Q.1)

o

(a) If ,X = 1 we recover the estimates of Corollary 5.6 (a) and Theorem 5.10 respectively.

(b) From part (b) of Theorem 7.4 we can immediately conclude the following lower bound which is slightly worse but more handy:

,X" - ,X > -P-Capo (9.1). - 'x+p ¢

7.3. Scaling invariance of eigenvalue estimates

We first describe the type of operators we are interested in. Let 0 be an open subset of ]Rd and let aa(J = a/3a E LToc(O). Consider the homogeneous bilinear form

ao(u, v) := L 1 aa(JD"'uDi3v dx, lal=IGI=m n

(7.7)

initially defined for u, v E C;:O(O). Assume that an is semi-bounded from below and closable. Let a be any closed extension of ao (not necessarily the closure of ao) and denote the form domain of a by F. Let H be the unique self-adjoint operator which corresponds to a in the sense of Theorem 2.2, put ,X := inf o-(H) and £1 := a - ,X + 1.

344 Andre Noll

Observe that ao is closable if

c~lllull~ ~ ao(u, u) + bllul1 2 ~ cllull~,

for some b E lR, c > 0 and all u E c;:x'(n), where II· 11m denotes the Sobolev norm of order m. In this case the above inequality carries over to all u E Ho(n) and the closure ao of ao has domain Ho(n). See also the discussion at the beginning of Section 6.

Define for each 'r > 0 the open set

nr : = {'rx : x En}

and the scaling transformation

'l/Jr : nr --+ n, 'l/Jr(x) = 'r~lX. Further define the self-adjoint operator Hr in Hr := L2 (nr ) as the operator which corresponds to the closed form

ar(u, v) := L 1n (aaf3 0 'l/Jr)DauDf3v dx, (7.8) lal=If3I=m r

which is defined for u,v E Fr := {Wo'l/Jr: W E F}.

Then

dom(Hr) = {u 0 'l/Jr : u E dom(H)}, and Hr(u 0 'l/Jr) = r~R(Hu) 0 'l/Jr,

where £ := 2m is the order of the differential operator H. For instance, if H is the Dirichlet Laplacian in L 2 (n), then £ = 2 and Hr is the Dirichlet Laplacian in L2(n r ). Now suppose that it is given a closed subspace 0 of (F,El(-, .)). Define

the operator HI] in HI] := g'H via the construction described in detail in Section 3.2. In the same way we obtain an operator Hf acting in an associated Hilbert space H? Denote the spectral bounds by >..1] and >..? respectively. Because of the homogeneity we have >"r = 'r~R >.. and >..? = 'r~R >..1], i.e.,

>..1] _ >.. = 'rR(>..? - >"r). (7.9)

Therefore an estimate for >..? - >"r also yields an estimate for the shift of the spectral bound of the operator H. Once one has proved a general estimate for the shift of the spectral bound one may apply this result for each pair Hr , Hf, r > O. Hence one can try to improve the estimate for )..1] - >.. by optimizing the right-hand side of equation (7.9) over r.

We shall make this idea concrete in the case of the estimates of Theorem 7.4. For this we use the zero-order capacity of Definition 7.2. In order to use it we have to assume that H is a strictly positive operator. First we show that the zero-order capacity has appropriate scaling properties.

Proposition 7.6. Let 9 be a closed subspace of (F,E1 (·, .)) and let u E F arbitrary. For r > 0 define

Or := {v 0 'l/Jr : V EO}, ur := u 0 'l/Jr.

Then the following is true:

(a) Cap~,(91') = rd-£Cap~(9), (b) For any subset A of 0 put

Then

{v E F : v = 0 m-a. e. on some neighborhood of A},

{v E F1' : v = 0 m-a. e. on some neighborhood of r A}.

Cap~,(F/A) = rd-£Cap~(F±).

Proof. (a) We first prove the identity

Fg,(u 0 'l/J1') = (Fgu) 0 'l/J1" (7.10)

To this aim, note that for any v, w E F we have

a1'(v 0 'I/J", w 0 'l/J1') = rd-£a(v, w),

implying

G-;:= {v E (]1': a1'(v,w) = 0 for all w E (]1'} = {Wo'I/J1': W E (].1.}.

Write u = v + w with v E (] and w E (].1.. Then

Fgr(u 0 'l/J1') = Fg,(v 0 'l/J1') + Fgr(w 0 'l/J1') = v 0 'l/Jr = (Fgv) 0 'l/J1' = (Fgu) 0 'l/J1"

which proves (7.10). It is enough to prove the assertion (a) for all u E dom(H). Using identity (7.10) we obtain

IIH,I/2 Fgr U1' 112 = (H,1/2 Fg,u", H,I/2U1' ) = (Fgr U1" H1'u1' )

r-£(Fgr(u 0 'l/J1')' (Hu) 0 'l/J1')

r- f ((Fgu) 0 'l/J1" (Hu) 0 'l/J1') rd-£ (Fgu, H u)

rd- e (Hl/2 Fgu, H 1/2U)

rd-£Cap~ (9).

For the proof of (b) note that obviously F"A = (F4)T' From the proof of the first part of this proposition we see that ((FA)1').1. = ((FA).1.)1" Consequently

o

As an application of Proposition 7.6 we show that the eigenvalue estimates of Theorem 7.4 cannot be improved by optimizing over r. If ¢ is a normalized eigenfunction of H with eigenvalue >.., then clearly

<iT := r- d / 2 ¢ 0 'l/J1'

346 Andre Noll

is a normalized eigenfunction of Hr with eigenvalue Ar = r-R A. Hence Theorem 7.4 (a) together with identity (7.9) yields

9 £ ArCaPfr (9f) A - A :s; r 0...L ' (7.11)

Ar - CaP¢r (9r )

and from part (b) we get

CapO- (9f) A9 - A > r£ A f-L ¢, .

- r r A; + f-LrAr - f-LrCaPfr (9f )) (7.12)

Theorem 7.7. The numbers on the right-hand sides of inequalities (7.11) and (7.12) coincide with the right-hand sides of (7.5) and (7.6) respectively. In particular they are independent of r.

Proof. By Remark 4.4 and Proposition 7.6 (a) we have

Cap~J (9r)...L) Cap~-d/2¢o,pJ (9r)...L)

r-dCap~O,pr ((9r)...L)

r-dCap~O,pr ((9...L)r)

r-£Cap~(9...L).

This together with the identity Ar = r-£ A gives us

£ ArCaPf, (9f) ArCap~(9...L) r Ar - CaPfr (9f) Ar - r-£Cap~(9...L)

8. Open problems

r- R ACap~(9...L)

r-R(A - Cap~(9...L))

ACap~(9...L )

A - Cap~(9...L)·

o

This concluding section describes some open problems which seem to be worthwhile for further research.


(1) Treatment of higher eigenvalues (a) Upper bounds. In the case of regular Dirichlet forms it is known that all eigenvalues of the perturbed operator admit capacitary upper bounds when the original operator is subjected to a domain perturbation on a compact set [McG96].

In Section 5.3 it has been proved that this is true in the general Hilbert space setting for the second eigenvalue, but the proof of this result was much more difficult than the corresponding proof for the bottom eigenvalue. Clearly, it would be desirable to have upper bounds for each eigenvalue of the perturbed operator. Since the minimax principle gives an expression for all eigenvalues, it should be possible to prove an analog of Theorem 5.15 and Corollary 5.16 for all eigenvalues. The idea is the following: By the minimax principle we have

)...; = inf{£ [u] : u E g, Ilull = 1,u -.l 'ljJl,'" ,'ljJk-d,

where )...~ are the eigenvalues of the perturbed operator He;; with corresponding eigenvectors 'ljJ k. Therefore

k-l

)...~ ::; £[Pe;;(h + L ajV>j]' j=l

where the ay's have to be chosen appropriately. (b) Lower bounds. In Remark 5.21 we have seen that in general one cannot expect capacitary lower bounds for higher eigenvalues by using the classical capacity. We also saw that inequality (5.14) becomes true when the classical capacity is replaced by the general capacity. Therefore one can conjecture that this did not happen by coincidence, but capacitary lower bounds are indeed true when using the general capacity. Unfortunately, the proof of the lower bound for the first eigenvalue (Theorem 5.10) does not carryover to the case of higher eigenvalues, so a more general method is needed to understand the behavior of higher eigenvalues in general.

(2) Capacity and scattering theory Given He;;, the domain perturbation of H with respect to the sub

space 9 (according to Definition 3.1), one can ask whether the pair (H, He;;) is a complete scattering system with respect to the identification operator id : HG '----> H, i.e., the question is whether the two-space wave operators exist and are complete. In [DC98] it has been proved that this is the case for the Dirichlet Laplacians (-~n, -~A)' A, n open subsets of ]Rd, if the symmetric difference n 6. A has finite capacity. One year later this result was generalized to Kato-Feller operators [DGCZ95]. In particular, completeness of the scattering system implies equality of the absolutely continuous spectra of H and He;;. So proving completeness is not only of interest for operators which are relevant for quantum mechanics. Since the proof of the above-mentioned results uses stochastic methods, it is not possible to generalize it to a larger

348 Andre Noll

class of operators, e.g., higher-order differential operators. On the other hand, in [Ede99] there is a very general approach for proving completeness which is based on purely functional analytic properties. With the aid of these results it seems to be possible to prove completeness for the pair (H, HQ) in the general Hilbert space setting.

(3) Numerical analysis The results of Section 5 gave estimates on eigenvalues in terms of the

capacity. For concrete applications one needs to compute these capacities, but this turns out to be difficult, even in the case of the classical capacity. On the other hand, it is quite easy to give estimates for these capacities by choosing good trial functions. At this point one could use the powerful tools of numerical analysis to find such trial functions. Besides giving quantitative bounds in concrete cases, numerical considerations can also be used to illustrate the abstract results.

( 4) N on-selfadjoint ness Although the results of Section 6 on differential operators made no re

strictions on the order of the operators in question, there remain important problems which cannot be treated with the methods described in this thesis because of our assumption on the self-adjointness. For instance in [BEGK99], a connection between domain perturbations and the Wentzell-Freidlin theory for non-self-adjoint generators was discovered. Therefore it seems to be of interest to analyze the behavior of the spectrum by considering domain perturbations of non-self-adjoint operators. In particular, operators whose spectrum is a discrete set which is contained in some sector of the complex plane could be treated by passing from symmetric forms to sectorial forms which are well studied, [Kat80].

In concrete cases the infimum A of the real part of the spectrum is often an isolated eigenvalue and there is no other eigenvalue with real part A. It is an open problem to examine the behavior of A under domain perturbations. To this aim the theory of capacity for self-adjoint operators would have to be translated to sectorial operators.

Appendix A. The minimax principle

Let H be a non-negative self-adjoint operator in the Hilbert space H. Then a(H) c JR, and the bottom of the spectrum of H is given by

A inf{(Hu,u): u E dom(H), llull = I}

inf{IIH!uI1 2 : u E dom(H!), llull = I}.

The number a E a(H) belongs, by definition, to the discrete spectrum of H if and only if a is an isolated eigenvalue of finite multiplicity. Otherwise A is said to be in the essential spectrum. For any finite-dimensional subspace 9 of dom(H!) and

n EN define

AH(Q) = sup{IIH!uI1 2 : u E g, lIuli = I},

An = inf{)..H(Q): dim(Q) = n}.

Then Al = A and (An)nEI\l is an increasing sequence of real numbers. The assertion of the minimax principle is the following: The least upper bound of (An)nEI\l equals the bottom Aoo of the essential spectrum of H. The sequence (An)nEI\l, omitting all values equal to Aoo (if there are such) coincides with the discrete spectrum in [A, Aoo) and each eigenvalue is repeated according to its multiplicity.

By an obvious modification one can also include self-adjoint operators which are merely semi-bounded.

Appendix B. Sobolev spaces of fractional order and the elliptic regularity theorem

For any s E JR define ks : JRd ----> JR by ks(x) := (1 + IxI 2)s. Further define positive measures /-ls on JRd by d/-ls := ksdx, dx being the Lebesgue measure in JRd. Each f E L2(JRd,/-ls) may be regarded as a tempered distribution in JRd. To see this, let S(JRd) be the topological vector space of all Schwartz functions and let (c,f>n)nEI\l be a sequence in S(JRd) that converges to zero with respect to the topology of S(JRd). Then, for each t E JR,

sup kt(x) Ic,f>n (x) I ----> 0 xEIRd

as n --> 00. Choose t E JR such that k- s- 2t E L 1 (JRd). Then, by the Cauchy-Schwarz inequality

< r Ifl 2k;j2dx r lc,f>nI2k~sj2dx JIRd JIRd

Ilfll~2(IRd.l-'s) ld lc,f>nI2k;k-s-2tdx

< Ilfll~2(IRd.l-'s) (sup lc,f>n(x)lkt(X))21ILs_2tll£1(IRd). xEIRd

This shows, that JIRd fc,f>ndx --> 0 as n tends to infinity. Therefore the map

Tf : S(JRd) --> C, Tf(c,f» = r fc,f>dx JIRd

is a continuous linear functional on S(JRd) , i.e., Tf belongs to the dual space S'(JRd) of S(l~d) which is called the space of tempered distributions. Denote by F the Fourier transform in S' (JRd) and recall that F is a continuous and bijective linear map in S' (JRd) whose inverse is also continuous. In terms of F the Sobolev space of order s E JR is defined by

HS(JRd) := F-l(L2(JRd,/-ls)) = {F- 1 f: f E L 2(JRd,/-ls)}.

350 Andre Noll

H S Cffi.d) becomes a Hilbert space with respect to the scalar product

(F- 1 f, F-1g)HS(J[/!.d) := (1, gh2(Kf.d,J.Ls)·

Let 0 be an open subset of JRd and denote the space of distributions in 0 by D'(O). A distribution T E D'(O) is said to be locally HS if for any x E 0 there is a neighborhood A of x and a distribution T E HS(JRd) such that T(¢) = T(¢) for all ¢ E C~(A).

Let N E N be fixed and suppose that fa E CCXJ(O) for each multi-index 0: with 10:1 ::; N. Assume that there is at least one fa with 10:1 = N which is not identically zero. These data determine a differential expression L by which we mean the map

L : D'(O) --> D'(O), L(T):= L fa DaT. lal::;N

The number N is called the order of L. L is said to be elliptic if

L fa(x)ya # 0 lal=N

for all x E 0 and y E JRd \ {O}. Observe that ellipticity only depends on those fa with lal = N. The lower order terms play no role.

Now the theorem on elliptic regularity reads as follows.

Theorem B.l. In the situation just described suppose that L is elliptic and T, T are distributions in 0 such that

(i) L(T) = T. (ii) T is locally HS for some s E JR.

Then T is locally Hs+ N .

Appendix C. Markov processes and Hunt processes

In this appendix we summarize the necessary background information on Hunt processes and introduce the notion of equivalence for Hunt processes. Almost all of the material can be found in [FOT94].

Throughout this appendix X will denote a locally compact separable metric space and m will be a strictly positive Radon measure on X of full support, i.e., m is a measure defined on the Borel sets B(X) of X which is finite on compact sets, strictly positive on non-empty open sets and supp(m) = X. Let X~ be the one-point compactification of X. When X is already compact, ~ is adjoint as an isolated point. A u-algebra B~ on X~ is given by

B~ = B(X) U {B U {~} : B E B(X)}.

Definition C.l. (a) A stochastic process with state space (X~, B~) is a quadruple

(0, M, (Xt)t~O, JlD) such that


(i) (Q, M, lP') is a probability space, (ii) X t is a measurable map from Q to Xl\. for each t ~ O.

(b) A filtration (Mtk,>o, i.e., an increasing family of sub-a-algebras of M, is called admissible with respect to (Q, M, (Xt)t>o, lP') if each X t is measurable from (Q, M t ) to (Xl\., BD.). It is called right continuous, if for each t ~ 0,

(c) Given an admissible filtration (Mt)t>o, a [O,oo]-valued function T on Q is called an (Mt)-stopping time if {T :::; t} E M t for each t ~ O. The a-algebra MT is then defined by

MT = {A EM: An {T:::; t} E M t for each t ~ O}.

Having defined these standard notions from probability theory we come to the definition of a Markov process as a stochastic process with additional properties.

Definition C.2.

(a) A quadruple M = (Q, M, (Xt)t>o, (lP'x)xEX,,) is called a Markov process on (X,B(X)) if (i) For each x E Xl\., (Q, M, (Xt k,::o , lP' x) is a stochastic process with state

space (Xl\., Bl\.). (ii) The function

X f-t lP'x{Xt E E}

is B-measurable for each t ~ 0 and each E E Bl\.. (iii) There is an admissible filtration (Mt)t>o such that

lP'x( {XHs E E}IMt ) = IP'x t ({Xs E E})

IP'x-a.e. for any x EX, t, s ~ 0 and E E B. (iv) 1P'l\.({Xt = 6.}) = 1 for each t ~ O.

(b) A Markov process is called normal if

lP' x ( { X ° = x}) = 1 for each x EX.

(c) A Markov process is said to be a strong Markov process with respect to the admissible filtration (Mtk::o if (Mtk:::o is right continuous and the equality

lP'fL({XT +S E E}IM T ) = IP'x T ({XS E E})

holds whenever J1 is a probability measure on Xl\., E E Bl\., S ~ 0 and T is an (Mt)-stopping time. Here the probability measure lP' fL on (Xl\., Bl\.) is defined by

lP'fL(E):= r lP'x(E)dJ1(x). ix"

(d) A Markov process is called m-symmetric if its transition function

pt(x,E) := lP'x({Xt E E}), x E X,t ~ O,E E Bl\.

(C.1)

352 Andre Noll

is m-symmetric in the sense that

for any non-negative measurable functions u and v.

All these definitions were necessary to introduce the main object of this appendix, the Hunt process which will be defined now.

Definition C.3. A normal Markov process M = (n,M,Xt,lP'x) on (X,B) is called a Hunt process if

(i) XXl(W) = ~ for each wEn. (ii) Xt(w) = ~ for each t :::: ((w) := inf{t :::: 0 : Xt(w) = ~}. The function ( is

called the life time of M. (iii) For each t E [0,00] there is a map 8 t : n ----> n such that Xs o8t = X Hs ,

s :::: O. (iv) For each wEn the path t f--7 Xt(w) is right continuous on [0,00) and has a

left limit on (0,00) (such a path is usually called cadlag). (v) There is an admissible filtration (Mdt2':o such that M is strong Markov with

respect to (Mtk:c:o. (vi) For any (Mt}-stopping time Tn increasing to T and for any probability mea

sure 11 on Xil. the following equality holds

lP'fL({ lim X Tn = XnT < oo}) = IP'fL({T < oo}), n-->oo

where IP' fL is defined in (C.1).

The connection between regular Dirichlet forms and Hunt processes consists in the deep fact that there is a one-to-one correspondence between these two objects, see Theorem C.18. But the Dirichlet form determines the associated Hunt process only up to equivalence, a phrase which we now want to make clear. First we have to introduce nearly Borel sets.

Definition C.4. Let M be a Hunt process.

(a) A set B C X is called nearly Borel if for each probability measure 11 on Xil. there are B l , B2 E Bil. such that Bl c B C B2 and

Here Bil. is the O"-algebra on the one-point compactification Bil. of B, which may be constructed from B as X il. was constructed from X in the beginning of this section.

(b) A nearly Borel set B C X is called M-invariant if for each x E B we have

IP'x({Xt E Bil.,Xt- E Bil. for all t:::: O}) = 1,

where X t - = limsit Xs.

Now equivalence between Hunt processes is defined as follows.

Definition C.5. Two Hunt processes M 1 , M2 are said to be equivalent if there is some nearly Borel set N c X such that

(i) X \ N is an invariant set for Ml and M 2 .

(ii) meN) = O. (iii) The transition functions of Ml and M2 coincide outside N.

Finally we need an optional sampling theorem for supermartingales, which is used in the proof of the stochastic representation of the equilibrium potential. First we have to define a suitable class of supermartingales for which this theorem holds.

Definition C.6. Let (0, M, (Yt)t>o, lP') be a stochastic process with state space ~ (equipped with the a-algebra of all Borel sets) and let (Mtk:~o be an admissible filtration.

(a) (0, M, (Mtk,~o, (Yt k::o , lP') is said to be a supermartingale if for each s > t"2 0:

lE{IYtI} < 00 and lE{YsIMd:s: Yt.

(b) A supermartingale (0, M, (Mtk:o, (Ytk:~o, JP') is said to be of class (DL) if for each t "2 0,

lim sup lE{IYTI : IYTI "2 n} = 0, n----+CX) TEAt

where At denotes the family of all (Mt)-stopping times T with T :s: t.

Using this definition the optional sampling theorem of Doob reads as follows.

Theorem C.7. (Doob's optional sampling). Let (fl,M,(Mt)t>o, (Yt)t>o,lP') be a - -

supermartingale of class (DL) which is right continuous. Further let T be a (M t )-

stopping time and 0 :s: t :s: s. Then

A proof can be found in [Dq, Appendix C.

C.l. Basics on regular Dirichlet forms

In this appendix we state without proof the necessary facts on regular Dirichlet forms which are needed to understand the previous sections. This is by no means a complete treatment of the subject, it is not even a complete survey. The reader should consult [FOT94], [Fuk80] or [MR92] for all the details and the proofs of the theorems. Most of this appendix is taken from [FOT94].

We start with describing the setup and fixing notation. Let X be a locally compact separable metric space and m be a strictly positive Radon measure on X of full support, i.e., m is a measure defined on the Borel sets 8(X) of X which is finite on compact sets, strictly positive on non-empty open sets and supp(m) = X.

354 Andre Noll

We use the notation L2(X, m) for the Hilbert space of all square integrable (equivalence classes of) real-valued functions on X equipped with the scalar product

(u, v) := L uv dm.

We write L=(X, m) for the Banach space of all m-essentially bounded real-valued functions on X, the space of continuous functions on X will be denoted by C (X), its subspace consisting of continuous functions with compact support by Cc(X). The pointwise maximum and minimum of two functions u, v is abbreviated by u V v and u /\ v respectively. Hence u+ := u V 0 and u~ .- -(u /\ 0) denote the positive and the negative part of u respectively.

Definition C.8. (a) A non-negative symmetric closed form (E, F) on L2(X, m) is called a Dirichlet

form if the following Markov property is satisfied.

u E F implies u+ /\ 1 E F and E[u+ /\ 1] ::::; E[u]. (C.2)

(b) A Dirichlet form (E, F) is called regular if F n Cc (X) is not only dense in (F,E1(·, .)) but also dense in Cc(X) with respect to the uniform norm 11·11=. Sometimes it is useful to replace the Markov property (C.2) by a smooth

version, explained in the next lemma.

Lemma C.9. Let (E, F) be a non-negative symmetric closed form. Then (E, F) fulfills the Markov property if and only if for each c > 0 there is a real function ¢E : lR. -+ lR. such that

(i) ¢€(t) = t for t E [0,1]. (ii) -c::::; ¢E(t) ::::; 1 + c for every t E R

(iii) h < t2 implies that 0 ::::; ¢€(t2) - ¢E(h) ::::; t2 - t 1 .

(iv) u E F implies ¢€ 0 u E F and E[¢e 0 u] ::::; E[u].

Remark C.10. Let H be the unique self-adjoint operator which corresponds to (E, F) in the sense of Theorem 2.2. Since H ~ 0 we can construct the semigroup (e~tH)t>o of H and the resolvent (H + 0:)-1, 0: > O. It is natural to ask about characterizations of the Markov property (C.2) in terms of the semi group and the resolvent. Before we can give the answer we have to introduce some more notation.

Definition C.H. A bounded linear operator T on L2(X, m) is said to have the Markov property if u E L2(X, m), 0 ::::; u ::::; 1 m-almost everywhere implies that o ::::; Tu ::::; 1 m-almost everywhere.

Theorem C.12. Let (E, F) be a non-negative symmetric closed form on L2(X, m) and let H be the self-adjoint operator which is associated to (E, F). The following are equivalent.

(i) (E, F) fulfills the Markov property (C. 2) . (ii) For each t > 0 the operator e-tH fulfills the Markov property of Definition

C.1l.

(iii) For each 0' > 0 the operator O'(H + 0')-1 fulfills the Markov property of Definition C.lI.

Theorem C.13. A regular Dirichlet form (E, F) possesses the following properties.

(a) If u, v E F n LOCJ(X, m), then uv E F and

E[uv] ~ Ilull~E[v] + Ilvll~E[u]. (b) Ifu E L 2(X,m), v E F are I-excessive, i.e.,

e-tHu~etu, e-tHv~etv, (t>O),

and u ~ v, then u E F and El[u] ~ El[V].

Every Dirichlet form admits a set function, the capacity, which is defined for arbitrary subsets of X.

Definition C.14.

(a) For U c X open define the capacity of U by

Cap(U) := inf{El[u] : u E F, u .2: 1 m-a.e. on U}.

If there is no such u the convention Cap(U) = 00 applies. The capacity is extended to arbitrary A c X by

Cap(A) := inf{Cap(U) : U => A, U open}.

(b) A statement is said to hold quasi-everywhere (q. e.) if it holds outside of some set of zero capacity.

(c) A function defined on X is said to be quasi-continuous (q. c.) iffor each c > 0 there is an open set U C X with Cap(U) < c such that ulUc is continuous.

(d) The zero-order capacity Capo(-) is defined in the same way as in (a) but with El[U] replaced by E[u].

The following result is an important consequence of the regularity of (E, F).

Theorem C.15. Suppose that (E,F) is a regular Dirichlet form. Then each u E F admits a quasi-continuous m-version U, i. e., there is a quasi-continuous function u which coincides m-almost everywhere with u. Two quasi-continuous m-versions of u coincide quasi-everywhere.

The notation u = v will always indicate that u is a quasi-continuous m-version of v.

Next we present a lemma which is often used to prove that certain statements which are assumed to hold m-a.e. are indeed valid quasi-everywhere. Part (b) can be proved with the aid of the Banach-Alaoglu and the Banach-Saks theorems.

Lemma C.16. Let (E, F) be a regular Dirichlet form on L2(X, m).

(a) If Un E F is a sequence such that Un -+ u in (F,El (·, .)), then there is a subsequence that converges to u quasi-everywhere.

356 Andre Noll

(b) Let Un E F such that sUPnEl\[ £1 [Un] < 00. If U E F such that Un ---+ u in L2(X, m), then there is a subsequence un] such that its Cesaro means i(un1 + ... + u nk ) converge to U in (F, £1(-, .)).

The most important properties of capacity are given in the following theorem.

Theorem C.17. Let (E,F) be a regular Dirichlet form on L2(X,m).

(a) Let AI, A2 C X arbitrary. If Al C A 2, then Cap(Ad ::; Cap(A2). (b) If B C X is a Borel set, then

Cap(B) = sup{Cap(K) : K C B, K compact }.

(c) If An, n E N, is an arbitrary sequence of subsets of X, then

(d) For arbitrary A C X the capacity of A may be computed as

Cap(A) = inf{£I[u] : u E F, it -2: 1 q.e. on A}. (C.3)

(e) The infimum in (C.3) is attained for a unique element eA E F, called the equilibrium potential of A. Moreover ° ::; eA ::; 1 m-a.e. and e'A = 1 quasieverywhere on A; eA is a i-excessive function, i.e., for each t > ° the following inequality holds.

e-tHeA ::; eteA,

where H is the self-adjoint operator corresponding to (£, F) in the sense of Remark C.lO.

(f) For arbitrary A C X there is a unique measure v A on X, called the equilibrium measure of A, satisfying the following properties. (i) supp(v A) cA.

(ii) v A charges no set of zero capacity. (iii) £1 (eA, U) = Ix U dVA for each u E F n Cc(X).

As an immediate consequence of (e) and (f) we see that for a closed set F C X the following equalities hold true:

(C.4)

We now want to describe the connections between Hunt processes and regular Dirichlet forms. Given an m-symmetric Hunt process M = (Sl,M,Xt,lPx ) on (X,B(X)), it is easy to construct a regular Dirichlet form: The transition function

Pt(x, E) := lPx ( {Xt E E}), (t > O,X E X,E E B(X))

of M determines a semigroup (St)t>o via

St!(x) := Ix f(y)pt(x, dy) = lEx(J(Xt)).


In this context the semigroup property is often given via the Chapman-Kolmogorov equation The above semigroup clearly fulfills the Markov property of Definition (C.lI). Since it turns out to be strongly continuous as well, there is a regular Dirichlet form on L2(X, m) which corresponds to (St)t>o. This Dirichlet form is called the Dirichlet form of M. It is not so easy to see that the converse of these considerations is also true:

Theorem C.IS. Given a regular Dirichlet form (E, F) on L2(X, m) there is an m-symmetric Hunt process M whose Dirichlet form is the given one (E, F). M is unique up to equivalence.

For a proof see [FOT94] Theorems 7.2.1 and 4.2.7. For the construction of the process it plays an important role that (E, F) is regular. According to Theorem C.15 every u E F admits a quasi-continuous m-version. This allows one to ignore successively sets of capacity zero.

In view of Theorem C.18 one can ask for a stochastic interpretation of analytical objects. In Theorem C.19 below we give a stochastic representation of the equilibrium potential eB where B c X is a Borel set. This theorem is identical to Theorem 4.1 where the proof is carried out.

Theorem C.19. Let (E, F) be a regular Dirichlet form on L2(X, m) with the corresponding Hunt process M = (O,M,M t ,Xt ,lP'x) and let Be X be a Borel set of finite capacity. Then

eB(X) = lEx{e-TB }, where TB := inf{t > 0 : X t E B} denotes the first hitting time of B.

A major part of this thesis is devoted to the analysis of the spectral consequences of domain perturbations of an operator or its associated form, and it is a natural question whether the restriction of a regular Dirichlet form (E, F) to a smaller domain F B consisting of functions which vanish on B (in a sense to be made precise), is again a regular Dirichlet form. This turns out to be rather difficult and depends on several properties of the set B eX. Fortunately everything works fine if B is closed. This is the content of part (a) of the following theorem, which is taken from Theorem 4.4.3 in [FOT94].

Theorem C.20. For any Borel set B C X define a closed subspace of (F, E1 (-, .)) by

FB := {u E F: u = 0 q.e. on B}. (a) If B is closed, then the restriction of E to the form domain F B is a regular

Dirichlet form in L2(X \ B,m). (b) Let PF-Ji be the orthogonal projection onto FiJ in (F,E1 (-, .)). Then

(PF-Jiu)(,) = lE(.){e-TBu(XTB )}·

For any open subset Y of X put B := X\Y. By Theorem C.20 (a) the restriction of E to F B corresponds to a unique self-adjoint and non-negative operator H Y

in L2(y, m). The Dynkin formula establishes a connection between the resolvents

358 Andre Noll

of Hand H Y . Before we can state this result we have to introduce measures of finite energy integral and their associated potentials.

Definition C.21. A positive Radon measure J.L on X is said to be of finite energy integral, if there is a constant 'Y ~ 0 with

Ix lui dJ.L :::; 'YV£l(U, u) (C.5)

for all u E F n Cc(X). The collection of such measures is denoted So.

For any measure J.L of finite energy integral the map u 1-+ Ix luldJ.L defines a continuous linear functional on (F'£l(·'·)). The Riesz representation theorem entails that there exists a function U1 J.L E F such that

for all u E F n Cc(X).

Definition C.22. The function U1 J.L E F is called the I-potential of J.L .

Example. Let f E L2(X, m) be a non-negative function. Then J.L := fdm is a measure of finite energy integral and its I-potential is given by U1 J.L = (H + 1)-1 f.

Now Dynkin's formula reads as follows.

Theorem C.23. Let (E,F) be a regular Dirichlet form in L 2(X,m).

(a) For any compact K C X and any I-potential u there is a unique measure J.LK E So supported in K such that

(~)(x) = lEx { e-r(K)u(XrK )}.

(b) (Dynkin's formula): Let f E L2(X, m) with f ~ 0 m-a.e. and denote the restriction of f to Y by fly. Then

(H + 1)-1 f = (HY + 1)-1 (fly) + U1J.LCf+1)-1 f

and

The measures of finite energy integral can also be used to compute the capacity of compact subsets of X.

Theorem C.24. Let K be a compact subset of X. Then

Cap(K) = sup {J.L(K): J.L E So, suPp(t:L C K, } J.L(X) < 00, 11U1J.Llloo :::; 1 .


Finally we need a highly non-trivial result on the set of irregular points. A point x E X is called regular for the Borel set Be X, if IP'x{TB = O} = 1, where TB = inf{t > 0 : X t E B} is the first hitting time of B; x is called irregular for B if IP'x{TB = O} = o. It is a consequence of Blumenthal's 0-1 law that x is irregular if and only if it is not regular.

Theorem C.25. For any Borel set B the set of irregular points in B zs of zero capacity.

For the proof see [FOT94] Theorems A.2.6 (i), 4.1.3, and 4.2.1 (ii).

Acknowledgments

I would like to thank my supervisor, Prof. Dr. Michael Demuth, for his support and his many mathematical suggestions. This work has also benefited from fruitful discussions with the members of the "Stochastic Spectral Theory" group in Clausthal: Sven Eder, Walter Renger, and Eckhard Giere. Moreover, I would like to thank Bruce Saint Hilaire for helping me to improve the English in several parts of the manuscript.

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Andre Noll Technical University of Darmstadt SchloBgartenstraBe 7 D-64289 Darmstadt, Germany e-mail: [email protected]


An Interpolation Family between Gabor and Wavelet Transformations: Application to Differential Calculus and Construction of Anisotropic Banach Spaces

Bruno N azaret and Matthias Holschneider

Abstract. In this paper we define an interpolation family of transformations, whose extremes are the Gabor and wavelet transformations, in order to extend the Cordoba-Fefferman results (see [CF]) and to define a differential calculus at the first order. This interpolation family is based on the representation through translated and modulated versions of an analyzing function, with the additional property that this family is naturally localized in paraboloids. This will allow us at the end of the paper to construct anisotropic Banach spaces of functions by pullback techniques.

1. Introduction

In signal treatment, there are mostly two transformations which are used. The first, the Gabor transformation, is based on the representation of the Weyl-Heisenberg group over L2(JRn) (see [AAGM]). More precisely, 9 E L2(JRn) being given, we define for all b, W E JRn,

gb,w(X) = eiw(x-b)g(x - b)

and, for all s E L2 (JRn), the Gabor transform of s with respect to 9 by

W~O)s(b,w) = (gb,w,S).

One can prove that W~O) is unitary and then invertible. Up to a constant, its inverse transformation is given by, for all r E L2 (T*JRn),

The second, the wavelet transformation, defined by Grossman and Morlet in [GMl], [GM2] at the beginning of 1980s for the analysis of seismic signals and developed by Meyer [eM] and his collaborators [Mal], [Daul], [Dau2], [Ja~, is

364 B. Nazaret and M. Holschneider

based on the representation of the affine group over L2 (IRn) (see [AAGM]). It replaces the frequency by a scale notion. Here, we define the wavelet transform of s with respect to 9 by: for all b E IRn, a > 0,

W[g, s](b, a) = Wgs(b, a) = (gb,a, s)

where

gb,a(X) = alng (X:b)

is a translated and dilated version of the analyzing function g. In the case of wavelets, the invertibility is more difficult to handle. The natural candidate to be the inverse is, as in the Gabor transformation, its adjoint, defined by:

Mgs(x) = { roo dadb gb.a(x)r(b, a) J]Rn Jo a

for r E L2 (IRn x IR+, d:db). The result is the following; defining

(OOda~ mg.h(~) = Jo -;;g(a~)h(a~)

we will say that 9 is admissible (condition AI) if there exists a constant c > 0 such that for all ~ =1= 0,

0< c- 1 ::; mg.g(~) ::; c < CXJ

and that 9 is strictly admissible if, for all ~ =1= 0, mg,g(O == 1. Then, if 9 is admissible, there exists a reconstruction wavelet h (that is such that Mh Wg = Jd). In the case where 9 is strictly admissible, 9 is its own reconstruction wavelet and Wg is unitary.

These two transformations do not present the same interest. The wavelet transform acts as a mathematical microscope. Indeed, suppose that the analyzing function is localized around a given point Xo with variance 1. Then, the function gb.a is localized around (xo - b) with variance a. This property allows the wavelet transformation to be used for analyzing pointwise and microlocal regularity (see [Holl], [Ho12], [Jafj, [HT]). Furthermore, the Fourier representation of the wavelet transform, obtained by the Plancherel theorem,

Wgs(b, a) = (2~)n in d~eibt;g(a~)s(~), shows that the wavelet can be interpreted as a time-frequency analysis. Its main disadvantage is that it possesses a bad angular resolution since, a going from 0 to CXJ, the support of g(a~) is translated in a conical set in the Fourier space.

On the contrary, the Gabor transform does not admit the pointwise regularity of functions since the width of the window (g, g) is constant. But, considering the Fourier representation of this transform,

W~O)s(b,w) = (2~)n in d~eib(t;-w)g(~_w)s(~), one can see that it isolates the central frequency direction.

Interpolation between Gabor and Wavelet Transformation 365

In order to avoid these difficulties, the first idea would be to perform all these operations on 9 (i. e., translations, dilations, and modulations), but it does not work because the obtained transformations are not L2-continuous. So, we have to make some restrictions. (This idea was already studied by Folland in [Foil.)

Let a E (0,1) be given. We define the analysis of order a of s with respect to 9 by the following: V(b,w) E T*IRn,

w~a)s(b,w)=\g~~,s) (1)

where g~~(x) = (w) n2" eiw(x-b)g ((w)a(x - b)). (2)

Here, the C= function (w) is given by:

{ (w) = Iwl for Iwl 2: 1, 1/2:::; (w) :::; 1 for Iwl :::; 1

(3)

(Iwl stands for the euclidean norm of w). In the same manner, we define the synthesis of order a of the function r defined on the phase space with respect to h: 'Vx E IRn,

(4)

Here, W~a) s analyzes s around b at the scale Iwl-a . Using the Plancherel formula, we obtain:

w~a)s(b,w) = ( l)n r d~(W)-n2"eib~g((w)-a(~_w))S(~). (5) 271' iran

Then, in Fourier space, the analyzing function family moves into paraboloids with opening a. Unfortunately, with this representation, we lost the inversion formula, though we will see later that there exists an approximate one. Now, let us remark that, for a = 0,1, we obtain respectively the Gabor and a wavelet type transformations. For a = 1/2, we have the FBI (Fourier-Bros-Iagolnizer) transformation introduced by Bros-Iagolnizer ([BI]) and Sjostrand ([Sjo]) for studying analytic microlocal classes and used by Cordoba-Fefferman ([CF]) for representing pseudo-differential operators as multipliers on the phase space at the first order.

In the first section of this paper, we study the transformations w(a), M(a)

and their continuity on the Schwartz space, and prove that "inversion" formulas hold modulo regularizing operators of any order, given a priori. Then, we define W~a) s for s being a tempered distribution by duality.

In the second section, we shall study the natural operator algebras generated by the transformation weal and its adjoint M(a), and look at the properties of multiplier operators on the phase space through w(a). Here, we generalise the Cordoba-Fefferman results for every exponent a E (0,1). We give also the action of Weal on diffeomorphisms and local change of coordinates.

Finally, in the last section, we use weal to characterize Sobolev spaces Hm(JRn)and construct anisotropic Banach spaces of functions. We finish with the study of an example, based on an L=-localization over the space.


2. The interpolation family 2.1. Analysis and reconstruction in S (JRn)

We study here the continuity of the transformations defined above on the Schwartz space. For that aim, let us introduce the localization functions, which are used in the definition of semi-norms in the Frechet space S(JRn). For k E JR and x E IRn, we set

K,k(X) = (1 + Ixl)-k . Then, the following classical technical lemma holds.

Lemma 2.1. Vk E JR, "ix,x' E JRn,

K,k(X + x') :::; O(l)K,dx)K,_lkl(x').

Remark. Throughout this article, 0(1) denotes a numerical constant, which may vary from one line to another.

Proof. In the case where k :::; 0, one has that 1 + Ix+x'i :::; (1 + Ixl)(l + Ix'l), which concludes the argument.

Suppose now that k > O. Obviously, one can assume that Ixl,lx'l ;::: 1. It is sufficient to show that the following quantity is bounded:

Ix'ik , Ixlk K,k(X + x ),

and is equal to

C~i + I,x',ex + II~ix'j)-k where ex denotes the unit vector with direction x. Let us proceed by reductio ad absurdum and then suppose that there exist subsequences (xp ), (x~) such that

. (lx~1 , 1 x~ I) p~o Ixpl + IXpl exp + Ixpl = 0,

hence Ix~I/lxpl ----7p--+oo 0 and lexp + IXpl-IX~11 ----7p--+oo 0, since Ix~1 is bounded below. In addition, (exp)pElN being bounded, one can extract from it a subsequence which converges to a unit vector e. Then, Ix~I/lxpl ----7 1, a contradiction. 0

We now give another technical lemma, which is fundamental in localization questions for the transformations w(a), and whose proof may be found in [Hol2].

Lemma 2.2. Let k > n, bE JRn, and a> O. Then, we have

IW[l\:k,l\:k](b,a)I:::;O(l)( 1) K,k(_b_). l+a n l+a

We can now prove the main result of the section, concerning the continuity of the transformations w(a) and M(a).

Theorem 2.1. w(a) : S(JRn)xS(JRn) ----7 S(T*JRn) andM(a) : S(JRn) xS (T*JRn) ----7

S(JRn) are continuous.

Proof. We begin by showing a result about the properties of rapidly decreasing functions.

Lemma 2.3. cP : ]R2n -. ]R is rapidly decreasing if and only if cP is rapidly decreasing with respect to x uniformly in y, and with respect to y uniformly in x.

Proof. The first assertion is obvious. Suppose now that cp verifies the two uniform growth conditions. Then, Vk, 1 E IN, :3Ck , Cl;

(Ixl 2: p, y E ]Rn) ==} Icp(x, y)1 :::; Ckp-2k

and (Iyl 2: q,x E ]Rn) ==} Icp(x,y)l:::; Clq-2k

then (Ixl 2: p, Iyl 2: q) ==} Icp(x, y)1 :::; min (Ckp-2k, CIQ-2k) .

One can assume that Ckp-2k :::; CIQ-2k. Then, one gets

Icp(x, y)1 :::; CkP-2k :::; JCkCIP-kq-k

and the conclusion. o

Now, we prove the continuity of w(a). In order to use Lemma 2.3, we show the two following points.

- Localization in b:

(w~a)s) (b,w) = In dx(w)n2"'e-iw(x-b)g((w)a(x_b))s(x).

Let k E IN be given.

< r dx(w) n2", K,d(w)a(x - b)) K,k(X) JlRn < (w)_n2"'W[K,k,K,k] (b,(w)-a)

< O(l)K,k (1 + (:)-a )

using Lemma 2.2. Moreover, 1 + (w)-a 2: 1, which implies the desired local-ization.

- Localization in w: Here we use the frequency representation of the transformation w(a).

hence

(w~a)s) (b,w) = In d~(w)-n2"'eibeg((w)-a(~_w))s(~).

< (w)_n2", r d~K,d(w)-a(~_w))K,k(~) JlRn < (w) ":t W [K,k, K,k] (b, (w)a)

< 0(1) (1 ~~~)a) n K,k (1 + ~w)a)

by the same lemma. In addition, since 0: < 1, one has that

"'k C + ~w)a ) ::; O(l)"'k (Iwl-aw)

which implies the conclusion.

Now, application of Lemma 2.3 gives us the continuity of Weal as a functional with values in the space of rapidly decreasing functions. The continuity as a functional with values in the Schwartz space comes from the two following identities: for Iwl > 1 and 1 ::; j ::; n,

This completes the proof of the continuity of W ea). In order to show the same result for Mea), we use the same arguments:

then

I(M~a)r)(x)l::; L'lRn dbdw(w)Y Ir(b,w)llh((w)a(x-b))1

::; r dw(w)-n2Q"'k(W) r db(w)nO!"'k ((w)a(b - x)) "'k(b) JlRn JlRn

::; 0(1) Ln dw ((w) ¥ : (w)-¥ ) n "'k(W)"'k (1 + ~)-a ) .

Now, if Iwl ::; 1, one has that H~:\ Q 2: ~Ixl, hence

"'k (1 + ~)-a ) ::; O(l)"'k(x).

Conversely, if Iwl 2: 1, one has that

"'k ( ~) )::; O(l)"'dx). 1 + w -a

Then, actually,

gives the continuity as a functional with values in the space of rapidly decreasing functions. In the same manner as for the analysis, the identity

(8.M ea )r) = M(a)(w T) + Mea) (Iwlr) J h h J 8 j h ,

for 1 ::; j ::; n, completes the proof of the theorem. D


2.2. "Inversion" formula

In what follows, we shall assume that a E (0,1). We have defined the transforma-tions Weal : S(lRn) x S(lRn) ____ S(T*lRn) and M(a) : S(lRn) x S(T*lRn) ____ S(lRn ),

which are continuous. In this section, we shall study the operator M~a)W~a), for g, hE S(lRn). In the sequel, we will note, for a function m, M(m) the multiplication operator by m.

Proposition 2.2. If s E S(lRn ), then "ix E IRn ,

(M~a)w~a) s)(x)

where m~~~ (~)

Proof. Let us recall that

F- 1 (M (m~~~) FS) (x)

r dw(w)-nag ((w)-a(~ - w)) h ((w)-a(~ - w)) . J]Rn

(w~a)s)(b,w) = J dx(w) n2" e-iW(x-b)g«(w)a(x - b)) s(x)

which gives us the following Fourier representation:

(w~a)s)(b,w) = (2~)n J d~(w) -~" eib~g((w)-a(~ - w)) s(~).

It follows that

_1_ Je r dbdw(w) n2" eiw(x-b)h «(w)a(x - b)) (21f)n J x J d~(w)-~"eib~g((w)-a(~-w))s(~).

Thus, by the Fubini theorem, one gets

and the result.

_1_ Jd~s(~)eiX~ (21f )n

x J dw(w)-nag((w)-a(~-w))h((w)-a(~_w))

D

Studying the Fourier multiplier m~~~, we obtain two approximate inversion formulas. The first presents the advantage of holding for all analyzing and reconstruction functions, and the second is more precise.

Theorem 2.3. Let g, hE S(lRn). Then, we have

M~a)w~a) = (21f)-n (g, h)IRn Id + T

where T is a smoothing Fourier multiplier in the Hormander class OPS';oa(lRn) (see [Horl, Hor2] for the definition of such classes). '

Proof. First, let us introduce some notation:

- et; denotes the unit vector with direction ~. - For w E lRn\{O}, we set e(w) = (w)-et(e - w) where e E §n-I. - f = FgFh.

In addition, we take ~ given, such that I~I 2: 2.

1st step Study of = el where el = (1,0, ... ,0). First, one has that (eI) = O. Moreover, a simple computation gives

\flwl 2: 1, \fy E lRn , D(w)· y = Iwl-a - 2 (w, y)(el - w) -Iwl-ety thus, D( eI) = -I d is invertible. We derive from this, by a local inversion theorem, that there exists r > 0 such that : B(eI,r) ----+ (B(eI,r)) := Vr(eI) is a diffeomorphism. Let us remark that one can choose r small enough such that B(O, 1) n B(~, rlW = 0.

Now, one makes a last remark; let R be a rotation such that R(eI) = e. Then, and e are conjugate by R (i.e., = R-IeR) which implies that the radius r can be chosen independently from e E §n-I.

2nd step Here, we prove Lemma 2.4 below. Let us split lRn into two parts DI = {Iw - ~I 2: rl~l} and D2 = lRn\D I. Then, we

set m~~2 = m2 + mI where

mj(~)= ldw(w)-netf((w)-a(~_w)) j=1,2. J

Lemma 2.4. mI is the symbol of an infinitely smoothing operator. In other words,

(I~I ----+ (0) .

Proof of the lemma.

where 1* is rapidly decreasing, non-increasing with respect to lxi, and such that If(x)1 ::; 1*(x). Let us split DI again into two parts, for w near or far from O. Then, let D~ = DI n {Iwl ::; I} and D'l = {Iwl 2: I}. The integral over D~ dearly define a bounded function of ( In order to estimate the integral over D'l, one writes that

hence

III dw(w)-na 1* ((w)-et(~ - w)) I ::; 0(1) 1 dwl~ - wi-no: 1* (I~ - wi I-a)

Iw-t;I2:rlt;1

and then, since f* is rapidly decreasing, one gets the result. D

3rd step Asymptotic expansion of m2.

Setting in the integral w' = w/I~I, one gets that

m2(O = 1~ln(1-a) r dwlwl- na f(I~ll-alwl-a(e~ - w)). J B(ef..r)

Now, one can set u = <l>ef. (w):

m2(~) = r 11 + I I_~~ )11~ln(l-a) f(I~ll-au). JVr(O 0' W e~ - w, w

The point here is to expand the function k(u) = J1+a JwJ i(ef.-w,w)J around 0. First, let us note that, for lui small, one can remove the absolute value. In addition, by the definition of <l>ef.' one has that

thus

1 + O'lwl-2(e~ - w,w) = 1 + O'lwla-1Iul(eu , ew ),

k(u) 1 - O'lwla- 1 (eu , ew)lul + O(lu12) 1 - O'(eu , ew)lul + O(luI2) 1 - O'(eu , e~)lul + luI 2E(U)

where feU) is uniformly bounded in ( Note that this estimate only holds in a neighborhood of 0, that is Vr(~). Let

us define the principal part in the entire space ]Rn. Since f is rapidly decreasing, this is equivalent to adding a rapidly decreasing function in ~. (The proof would be the same as the proof of Lemma 2.4.) Then, one may easily prove that:

m2(~) = ~n duf(u) - 0' (~n du(e u , edf(u)) 1~la-l + 0 (1~12(a-l)) for I~I large enough.

4th step Conclusion of the proof. One has obtained:

m (a) (0 = 1 duf( u) + r(et) (~) g,h g,h IR"

where r~a~ (~) = 0 (1~ll-a) for I~I 2: R. Now, let X be a cut-off function with

support in B(O, R), and consider the following decomposition:

rea) = Xr(a) + (1 - x)r(a). g,h g,h g,h

The first part xr~~~ is smooth and compactly supported, and so is the symbol of

an infinitely smoothing operator. By the study above, the second part (1 - x)r(a~ g, is the symbol of a smoothing operator of order 1 - 0'. To complete the proof, it


remains to show that the derivatives verify the Hormander class estimates. For that aim, let us write that, for all f3 E INn,

oj3m~~~(~) = r d~(w)-(n+J3)a f ((w)-a(~ - w)). }[{n Applying exactly the four steps of the demonstration above, one may obtain the desired estimate. D

Now, we state a more precise result, which needs an admissibility condition on the analyzing function g. (Another condition will appear later for the L 2-theory.) This result proves the existence of a reconstruction function modulo a smoothing operator of arbitrary order, this order being given a priori.

Theorem 2.4. Let 9 E S(I~n) be such that g(O) -=I- 0, and let N E IN. Then, there exists a function hN E s(~n) such that

M(a)w(a) = I d + RN hN 9

where RN E OPS::61-a)(~n).

Proof. In the proof of Theorem 2.3, one has seen that

m~~~ (~) = m2(O + ml (~) with ml(~) = 0 (1~I-CXl), (I~I ---+ (0) and that

m2(O = 1 duk(u)(gh)(I~ll-au)I~ln(l-a). VetO

For the result above, one expanded k to the first order. Let us extend this expansion to further orders. One then obtains terms such as

1 dUPef.(u)gh(u). Vr(l;)

Let us set f = gh and let us choose f with compact support in Vr(~) such that leO) = I, and Vf3 E JNn , verifying N > 1f31 ;::: I, one has Oj3leO) = O. Under this assumption, choosing r small enough such that g does not vanish on Vr(~)' one

can set ~(u) = ~~:;. Actually, all terms of order at most N in the asymptotic

expansion vanish, and hN is the desired reconstruction function. Actually, the estimate on the derivatives is obtained in the same way as in Theorem 2.3. D

2.3. Analysis of distribution This section is devoted to the definition of the transformation Weal for tempered distributions, in order to construct later anisotropic microlocal classes. For that aim, we proceed by duality, using the continuity of the analysis and the reconstruction. Then, let 9 be an analyzing function in the Schwartz class and TJ be a tempered distribution. We set, for r E S(T*~n),

\r, w~a)TJ) = \M~a)r, TJ) . (6)


Then, the continuity of weal implies that W~a)1] is a tempered distribution on

the phase space T*JRn. Furthermore, the continuity of the functional W~a) is a mere consequence of equation (6). In fact, we obtain a more precise result for the

distribution W~a\l.

Proposition 2.5. Let 9 E S(JRn) and 1] E S' (JRn). Then, in the sense of distributions, W~Q\I is equal to the Coo -function defined on T*JRn by:

O(b,w) = (gb.w,1]).

The proof follows immediately from the definitions. This proposition shows that even if the distribution is very irregular, its

transform is smooth. Here, the regularity properties of distributions are closely related to the localization of the coefficients through the analysis by W~Q).

In the same manner, we define also, for h E S(JRn), the continuous functional M~Q) from S'(T*JRn) to S'(JRn) by: 'iO E S'(T*JRn), 'is E S(JRn),

(s,M~a)O) = (W~a)s,O). (7)

Remark that, using the inversion formula in Theorem 2.3, M~Q)O can be very irregular.

3. w(a) transformations and operator algebras

Here, we generalize the Cordoba-Fefferman results ([CF]). A pseudo-differential

operator T being given, we want to write it in the ~rm T = M~a)TW~Q) up to a

smoothing operator and at arbitrary order, where T acts on the phase space and has an asymptotic expansion in the different levels of a graded algebra, which we shall present and study here. Our first point is to define the "identity" part of this algebra.

3.1. A graded kernels operator algebra Definition 3.1. Let K = Op(K) be an operator acting on functions of T*JRn. We will say that K belongs to OPS6a) (T*JRn) iff

IK(b,w; b',w')1 = rp ((w)alb - b'l, (w)-al w - w'l) where rp : JR2 -+ JR is rapidly decreasing.

We begin by giving the equivalent following definition.

Proposition 3.2. In both dilation terms of Definition 3.1, we can replace w by w'.

In order to prove 3.2, let us give a technical lemma.

Lemma 3.1. A function f is localised in (w)-a(w - w') if and only if it is localised in (w,)-a(w - w').

Proof. It is sufficient to prove one of the assertions. Then, let us suppose that f is localised in (w)-""(w - w'). One has that

Here, four cases can happen. lrst case Iwl, IW'1 2: 1. Then, one writes that

(w') IW'1 Iw - w'l (w) ~:::; 1 + Iwl

< 1 + Iwl-""Iw - w'l < 2Iwl-""lw - w'l

since one can suppose without loss of generality that (w)-""(w - w') 2: 1. Consequently, one obtains

which gives the result. 2nd case Iw I, IW'1 :::; 1. It is immediate because of the boundedness of the functions.

3rd case Iwl :::; 1, IW'1 2: 1. In that case, (w') is equivalent to Iw - w'l since a < 1, which implies the desired estimate. 4th case The proof is analogous to the third one. D

Proof of Proposition 3.2. By Lemma 2.3, we can look at the localizations (in b and w) separately. In addition, Lemma 3.1 implies that the localization in the w variable is solved. Let us suppose having the localization in (w)""(b - b' ) and in (w)""(w - w') (and so equivalently in (w')""(w - w')). One has that

Furthermore, in the proof of Lemma 3.1, one proves that

which ends the proof.

Proposition 3.3. OPSa"") (T*IRn) is an algebra.

Proof. Let K = OP(K) and K' = Op(K') in OPSa"")(T*IRn). Then, K 0 K' Op(K") with

D

K"(b,w;b',w' ) = Je r db"dw"K(b,w;b",w")K'(b",w";bl,w' ). (8) JT*'Rn

Now, let k, 1 be some integers;

IKI/(b, w; b', w') I < Jr r db" dJ.J' "'k ((w)a W' - bl) "'z ((w) -alw" - wi) JT*[tn X"'k ((w,)alb" - b'l) "'z ((w')-<>Iw" - w'l)

< r db"",d(w)alb"-bl)",d(w,)alb"-b'l) J[tn X r dw"",z ((w)-alw" - wi) "'z ((w')-alw" - w'I). J[tn

We study each part separately. First, one has that

On the other hand, the integral in b" gives

! dw""'k ((w)alb" - bl) "'k ((w,)alb" - b'l)

:::; O(1)(W)-na"'k ((WI) I~-:;~) ,,), hence one obtains the following estimate:

"( , ')1 () ( Iw-w'l ) ( Ib'-bl ) IK b, w; b ,w :::; a 1 "'z (w,)a + (w)a "'k (w,)-a + (w)-a . (9)

Let us remark first that if one of the w, w' variables has its norm less than or equal to 1, then the argument is concluded. It remains to show that equation (9) implies the desired localization on high frequencies. This shall be done in the following lemma, which ends the proof of the proposition. 0

Lemma 3.2. Let K : (T*lRn)2 ------> lR be such that:

K(b W' b' w') _ ( Iw - w'l Ib - b'l ) , " - 'P Iwla + Iw'la' Iwl-a + Iw'l-a

where 'P is rapidly decreasing. Then, K verifies

K(b,b';w,w') = 'lj; (lw'lalb - b'l, Iw'l-alw - w'l)

with 'lj; rapidly decreasing.

Proof. We first study the Iwl-alw - w'llocalization. The estimate

Iw-w'l 1 ""'-:'---:--'-:-- = Iwl-alw - w'l > 3a (lwl-a lw _ w'l)l-a Iwl a + Iw'la 1 + Iw'la -

Iwla

holds, hence

a ( Iw - w'l ) l-=a W -a W - w' < 3 I-a I I I I - Iwla + Iw'la

which implies the Iwl-alw - w'llocalization (and also the Iw'l-alw - w'llocalization). Now, let us look at the Iwlalb-b'llocalization. In the same manner as above, one has that

Ib - b'l -lw'lalb _ b'l 1 ~ 3-a lw'l a lb _ b'I(lwlalw _ w'I)-a, Iwl-a + Iw'l-a - 1 + Iw'la

Iwla

hence

Iw'lalb - b'l :::; 3a (lwl a lw _ w'l)a Ib - b'llw'l-a Iwl-a

and one obtains the result by the Iwl-alw - w'llocalization. o

This algebra is a basic component in the sense that, as the following result shows, the typical elements of ops(a) (T*JR.n) are the cross kernel operators W~a) M~a).

Proposition 3.4. Let g, h be in S(JR.n). Then,

weal M(a) = Op[p(a») E OPS(a)(T*JR.n ) 9 h g,h 0

Proof. From Fubini's theorem, one immediately obtains that W~a) M~a) = Op [(gb,w, hb, ,w' )). Let us now look at the localization of this kernel. First, one treats the Iw - w'llocalization. One has the following Fourier representation:

(a)(b b' ') Pg,h ,Wj ,w

= «W)~~~~r¥ In~n d~ei~(b-b')g«w)-a(~ - w)) h«w,)-a(~ - w')),

hence

Ip~~~ (b, Wj b', w') I :::; «W)~~~~t¥ JlRn d~ Ig«w)-a(~ - w))llh «w')-a(~ - w'))I.

Now, let k E IN be given,

Ip~~~ (b, w; b' , Wi) I

< 0(1) (w) (Wi)) ~ n2" Lnd~~k( (w) ~a (~ - w)) ~k( (Wi) ~a(~ - w'))

< 0(1) C::J n2" W[~k' ~kl (Wl)a(b - b' ), C::)) a)

< 0(1) ( <::)) n2" ~k (w)~ ~ 1~,)a ) .

On the other hand, for the Ib - b'l localization, the same computation gives the following estimate: Vk E IN,

Ca) ( . I ') _ () (Wi)) n2" ( b - b' )

Pg.h b, W, b ,w - 0 1 (w) ~k <w)~a + (w')~a .

Finally, the result follows from Lemmas 3.1 and 3.2. o

We also give another result, which follows obviously from the proposition above, but has a great interest. It is the equivalent of the reproducing kernel equation for the wavelet transform.

Proposition 3.5. Let g, h be in S (JRn) and let N be an integer. We assume that g(O) i- O. Then,

3KN E OPSSa)(T*JRn ); W~a) = KNw~a) + W~a) RN

where RN is a smoothing Fourier multiplier belonging to OP S:61~a) (JRn).

Proof. It suffices to take KN = W~a) M~';], where hN is a reconstruction function of g at order N. 0

Now, we define the other components of the graded algebra. First, we introduce weighted functions. Let us consider a function r : T*JRn -----+ C, for which we suppose that it verifies the following growth condition:

(10)

Now, we define the set of kernels on the phase space S~a) (T*JRn) by:

s E s~a) (T*JRn ) <¢=}, Is(b, w; b' , Wi) I ::; r(b' , w')F(b, w; b' , Wi)

where F E S6a) (T*JRn), F 2': O. In the case where r(b,w) = (w)m (which verifies

the condition (10) by Lemma 3.1), one notes this set S;;:) (T*JRn). Condition (10) immediately implies the following lemma, concerning dependence on the variables.


Lemma 3.3. We assume that the function r verifies the condition (10). If s E S~a) (T*IRn), then there exists F E S~a) (T*IRn), F 2 0 such that:

Is(b,w;b',w')I:S r(b,w)F(b,w;b',w').

Now, for s E S$a) (T*IRn), we define the operator

Op(s)· u(b,w) = r db'dw's(b,w;b',w')u(b',w') JT*&.n

and we denote the set {Op(s), s E S$a) (T*IRn )} by OPS,(a) (T*lRn). Then, the

following composition result holds.

Proposition 3.6. Suppose that the functions r1, r2 both verify condition (10). Then,

OPs(a) (T*IRn) 0 ops(a) (T*IRn) C ops(a) (T*IRn). ,} T2 1'1'2

Proof. Let Sl E S$~) (T*IRn) and S2 E S$~) (T*IRn). Then, Op(sd OOp(S2) Op(S3) with

s3(b,w;b l ,W") = j db'dw'Sl(b,w;b',w')S2(b',w';b l ,w"). T*IRn

One must prove that S:1 E S$~)2 (T*IRn). By definition, there exist F1, F2 E

S~a) (T*IRn) such

IS3(b,w;bl,w")I:S rl(b,w) r db'dw'r2(b',w')F1 (b,w;b',w')F2(b',w';b l ,w"). JT*IRn

By Lemma 3.3, one may write

IS:1(b, w; b", w") I :S (r1 r2) (b, w )F3 (b, w; b", w")

where F.1 is defined by Op(F3) = Op(Fd 0 Op(F~), with F1, F~ E S~al (T*IRn). Proposition 3.3 completes the proof. D

Remark: Lemma 3.3 above means that the elements of OP S~a) (T*IRn) are products of multipliers on the phase space which have some growth by elements of OPS~a)(T*IRn). (We shall see the importance of this remark in the next section, dealing with decomposition of pseudo-differential operators through weal transformations. )

We finish the section by defining the graded algebra we promised at the beginning. Let us set

OPs(al(T*IRn) = U OPSr::l(T*IRn). mEIR!.

Proposition 3.7. OPs(al(T*IRn) is a graded operator algebra.

3.2. Representation of PD~ through W(cx) transformation

Let T E OPS~8(lRn) be a pseudo-differential operator with symbol O"T in a Hormander class (see [Horl, Hor2] for the definition of these classes). Then, setting

Tp = W~a)T M~~, where P E IN, and g, hp verify Theorem 2.4, we get

T = M~a)Tpw~a) + Rp

where Rp is a smoothing operator of order P(l-O'). We shall study in this section the operator T. For that aim, we introduce, for fL, v two multi-indices and for a real m, the quantity

m(fL, v) = m - (lfLlb - 0') + Ivl(O' - 5)).

Theorem 3.8. We assume that 0 :S 5 < 0' < 'Y :S 1. Then, for all N E IN, we have the decomposition

N

T= L L K}:/) k=O IILI+lvl=k

where K~rv,;k) E OPS~a?ILv/T*lRn).

Proof. It follows from a direct computation that T = Op [(gb.w, Thb',w')] with

Thb'.w'(X) = -( l)n r d~(w') - n 2" eif,(x-b')O"T(X, Oh ((w') -a(~ - w')) . 27r i)Rn

Now, let us make a Taylor expansion of the symbol O"T around (b', w').

where

N-l

L L CIL.v8~8tO"],(b',w')(x - b')IL(~ - w')V k=O IILI+lvl=k

+ L CIL.Vp(x, b';~, w')(x - b')IL(~ - w')V IILI+lvl=N

p(x, b';~, w') = t dt(l - t)N-18~8t O"T (b' + t(x - b'), w' + t(~ - w')) . in First, we treat each term of the main part in the expansion:

C IL.v8!:; at O"T (b', w') r d~ (w') - "2° eil;(J;-b') (x - b')IL (~ - w'th ( (w') -a (~ - w')) . i)Rn Let us set h2(W) = (w)Vh(w). One gets

C IL.v8!:;8t r d~ (w') - "2" +alvl eif,(x-b') (x - b')IL h; ( (w') -a (~ - w')) . JIRf!

Now, one sets h(w) = (-i8!:;) h2(W), which gives

CIL .v(w,)cx(l v l-ilLl)8!:;8tO"T(b',w') r d~(w')- "2" eif,(x-n'h ((W')-a(~ - W')) . JIRH


Then, the matrix elements are written as

C (w')"(lvHILlJaILEY O",,,(b' w') (gb h b, ,) J-L,V x ~ l' ,W' ,W

and we obtain the desired result by estimates on the symbol O"T. Now, one treats the remainder. The expression one needs to estimate is the

following:

L cIL,v 1 dt,(W,)-n2"'ei~(x-b')(x-b'Y'(t,-w'tp(x,b';t"w') IILI+lvl=N IRn

xh ((w')-"(t, - w')).

This estimate is a sum of type L: RIL,vhb' ,w' where R IL .v is a pseudo-differential operator. We will obtain the conclusion if we prove that, for each term of the sum, there exists some real L such that (w') -L RIL,vhb' ,w' and its Fourier transform are rapidly decreasing, in particular if the family

is bounded in S(Il~n). Setting h(w) = (w)Vh(w), one gets the expression

L cIL,v 1 dt,(W,)-n2a+"lvlei~(x-b')(x - b')ILp(x,b';t"w') IILI+lvl=N IRn

which gives, by applying the inverse transformations,

IILI+lvl=N

x r dt,eix(w')-a(~_W')xIL p ((w') -"x + b', b'; t" w') h ((w') -"(t, - w')) . JJF.n

Changing the integration variable t, by (w')-"(t, - w'), one obtains

IILI+lvl=N

x r dt,eiX~xILh(t,)p((w')-"x+b',b';(w')"t,+w',w'). JlRn

In order to simplify the expressions, we set

fb',w'(X, t,)

(w)"(IVHIL1) p ((w') -"x + b', b'; (w')"t, + w', w')

(w)"(lv l- 1IL1) 11 dt(l - t)N- 1 a:::arO"T (b' + t(w')-"x,w' + t(w')"t,) .

Now, let us consider an integral with parameters

Kb"w'(x) = J dt:,eixC,kb"w'(x,t:,),

Then, the following lemma holds

Lemma 3.4. We assume that (W')~Lkb"w' is rapidly decreasing with respect to t:" and all its partial derivatives in t:, too, uniformly in x, b' , Wi, Then,

is a bounded family in S(JRn).

Proof. Let (S'\)'\EA be a family in S(JRn), and set S.\ = :F~1 (s.\). By the continuity of the inverse Fourier transform, if (S.\hEA is bounded in S(JRn), then (S.\h.EA too, and in particular, in the space of rapidly decreasing functions. Let us set Sb' ,w'.>Jt:,) = (Wi) ~L kb' .w' (A, O. Consequently, Sb'.w' ,.\ is rapidly decreasing with respect to x, uniformly in b' , Wi, A. One obtains the conclusion by specializing A=X. D

Let us come back to the proof of the theorem. First, one remarks that Lemma 3.4 cannot be applied to the derivatives of K b, ,w" Indeed, the derivatives in x does not allow the specialization of the parameter A. We need some additional work before. One has that

8~Kb"w'(X) = L C~,f31 dt:,eixf,e~E8~kb"w'(X,t:,). O:O:IEI:O:lf3l ]Rn

Then, it is clear that the right assumption on kb, ,w' is: \:1(3, E E INn, \:Ik E IN, 3Ck,b',w';

\:Ix,t:"b',W' , 18~8;;kb"w'(x,t:,)I::; Ck,b',w,(w,)LKk(t:,).

Since h is in the Schwartz space, it is sufficient that fb' ,w' verifies the following assumption:

3L E JR; \:1(3, E E INn, 3L' E JR, 3C6 .£ E JR;

\:Ix, t:" b' , Wi, 18~ 8Ub' ,w' (x, t:,) I ::; C!3.EKL' (t:,) (Wi) L.

Then, let (3, E E INn be given. With the estimates on the symbol (JT, one gets that

8~8Ub"w'(X, t:,) (w')"'(IEI+lvl~lf3l~lfLl) 11 dt(l - t)N~ltlf3I+IEI

x8;::+ f3 8'/.+E(JT (b' + t(W')~"'X,W' + t(w')"'t:,) ,

18~8Ub"w'(x,t:,)1 < C(W')"'(IEI+lvHf3l~lfLl) 11 dt(l- t)N~l

XK~rn+'"Y(lvl+IEI)~b(lfLl+If3I) (Wi + t(W')"'t:,).

We suppose that I~I, Iw'l 2: 1 (low frequencies do not modify the result). Let us set l = m - ')'(Ivl + lEI) + 8(llil + 1,61) and distinguish two cases. ~ l > 0: then, writing that

K,~l (w' + (w')Ot~) < 0(1) (lw'l + Iw'I"~/

one obtains that

< O(l)lw'l"l (lw'll~" + ~)l < O(l)lw'l"llw'l(l~a)ll~ll

IB~ BUbl.wl (x,~) I < 0(1 )K,l (~) (w) rn(IL,v)~(kICr~a)+IJJI(,,~8))

< O(l)K,l(~)(w)rn(lL.v)

with the assumptions on (x, ,)" 6, and the argument is concluded. ~ l < 0: this case is more difficult to treat. One has to show that there exists some real L' such that

Iw'l~ll~IL'lK,l (w' + tlw'la~) is uniformly bounded with respect to t, w', ~. This quantity equals

1

(lw'l~ll~lL' + I~IL' lewl + tlw'I"~l~I)~l' (11)

One shall proceed by reductio ad absurdum. Let us suppose that there exist sequences (tp), (w~), (~p) such that (11) tends to +00 as p goes to 00. Then, one has that

and that

Since the sequences (tp) and (ewl ) are bounded, one can assume that they converge p

respectively to to E [0,1] and e E §n~l. It follows that

lim l~pllw~la~l 2: tal p->oo

(here, we adopt the convention 1/0 = (0). Choosing L' > l2a in

l~pILllw~l~l = (l~pllw~l~l)LI Iw~l(l~a)L/~l

one obtains a contradiction. In the same manner as in the case l > 0, one gets the desired estimate. The theorem is proved. 0

Now, we give a result which makes precise the form of the main part and the order of the remainder terms. Its demonstration obviously follows from the proof of 3.8.

Theorem 3.9. Under the same assumption as in Theorem 3.8, T expands as follows:

T= L K1~J 11L1+lvl<N, rn(lL,v»rn~N minCr~a,a~8)

with /CitJ E OPS:::!/A,//) (T*ffi.n) , Furthermore, the kernels are written as

KL~)(b, w, b',w') = kf!J(b', w') (gb,w" hb"W)

where Fh = ( ~iof) (C' Fh).

Then, one obtains a decomposition result of the symbol aT through the weal at the first order, by using Proposition 3.5. In addition, we can remark that we need not a precise reconstruction function (Theorem 2.3 is then sufficient). For higher orders, the result does not hold anymore. Indeed, some kernels appear which, though they are localised, are not absorbed in the analysis with respect to g.

3.3. Multipliers on phase space

We study here the most simple operators which are generated by the weal transforms, that is the multipliers through these transformations. Let m : T*ffi.n ----+ C be a function defined on T*ffi.n. For Q; E (0,1), and g, h in the Schwartz class verifying the hypothesis of Theorem 2.3, we set

Open) (m) = M(n) M(m)WCa) g,h h 9 .

These operators have a great importance in our first-order differential calculus. Indeed, the result of the last section can be rewritten (at the first order) as follows:

Proposition 3.10. We adopt the notation of Section 3.2. Then, T E OPS~8(lRn) being given and assuming that l' > Q; > <5, we can write that:

T = op~~2 (aT) + M~a) /Crest w~a)

where /Crest E 0 P S:::~rninCr-a,a-8,1-Cl')' Remark: We shall see in the section dealing with function spaces that the rest is really smoothing, since the multiplication by a term such as (w) -k with k 2: 0 sends the Sobolev spaces Hrn(ffi.n) to Hm+k(ffi.n) after transformation by w(a).

Our aim here is to obtain a result, which would allow us to write that, m, m' being given functions defined on T*ffi.n, we have the identity

O Ca)( ) 0 (a)(,) 0 (a)( ') T Pg,h m 0 Pg,h m = Pg,h m· m + rest

where Trest is smoothing in some sense. We have already seen in Proposition 3.10 that if m' is in some Hormander class of symbol, this holds, using the first order composition theorem of pseudo-differential operators. Now, we give weaker conditions on the multiplier m' for this identity to hold.

Theorem 3.11. Let Q; E (0,1), and g, h be in the Schwartz class. We assume that the multiplier m : T*ffi.n -----+ C verifies: if 1,61 + bl :s; 1,

lof,;J m (b + b', w + w') I :s; reb, w) ((w)" Ibl + (w) -alw'l) K (12)

where K E IR is given and r(b,w) = o(m(b,w)) as Ibl + Iwl goes to 00 and verifies condition (10). Then, the following identity holds:

W~<»M~<» M(m) = M(m)W~<»M~<» + R

with R E OPS~<» (T*lRn).

Proof. The Taylor formula, applied to m, gives:

m(b',w') = m(b,w) + 101 dtDm (b + t(b' - b),w + t(w' - w))· (b' - b,w' - w).

Let us note the remainder Q. By hypothesis (12), one gets the estimate for Q,

IQ(b,wjb',w')1 ~ r(b,w) ((w)<>ib' - bl + (w)-<>Iw' - wl)K+1 101 dt t K .

Consequently, one obtains that

[W~<»M~<»,M(m)] = R

where the kernel R of R verifies

IR(b,wjb',w')1 ~ O(I)r(b,w) ((w)<>lb - b'l + (w)-<>Iw' - wl)K+1 (gb,w,hb',w')'

This completes the proof. D

Theorem 3.12. Let us suppose that m, m' satisfy condition (10) and that m' verifies (12). Let us note R E OPS~o<>(lRn) the pseudo-differential operator of inversion Theorem 2.3. Then, for all N' E IN, there exists r verifying condition (10) with the additional property that r(b,w) = o(mm'(b,w)) as Ibl + Iwl goes to 00 and there

exists RN E OPS{:?"-'mm' (T*lRn) U OPS~<» (T*lRn) such that:

Op(<»(m) 0 Op(<»(m') = Op(<»(mm') + M(<»RNW(<» + Op(mm')RN . g,h g,h g,h h 9

The proof is based on the following lemma

Lemma 3.5. We take the notation of Theorem 3.12. Then, for all N E IN, there exists fiN E OPS~~l' such that

W~<» = RN W~<» + W~<» RN.

Proof. Let us set R = W~<» RM~<». Through the pseudo-differential operator R belonging to a limit class with respect to Theorem 3.8, one may show, following the proof of 3.8, that R E OPS~~1 (T*lRn) (details are left to the reader). Then, a recurrence using inversion Theorem 2.3 ends the proof. D

Proof of Theorem 3.12. It follows immediately from Theorem 3.11 and Lemma 3.5. D

3.4. w(a) transformations and diffeomorphisms

In this section, we shall study the action of diffeomorphisms on the coefficient space. For that aim, we want to find an operator 1l1[<p] acting on functions of the phase space such that, up to a smoothing operator:

w~a)[<p] = 1l1[<p]WJa ) (13)

where 9 E S(lRn) is an analysing function, [<p] is defined by:

[<p] : s(y) 1-* J (<p~l(y)) S (<p-l(y)) .

Here, J denotes the Jacobian of <po The adjoint of [<p] is *[<p] : rex) 1-* r (<p(x)) , in the following sense: for all r, S : lRn ---+ IRn,

r dx([<p]s) (x)r(x) = r dxs(x) (<1>* [<p]r) (x). J~n J~n

Here, we need to change slightly the definition of the weal transformation, in order to compensate the loss of covariance in the transformation. We introduce in the definition a matrix parameter, which gives a certain latitude in the relative direction between the frequency modulation and dilation. The analysing family then becomes, for 9 E S(lRn), (b,w) E T*lRn, C E GLn(IR),

gb,w,C = (C) n2" eiw(x-b)g ((c)a(x - b)).

The reader may be easily convinced that it does not change any of the results we give before, but offers us the possibility to do linear deformations on the tangent space.

Let us now go back to the diffeomorphism action on a w(a) transform. In order to find 1l1[<p], we make in the following theorem an asymptotic expansion of the matrix elements K(a)[g, <p] of the operator WJa)[<p]W~a), for 9 E S(lRn).

We begin by setting some notation:

• 8x[<p] : s(y) 1-* J(<p \x»s (8<p;ly) (it describes the action of <p on the tangent space).

• For N 1 , N2, N3 integers, we define N(Nl' N2, N 3) being the following set:

{(k1 ,k2,k3) E IN3,O:::; kl < N 1 ,2:::; k2 < N 2,O:::; k3 < N3,kl + k3 ~ I}.

• For a function F defined on (T*lRn)2, we will say that F satisfies the (*) condition if:

(*) web, Wj b', w')/ :::; p ((w)a (b - <pCb')) , (w) -a (w - 8<p<P(bl)W'))

where the function p is rapidly decreasing.

Then, the result is the following:

Theorem 3.13. We assume that 0' > 1/2. Then, the kernel K(n)[g, ip] is written as follows: for all integers N 1 , N2! N 3 ,

K6") [g, ip] + ~ K(n) [g,ip](C'w,)k3(1-k 2 n)-k, a

L k,.k2.k3 (k,.k2,k3)EN(N"N2,N3)

+R(a) (C'w,)N3(I-N2a)-N,Ct N,. N2,N3

where the kernels K(a) and the remainder R(a) satisfy the (*) condition. Furthermore,

Remark: If we only expand at the first order, we find exactly the adjoint of the searched operator w[ip]:

w*[ip]: r(b,w,C) f--+ Jtb)r (ip-l(b),toip""(b)W,CtOip~(lb))'

Remark that the fact that the functions really depend on the matrix parameter is not important for the estimates because of the boundedness of Oip. In addition, if ip is affine, the relation (13) exactly holds (i.e., without a smoothing term).

Proof. The matrix elements of WSa)<l>[ip]M~a)is

Let us set

One has that

K(a) [g, ip](b, w, C; b', w', C') = (gb.w,C, <l>[ip]gb' ,w' ,c') .

g ((c)a (ip-l (ip(b) + (c)-ax) - b)) J (ip-l (ip(b) + (c)-ax))

X eiw ( tp-1 (tp(b)+(C) -u x ) -b-8",,-1 (",,(b))(C) -ax) . (14)

The first order in the asymptotic expansion is given by the limit, as Iwl tends to infinity, of the quantity (14). Let us remark that, since 0' > 1/2,

(15)

in the COO-topology. Now, one treats the rest of the expansion. The non-oscillating term expands

without any difficulties as follows:

(o<l>tp(b)[ip]g) (x) + LAk(b)g(x)(C)-ka (16) k>1

where Adb) are some differential operators (uniformly bounded in b). One writes for the oscillating term

eiw ('P- ' ('P(b)+(C) -ax )-b-arp~/b) (C) -ax) . '" (a -1)13 13(C)-I13l o 1W L..- 1 [J 1 < N rp x ~ ( b ) e- cp(h) +rNX"W

where rN(b,x,w) = 0(1)lxI N+1(C)-(N+l)a. Moreover,

iw"'. .(a· -1)13 xi'(C)-li31c. e L..-I·JI:S·"" rp "'(0)

L ~~ (w, L (OCP_1):(b)X 8(C)-Lr3 la)k kSN' 1r3ISN

+rN.N'(b,x,w).

One gets for the oscillating part

·k L ~! L (ew, (Ocp-1)6 x 13 )' (C)hl(l- Lr3 la) + rN.N,(b, x, w). kSN' 1!3ISN.I~rlSk

As a conclusion, this term is equal to

where

Fk,k' (b, x,w)(C)k'(l-ka) + rN,N,(b, x, w)

Fk.k' (b, x, w) = L C/,.6 (ew , (Ocp-1 ):(b) x (3 )'

/ .. 6

(17)

is polynomial growing in x and uniformly bounded in b, w, since cp is bounded with all its derivatives. Furthermore, the remainder satisfies the estimate

IT N.N' (b, x, w) I :::; 0(1) (C) (N' +l)(l-(N +1)0') Ixl(N' +l)(N+l).

Putting (16) and (17) together, one gets that

1>b,w,c[cp]g

with

Now, writing that

where

01> 'P(b) [cp]g + L Fk2 ,k3 (b, x, W)A k1 (b )g( C) k3 (1-k2a)-k, 0:

N +RN" N2 ,N3 (b, x, w)

one gets the announced expansion. The estimate on the kernels K(a) follows from the boundedness in S(JRn) of the family

{ D(C)-aE; -1 T;(b)Fk2,k3(b,W)Akl(b)9}' "''P(b)w

Actually, it remains to compute the first-order term in the expansion. By (15), one has

K6a ) (b, w; b' , w') = \gb,w,C;b' ,w' ,C', 8<P",(b') [cp]g)

/ gb,w,C, (8<P",(b') [CP]gL(b') t8,n- 1 w' C't8,n ,) \ .,.. , .,.. 'PCb')' ""'P(b )

which gives the conclusion. o

4. Application to construction of directional regularity Banach spaces

4.1. The Hm-theory

We begin by giving the characterization of Hilbert-Sobolev spaces through the w(a) transformation coefficients. Those spaces need another admissibility condition on the analyzing function.

Definition 4.1. We will say that a function 9 E S(JR.n) satisfies the (A2) condition if and only if

::Ic > 0; V.t' E JR.n , 0< c- 1 < m(al(c) < c < 00. - g,g <" - (A2)

Remark: By Theorem 2.3, the boundedness is not the problem. But, it may happen that the symbol vanishes somewhere. Basically, the best examples of functions satisfying (A2) is the Gaussians.

Before studying Sobolev spaces, we define the w(al transform of functions in L2(JR.n).

Proposition 4.2. Let g, s E S(JR.n). Then,

Ilw~al slli2(JRn l = / s, mgCagls) . \ ' £2(JRn)

Proof. It immediately follows from the fact that the adjoint of W~al is M~a). 0

By the way, W~a) is continuous from S(JR.n) to L2(JR.n ) for the L2 norm. From

the density of the Schwartz space in L2(JR.n), it follows that we can extend W~al to

L2 (JR.n) by continuity. Moreover, if 9 satisfies (A2), the fact that the symbol m~~J is bounded below gives us the following characterization result:

Theorem 4.3. Let 9 E S(JR.n) verify (A2). Then,

s E L2(JR.n ) {==;> w~a) s E L2(T*JR.n ).

The same manner of characterization holds for Sobolev spaces.

Theorem 4.4. Let 9 E S(JRn) verify (A2). Then,

s E Hm(JRn ) ~ w~a)s E L2(T*JRn ,dbdw(w?Tn) = L~(T*JRn).

Proof. In the same way as in the proof of Proposition 4.2, one has

\/g,s E S(JRn ), [[w~a)S[[£2(T*IRn dndw(w)rn) = /s,m~ads) , \, L2(IRn,11;1 2rn dl;)

where

With the same arguments as in Theorem 2.3, one may prove that the asymptotic expansion is modified by the addition of a term as Cm[.;[l-a and then that this multiplier is bounded. It follows from a density argument that, if s E Hm(JRn), then W~a) s E L~(T*JRn).

Let us show the converse assertion. It leads us to show that the admissibility condition is not modified by the addition of (w)2rn. If m ;::: 0, then m~~d (.;) ;::: m~~d(.;), which gives the result, since g verifies (A2). Now, assume that m < 0

and proceed by reductio ad absurdum. Suppose that m~~d is not bounded by

below. In that case, there would exist a sequence (';j )jEIN such that m~~d (';j) ---+ O.

In addition, the expansion of m~~d shows that (';j) is bounded. Then, up to a

sequence, it converges to .; E JRn verifying, by continuity of m~~d, m~~d (.;) = 0,

which implies that, for every w, g ((w) -a (.; ~ w)) = 0, hence m~~d (.;) = 0 and is a contradiction. 0

4.2. Admissible Banach spaces

The HTn-theory being built, we are now able to define anisotropic Banach spaces with the Weal transformations by pullback technics. Indeed, these function spaces shall be included in the Sobolev spaces scale. For that aim, we consider a Banach space B(T*JRn) of locally integrable functions on the phase space T*JRn. Moreover, we suppose that:

S(T*JRn) c B(T*JRn) C S'(T*JRn).

Let us now introduce the following definition.

Definition 4.5. Let B(T*JRn) be a Banach space defined as above, and a E (0,1). B(T*JRn) is said to be a-admissible if and only if it satisfies the following two conditions:

3m, m' E JR; L~(T*JRn) C B(T*JRn ) C L~,(T*JRn). (Bl)

\/K E Sga)(T*JRn),Op[K]: B(T*JRn ) ----+ B(T*JRn ) is continuous. (B2)

Now, in order to define anisotropic Banach spaces, we need to study the dependence with respect to the analyzing function.


Proposition 4.6. Let et E (0,1). Consider an a-admissible Banach space B(T*IRn), 9 E S(IRn) an admissible analyzing function and s E S' (IRn) a tempered distribution such that W~a) s E B(T*IRn). Then, for all h E S(IRn), W~Q) s E B(T*IRn).

Proof. Since 9 is admissible, by Proposition 3.5, one gets for all integer N the existence of KN E S~a)(T*IRn), and RN E S:(\l-a) (IRn) verifying that

Wha) = Op[KNlw~a) + W~a) R N .

Then, by condition (B2), one has that Op[KN1W~a)s E B(T*IRn). On the other

hand, since W~a) s E B(T*IRn) c L~,(IRn), one knows by condition (B1) that the distribution s is in the Sobolev space Hrn' (IRn). As a consequence, for N large enough, RNs E Hrn(IRn), and then W~a) s E L~(IRn) c B(T*IRn), which completes the proof. 0

This result proves that the following definition is well posed, that is does not depend on the analyzing function.

Definition 4.7. Let et E (0,1). We consider an et-admissible Banach space and 9 an analyzing function in the Schwartz space. We define the Banach space B(IRn) of distributions on IRn as follows: for a tempered distribution s,

s E B(IRn) ~ w~a) s E B(T*IRn).

We define the norm in B (IRn) by:

IISIIB(lRn) = Ilw~a)sIIB(T*lRn).

Remark Proposition 4.6 and Theorem 4.4 show that if we take another admissible analyzing function h in the definition of B (IRn), then the norm induced by h is equivalent.

The next step in building these Banach spaces is to give more developed conditions on B (T*IRn), for this space to be admissible. There are four conditions:

S (T*IRn) C B (T*IRn) c S' (T*IRn) .

IllsIIIB(T*lRn) = Il s IIB('1'*lRn). Condition (C2) means that B (T*IRn) is a Banach lattice.

We can estimate for Sb',w' (b, w) = s (b + (w)-ab', w + waaw') , II Sb' ,w' II B('1'*lRn ) s: p(b', w') II s II B(T*lRn) where p is at most polynomial growing.

11 E Loo (T*IRn) ,s E B (T*IRn) ===? I1S E B (T*IRn) , IIM[1111IB(T*lRn)--->B(T*lRn) = 1II1IIu=(T*lRn).

(C1)

(C2)

(C3)

(C4)

This condition says that the multiplier algebra of B (T*IRn) contains the bounded functions.

These conditions are sufficient to imply the invariance condition (B2), as it is shown in the next lemma.


Lemma 4.1. Suppose that B (T*JRn ) verify condition (C1)-(C4). Then, condition (B2) holds for B (T*JRn).

Proof. Let s E B (T*JRn), and K = Op(K) E S6a) (T*JRn). Then, by a change of variables, one may write

Ks = 1r*~n db'dw'(Ks) (b + (w)-ab',w + (w)c>w').

Now, passing to the absolute value, then to the norm on both sides, and using hypothesis (C3), one obtains

IIKsIIB(1'*~n) ~ IlsIIB(1'*~n) r db'dw'K(b+(w)-CXb',w+(w)CXw')p(b',w'). JT*~n

Since K E S6C» (T*JRn), the last integral is finite, and the lemma is proved 0

The end of the section is devoted to a short study of the multiplier operators in admissible Banach spaces. Let B (T*JRn) verify conditions (C1) (C4), and let r be a weighted function. We assume that r satisfies both (10) and (12). Then, we can define the function space

rB (T*JRn) = {rs, s E B (T*JRn)}

on which we put the norm IlsllrB = Ils/rIIB. Proposition 4.8. With the assumptions above, r B (T*JRn) is a-admissible.

Proof. Let K E OPS6cx) (T*JRn) and let s E B (T*lRn). Then,

Krs(b, w) V' Krests(b, w) ----'---'----'- = ''vS + ------,------'-----'--r(b,w) r(b,w)

with Krest E OPS~~') (T*JRn), and with r' = o(r), by Theorem 3.11. It remains to show that the remainder term is in B (T*JRn), which is obvious by the estimate in OPS,(~) (T*JRn). 0

As a consequence, we can define the space Br(JRn) by pullback from r B (T*JRn). Then, we obtain the following continuity result.

Proposition 4.9. Let r verify (lO), (12), and let B (T*JRn) be an admissible space. Then, Op~a~(r): B(JRn) --+ Br(JRn) is continuous.

Proof. Immediate. o We finish by giving the analogue of Theorem 3.12 for admissible Banach

spaces.

Theorem 4.10. Let m, m' verify (10), (12), and let B (T*JRn) be a-admissible. Then, there exists s = o(mm') and R continuous from B(JRn) to Bs(IR,n) such that:

Op(CX)(m) 0 Op(a) (m') = Op(a) (mm') + R. g.h g,h g.h

Proof. Adopting the notation of Theorem 3.12, it suffices to take 8 equal to max(r,lwl",-lmm'). 0

4.3. An example We finish this paper with the study of an example based on the Loo-spaces. We define the following function space on T*ffi2 (for simplicity, we only consider the case n = 2): for m 2: 1,

r E Bm (T*ffi2)' {==} Ir(b,w)1 :S O(I)'Pm(w) uniformly in b

where 'Pm(w) = (1 + IWll2m + IW212)1/2

and we put on Bm (T*ffi2) the norm 11811m = 118/'Pmllv=. Then, we have the following result

Theorem 4.11. Brn (T*ffi2) is an a-admissible Banach space for m < I/a.

Proof. First, Brn (T*ffi2) obviously verifies condition (BI). It remains to verify condition (B2) for Bm(T*ffi2). Then, let r E Brn(T*ffi2) and K = OP(K) E

OPS6"')(T*IR?) be given.

(Kr) (b,w) = r db'dw'K(b,w: b',w')r(b',w'). JT*1R 2

By definition of OPS6"') (T*ffi2) , one has that, for all k, l E IN,

(Kr) (b,w):S r db'dw'Kk ((w)"'(b - b')) Kl ((w)-"'(w - w')) r(b',w'). JT*1R 2

Let us treat the integral in the b' variable.

r db'Kk ((w)"'(b - b')) = r db'Kk ((w)"'b') = (W)-2", r Kk(b') = O(I)(w)-2n. JIR 2 JIR 2 JIR 2

It remains to estimate the w' integral. First, one remarks that one can take Iwl 2: 1 without loss of generality. Then, the inequality becomes

(Kr) (b,w):s O(I)lwl-2", r dw''Pm(W')Kl (Iwl-"'(w - w')). JIR 2

Let us set for simplicity

frn(w) = r dw''Pm(W')Kl (Iwl-n(w - w')). JIR 2

Here, we split ffi2 into two parts, Os = {(w)-"'Iw - w'l :S (w)<} and OM = (OMt, E being a given positive real. Then, one has that

f:n.s r dW'Kl (Iwl-"'(w - w')) 'Pm(w') In's

< KlI (w) r dw" Kl2 (w") 'Pm (w + Iwl"'w") J1wll l2:lwl'

for llarge enough. Then, one can see that f:n.8 is rapidly decreasing for all E > O. Now, we estimate the second part of fm(w). One writes that

f~Ll\J(w) Iwl 2a r dw" fq(W")'Pm (w + Iwlaw") Jlw"I~lwl'

< Iwl 2a (r dW"fq(W")) ( sup 'Pm (w + IWlaw,,)) J~2 Iw"I<::lwl'

( 2m 2 ) 1/2 < O(1)lwI 2a Ilwla+E + IW111 + Ilwla+E + IW211 + 1

Hence, one needs to prove that

f:n.M(W) ::; O(1)lwI 2a 'Pm(w).

Since Iwl :2: 1, one has to treat three cases.

1st case: IW11, IW21 > 1 and IW11 :2: IW21. In that case, one has that Iwl2 < 21wl12 and, for E small enough that a + E < 1. Then, one obtains that

1/2 < O(1)lwI 2a (11wl + IWl112m + Ilwl + IW2112 + 1)

< O(1)lwI2a (IWl1 2m + Iwl2 + IW212 + 1) 1/2 < O(1)lwI 2a 'Pm(w)

and one gets the result for the first case.

2nd case: IW11, IW21 > 1, IW11 < IW21. Here, the estimate becomes

f~LM(W) < O(1)lwI 2a (IW112m + IwI 2m (a+E) + IW212 + 1) 1/2

< O(1)lwI2a 'Pm(w)

since am < 1, and the result follows.

3rd case: one of Iw 11, IW21 is lower than 1. Here, the conclusion is immediate, since a < 1.

The theorem is proved. o

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A. Grossmann, J. Morlet, T. Paul. Integral transforms associated to square integrable representations I: General results. J. Math. Phys., 26:2473-2479, 1985.

A. Grossmann, J. Morlet, T. Paul. Integral transforms associated to square integrable representations II: Examples. Ann. IHP, Phys. Th., 45:293-309, 1986.

M. Holschneider. Analyse d'objets fractals par transformation en ondelettes. These de doctorat, Marseille 1989.

M. Holschneider. Wavelet analysis of partial differential oprators. In Demuth, Schrohe, Schulze, Sjostrand (eds), Schrodinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras. Advances in Partial Differential Equations. Akademie Verlag Berlin, 1996.

M. Holschneider, Ph. Tchamitchian. Pointwise regularity of Riemann's nowhere differentiable function. Inventiones M athematicae, 105: 157-175, 1991.

1. Hormander. The analysis of partial differential operators 1. Springer-Verlag, 1982.

L. Hormander. The analysis of partial differential operators 3. Springer-Verlag, 1985.

S. Jaffard. Estimations holderiennes ponctuelles des fonctions au moyen de leurs coefficients d'ondelettes. C.R. Acad. Sc. Paris, Ser. I, 308, 1989.

S. Mallat. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. and Mach. Int., 11:674-693, 1989.

J. Sjostrand. Singularites analytiques microlocales. Asterisque, 95, 1982.

Bruno N azaret Department of Mathematics University of Cergy-Pontoise 2, Avenue Adolphe Chauvin F-95302 Cergy-Pontoise, France

Matthias Holschneider Department of Mathematics University of Potsdam PF 60 1553 D-14415 Potsdam, Germany


Formes de torsion analytique et fibrations singulieres

Xiaonan Ma

Abstract. In this paper, we generalize to a relative situation a result of Bismut [B2] on the Quillen metric associated to a singular fibration. To establish our results, we make essential use of the immersion theorem of Bismut for analytic torsion forms [B3].

Resume. L'objet de cet article est d'etudier des formes de torsion analytique au voisinage de fibres degenerant en diviseurs a croisements normaux. On calcule la partie finie de la renormalisation a l'aide des formes de torsion analytique sur la normalisee de ces fibres. Ce resultat etend une formule de Bismut [B2] a une situation en famille.

Introduction

Les formes de torsion analytique de Bismut-Kohler sont l'extension de la torsion analytique de Ray-Singer pour une submersion holomorphe. Ces formes sont contenues dans la definition de l'image directe de Gillet et Soule [GS] en geometrie d'Arakelov.

Soit 7r : X --> S un morphisme de varietes compactes complexes qui est une submersion sur Ie complementaire d'une sous varieteI: de codimension 2, ou 7r a des singularites ordinaires. Soit ~ = 7r(I:). Soit ~ un fibre holomorphe sur X. On pose .\(j*O = det(R-7r.;)-I. Dans [B2J, Bismut a calcule Ie comportement de la metrique de Quillen sur .\(j*';) pres de ~. Apres extraction d'une divergence logarithmique, il a decrit la metrique limite en fonction de la met rique de Quillen sur la normalisation des fibres singulieres.

Le but de cet article est de generaliser un result at de Bismut [B2, Theoreme 0.2] a une situation en famille. On Ie donne comme Ie Theoreme 2.1. Pour montrer Ie Theoreme 2.1, on procede comme en [B2]. C'est a dire, on utilise la technique de la deformation du cone normal [BaFMJ, et Ie theoreme d'immersion de [B3, Theoreme 0.1]. Comme tous les calculs sont presque les memes que dans [B2J, on omit toujours les details des calculs. On se concentre souvent de verifier que les cohomologies intermediaires soient localement libres, et les relations des classes de Bott-Chern correpondantes.

396 Xiaonan Ma

Cet article est organise de la maniere suivante. Dans la Section 1, en utilisant Ie result at de [B3] , on calcule Ie comportement des formes de torsion analytique pres des fibres singulieres. Dans la Section 2, on enonce Ie Theoreme 2.1 qui, a l'aide des formes de torsion analytique sur la normalisee de ces fibres, calcule sa renormalisation. Dans la Section 3, on montre Ie Theoreme 2.1 dans Ie cas Z = Zl UL Z2. Dans la Section 4, on montre Ie Theoreme 2.1 dans Ie cas general. Dans la Section 5, on verifie la compatibilite du Theoreme 2.1 et [BerB, Theoreme 3.1].

Remerciements

Je tiens a remercier Ie Professeur J.M. Bismut pour ses encouragements et pour d'utiles discussions. Je voudrais remercier aussi ICTP, Trieste, specialement Professeur M.S. Narasimhan pour hospitalite.

1. La divergence logarithmique des formes de torsion analytique pres des fibres singulieres

Cette section est organisee de la fa<;on suivante. Dans a), on definit une fibration singuliere. Dans b), on donne des hypotheses et des notations. Dans c), on applique Ie Theoreme d'immersion de [B3] a notre situation. Dans d), on calcule la partie divergente des formes de torsion analytique pres des fibres singulieres. Dans e), on indique les formules d'anomalie pour la renormalisation des formes de torsion analytique.

a) Dne fibration singuiiere

Soit V, B deux varietes complexes compactes Kiihleriennes. Soit S une variete complexe de dimension 1. Soit ~ une so us variete complexe de V de co dimension 2, soit ~ un ensemble de points isoles dans S. Soit 7r : V -+ B x S une application holomorphe de fibre compacte Y. Soit P1 : B x S -+ B et P2 : B x S -+ S les projections naturelles. On note 7r1 = P1 07r et 7r2 = P2 07r.

On suppose que 7r : V \ ~ -+ B x S (resp. 7r : ~ -+ B x ~) est une submersion et que si x E ~, il existe un systeme de coordonnees holomorphes (xl, ... ,xn) pres de x, une coordonnee (sl) pres de 7r2(X), un systeme de coordonnees holomorphes (b 1 , ... , bm ) pres de 7r1 (x) tels que localement

~ = (x 1 = 0, x2 = 0), ~ = (Sl = 0),

( 1 n) _ ( 3 m+2) 7r1 X , ••. , x - x , ... , x , (1.1)

( 1 n) _ 1 2 7r2 X , ... , x - x x .

Pour b E B, s E S , so it Xb = 7rl1(b), Zs = 7r2 1(S). Alors 7r1 : V -+ Best une submersion holomorphe de fibre compacte X.

On note aussi Z = 7r21(~), on considere les immersions i : ~ -+ V,js : Zs -+

V, j : Z -+ V. Soit VL; l'eclate de V Ie long de ~, et soit p : VL -+ V la projection canonique. Soit p : Z -+ Z la normalisee de Z. Alors Z est un diviseur lisse de VL;. On pose ~ = p-1(~). Alors p : ~ -+ ~ est un revetement double. Soit K, Ie

Formes de torsion analytique et fibrations singulieres 397

Z2 -fibre sur ~, y;, = det p*0i:' Alors Z, Z, f: et E se fibrent sur B x ~ de fibres

compactes ZB,~ZB' f:~ et ~B. Pour b E B, s E 6., si on note Y(b.s) la normalisee

de y(b.s) , on a ZB = y(b.s)'

b) Hypotheses et notations

Soit ~ un fibre vectoriel holomorphe sur V. Dans la suite, on suppose que les images directes R·7r*~, R·(7rjp)*(jp)*~,

R·(-rri)*i*~, R·(7rip)*(ip)*~ et R·(7ri)*(i*~ @ y;,) sont localement libres sur B x 5, B x 6., B x ~, B x ~ et B x ~. Soit H(Y, ~IY) la cohomologie de ~IY Ie long des fibres Y. Alors ils sont des fibres holomorphes Z-gradues. Plus exactement R·7r*~ = H(Y,~IY)'

Soit gTV une metrique Kiihlerienne sur TV, soient gTX, gTY et g1'Z s les metriques induites par gTV sur TX, TY et TZs sur V, V \ ~ et V \ Z. Soit hE une metrique hermitienne sur ~. On munit la metrique triviale sur y;,. De plus, les varietes Z et E sont lisses, la met rique gTV induite donc des metriques g1'ZB, gT"2:,B sur TZIB x ~,T~/B x~.

Pour bE B x (5\~), soit O(Yb , ~IYb) Ie complexe de Dolbeault des formes Coo sur la fibre Yb a valeur dans ~. On munit O(Y, ~IY) de la met rique hermitienne L2 associee a gTY, hE, [B3, §2.6]. On identifie H(Y, ~IY) aux formes harmoniques dans Ie complexe de Dolbeault relatif O(Y, ~IY)' So it hR7r • E la metrique L2 associee sur W7r*~ sur B x (5\~) [B3, §2.6]' [BerB, §lc)]. De meme, on note h R (7ri).(i·ErZ;nc) et hR ( 7rjp). (jp)' E les metriques L2 sur R· (7ri)* (i* ~ @ y;,) et R· (7rj p)* (j p)* ~ sur B x .6. induites par g1'ZB ,g1'"2:,B et hE,.

Si K est un fibre vectoriel holomorphe avec une met rique gK sur B, si Q est un polynome caracteristique, on designe par Q(K, gK) E pB la forme de ChernWeil associee a la courbure de la connexion holomorphe hermitienne sur K.

Soit pB l'espace des formes reelles Coo sur B qui sont la somme de formes de type (p,p). Soit pB,O l'espace de 0' E pB qui s'ecrivent 0' = 8(3 + 8, ou (3" sont COO sur B.

Comme Best compacte Kiihlerienne, pB.D est un sous espace vectoriel ferme de pB pour la convergence uniforme [B3, §6.5].

Definition 1.1. Soit 11.11 k(k E N) la norme sur pB I pB,D dejinie de la maniere suivante.· pour 0' E pB! on pose

8a inf sup I ~ (0' - (3) (x) I·

.SEPB,a xEB.lalSk uxa (1.2)

Pour s E 5\~, soient w Zs , wV , wZ et w"2:, les (1, I)-formes reelles fermees sur Zs V, Z et ~ associees aux metriques gTZs, gTV, gTZ et gT"2:,.

Soit T(wZs,hE,) E pB, T(wV,hE,) E pBX(S\b.), T(wZ,hE) E pBxb. et T(w"2:"hi ' E0",) E pBxb. les formes de torsion analytique construites dans BismutKohler [BK] sur B, Bx (5\~), Bx 6. et Bx 6. associees a (7rjs, wZs , hE), (7r, wV , hE),

398 Xiaonan Ma

et les formes T(wZs, hE), etc, verifient des equations analogues. Pour s E 5, so it Ts : B ---+ B x 5 l'application de y E B a (y, s) E B x 5.

Alors pour s E 5 \~, on a

(1.4)

c) Theoreme d'immersion pour les formes de torsion analytique

Soit Gr(1T2) Ie graphe de 1T2 dans V x 5. Alors Gr(1T2) est un diviseur lisse dans V x 5. Soit [Gr(1T2)] Ie fibre en droite sur V x 5 associe a Gr(1T2). Soit ll'identification x E V ---+ lex) = (X,1T2(X)) E Gr(1T2)' So it m l'immersion Gr(1T2) ---+ V x 5. Soit n : V x 5 ---+ V, n' : V x 5 ---+ 5 les projections canoniques. On deduit facilement que

[Gr(1T2)]IGr(7r2) = n'*T5IGr(7r 2) , NGr(7r2)/VXS = n'*T5IGr(7r2)' (1.5)

Soit js l'immersion Zs ---+ V. Soit [Zs] Ie fibre en droite sur V associe a ZS' Soient 0"[Gr(7r2)], O"[Z,] les sections canoniques des fibres en droite [Gr(1T2)], [Zs]. Si s E 5, alors [Gr(1T2)]Vx{s} = [Zs] et 0"[Gr( 7r2)]vx{s} = O"[Z,]. Sur V X 5, on a une suite exacte de faisceaux

(1.6)

aIGr(~2)1 * * ----+ Ovxs(n 0 ---+ m*OGr(7r2) ((nm) ~) ---+ O.

Pour s E 5, la restriction de (1.6) a V x {s} est exactement

(1. 7)

On note (TJ, v) Ie complexe sur V x 5,

(1.8)

Rappelle que l'application (1Tl 0 n, n') : V x 5 ---+ B x 5 est une submersion de fibre compacte X. Soit H(X,TJlx) l'hypercohomologie de (Oxbx{s}(TJIXbx{s}),v). Alors par (1.6), H(X,TJlx) est isomorphe a H(Y,~IY) = R·1T*~, et H(X,TJlx) est un fibre vectoriel sur B x 5.

Soit gTS une met rique hermitienne sur T5. So it g[Gr(7r2 )] une met rique hermitienne sur [Gr(1T2)]' Soit g-[Gr(7r2)] la met rique hemitienne sur [-Gr(1T2)] induite par g[Gr(7r2 )]. Soit h TJ = ht,®g[-Gr(7r2 )]ffiht, la met rique sur TJ = n*~®[-Gr(1T2)]ffin*~.

Soit hH(x.TJlx) la met rique L2 sur H(X, TJlx) associee aux metriques gTX, hTJ definie par [B3, §3.2] et ch(H(X,TJlx), hR7r·t"hH(X,TJ1x)) E pBx(SV~)/pBX(S\t::..).()

la classe de Bott-Chern associee a [BGSl, §If)]. Par [B3, §3.2]' on peut construire des formes de torsion analytique T(n*wv, h1J) E pBxs / pEXS.O, telle que

8.8 T(n*wv, h1J) = ch(H(X, 1]lx), hH(X,1J1x)) - r Td(T x, gTx)ch(1], h1J). 2m J xx{s}

Sur V \ ~, on a une suite exacte de fibres vectoriels holomorphes

0---+ TY ---+ T X ---+ 7r~TS ---+ O.

(1.9)

Soit Td(T X, gTX, 7r~gTS) la classe de Bott-Chern associee a la suite exacte cidessus, on a

Pour s E S \ .6., on pose

Is = T; [T(n*wV, h1J) - T(w v, hE,) + ch(H(X, "7lx), hH(X,r1I x ), h R1f*E,)]. (1.10)

On a l'analogue de [B2, TMoreme 5.6]' Theoreme 1.2. Pour s E S \.6., l'identite suivante est vraie dans pE / pE ,0:

! Td(T X, gTX)ch(~, hE,) 2

Is =. Td([G ( )] [Gr(1f2 )]) log(110'[Gr(1f2)]II ) X x {s} r 7r2 ,g

(1.11)

j Td(T X gTX 7r*gTS) _ "2 ch(C hE,)

Y, Td([Gr(7r2)],g[Gr(1f2 )]) <", .

Preuve. En appliquant la for mule de [B3, Theoreme 0.1] a l'immersion js : Zs ---+ V (s E S \ .6.) qui est exactement l'immersion Gr( 7r2) n V x {s} ---+ V x {s}, et a la suite exacte de faisceaux (1.6) sur V x {s}. En procedant comme en [B2, Theoreme 5.6], on a (1.11). 0

d) La divergence logarithmique des formes de torsion analytique pres des fibres singulieres

On pose

E(x) = x - sinh(x) 2x(1 - cosh(x))

Alors si (( s) est la fonction zeta de Riemann,

E(x) = - L n?;:O.n pair

(n + 2)(( -n - 1)xn

(n + 1)!

On identifie E au genre additif correspondant.

E(O) = ~.

(1.12)

(1.13)

400 Xiaonan Ma

Soit h,R7r.t. une met rique hermitienne Coo sur R-7f*1:, sur B x S. Soit ch(R-7f*1:" h R7r .t., h,R7r.t.) E pBX(S\t::.)/pBX(S\t::.),a la classe de Bott-Chern de [BGSl, §If)] telle que

28.8 clI(R-7f*1:" hR7r.t., h,R7r.t.) = ch(R-7f*1:" h,R7r.t.) - ch(R-7f*1:" hR7r.t.). (1.14) Z7f

Soit O"[t::.] la section canonique du diviseur [.6.]. Soit g[t::.] une met rique sur [.6.].

Theoreme 1.3. Pour So E .6., la limite

lim {r; [T(wV, ht.) + clI(R-7f*1:" hR7r ,t., h'R7r't.)] S\t::.3s->so

(1.15)

+ ~ log(IIO"[t::.] II;) r Td(TI:B)E(NL:/v )ch(f:,)}. 2 JL:B

existe dans pB / pB,a. On note cette limite comme T(w Z, ht., h'R7r,t.) E pB / pB,a.

On considere T(w Z, ht., h'R7r,t.) E pB / pB,O comme les formes de torsion analytique de Zso --+ B.

Avant demontrer Ie Theoreme 1.3, on verifie que ce Theoreme est compatible a [B2, TMoreme 5.9], [BerB, TMoreme 3.1].

Soit A(I:,) et A(R-7f*1:,) les fibres en droite sur S qui sont les inverses des determinants de H (Z, 1:,1 z) et H (B, R- 7f * 1:,1 B ). Alors par [KM], Ie fibre en droi te A(I:,) Q9 A-I (R-7f*1:,) a une section canonique non nulle 0" sur S. Soit II 11'\(t.) et II 11,\(Ro 7r,£,) les metriques de Quillen sur A(t:,) et A(R-7f*1:,) sur S \.6. associees a gTZs,ht. et gTB,hR7r .f,. Soit II 11,\(f,)0,\-1(R07r,£,) la metrique correspondante sur A(I:,)Q9A- I (R-7f*t:,) sur S\b.. Alors pour s E S\.6., en appliquant [BerB, TMoreme 3.1], on a

log 110"11~(£,)0,\-1(R07r*£,) = - fBX{s} Td(TB,gTB)T(wV, hI',)

+ fz Td(TZ,TB,gTZ,gTB)ch(l:"ht.). 8

(1.16)

lci, comme en (1.11), la classe de Bott-Chern Td(·,·) correspond a la suite exacte

o --+ TY --+ T Z --+ 7f~T B --+ O.

En utilisant [B2, TMoremes 2.1 et 5.9] et [BGS2, TMoreme 1.23] (se referer aussi a §5), quand s E S \ b. --+ So E .6.,

- r Td(TB,gTB) [T(wV, hE.) + clI(R-7f*!:" hR7r ,f" h'R7r'E.)] J Bx{s}

- ~ log(IIO"[t::.] II;) ~ Td(TI:)E(NL:/v)ch(l:,)

a une limite. C'est a dire, Ie TMoreme 1.3 est compatible a [BerB], [B2]. D

Preuve du Theoreme 1.3. Dans notre cas, par une modification simple de l'argument de [B2], on sait que les resultats analogues de [B2, TMoremes 2.2 et 3.3] sont aussi vrais.


En procedant comme en [H2, Theoreme 5.7], quand S E S \ ~ ----7 So E ~, l'asymptotique du terme a droite de (1.11) dans pB / pB,O est exactement la me me for mule que celle de [H2, (5.20)].

Comme T(n*wv, h'l) E pBXS, on a montre Ie Theoreme 1.3. D

Par Ie Theoreme 1.3, dans pB / pB,O, on pose

(1.17)

e) Formules d'anomalie pour des formes de torsion analytique

Soit gl1'V,h'E"hI/Rrr,E, des metriques sur TV,~,R·7r*~. Pour So E ~ fixe, on peut construire aussi T(W'Z, hiE" hl/RIT'E,) E pB / pB.O com me dans Ie Theoreme 1.3.

Par [H2, Theoreme 1.1], Ie fibre vectoriel TY sur VE \ P(NE / V ) :::: V \ 1:

s'etend en un sous-fibre v(TV/ B x S) de p*TV sur V~::. Alors sur Z, on a TY = v(TV/B x S). Soit gv(1'V/BXS), g'V(1'V/BXS) les metriques induites par g1'V, g'1'V

sur v(TV/ B x S). Soit g(['Y la metrique sur TY induite par gITV. So it

Td(v(TV/ B x S), gv(TV/BXS), g,v(TV/BXS)) E pVE / pvE.o,

2h(~,hE"h'E,) E pv/pv,o, Td(TY,gTY,gl1'Y) E pz/pz,o,

ch(R-7r *~, h'RIT,E" hl/RIT,E,) E pBXS / pB XS,O

les classes de Bott-Chern [HGS1]. On a

28.8 2h(~, hE" hiE,) = ch(~, hiE,) - ch(~, hE,). Z7r

(1.18)

et les autres formes ci-dessus verifient des equations analogues. Alors par (1.15), [H2, Theoreme 2.1], [HK, Theoreme 3.10]' on a

Theoreme 1.4. On a l'identite suivante dans pE /pB.O

T(W'Z, hiE" hl/RIT,E,) _ T(wz, hE" h,RIT.E,) = c1(R-7r*~, h'RIT,E" hl/RIT,E,)

-h (Td(TY, g1'S;', gITY)ch(~, hE,) + Td(TY, g'TY)ch(~, hE, h'E,))

-2 r {ch(~, hE,) r Td(v(TV/ B x S), gv(TV/ BXS), g'v(TV/ BXS)) lEE 1 P(N):/F)

+2h(~,hE"h'E,) r Td(v(TV/B x S),g'V(TV/BXS))}. (1.19) lp(NE / v )

2. Comparaison des formes de torsion analytique

Pour S E ~ fixe, soit zs la normalisation de Zs et p : Zs ----7 Zs est la projection canonique. On note k : 1:s ----7 Zs l'immersion triviale. Dans la suite, on omit l'indice s.

402 Xiaonan Ma

Alors p*Oz((jp)*O est un Oz-faisceau sur Z, et il existe un morphisme naturel Oz(j*~) ----> p*Oz((jp)*O et un morphisme de difference p*Oz((jp)*O ---->

k*(Odi*~ ® h:)). Sur Z, on a une suite exacte de faisceaux [B2, (5.53)]

(L, v) : 0 ----> Oz(j*O ----> p*Oz((jp)*O ----> k*(Odi*~ ® h:)) ----> O. (2.1)

D'apres (2.1), on a une suite exacte de fibres vectoriels holomorphes sur B

(Re1r*L,v) : ... ----> Ri-l(1ri)*(i*~ ® h:) ----> Ri(1rj)*~ ----> Ri(1rjp)*(jp)*~ ----> ...•

(2.2) On munit Re1r*L de la metrique h R7r*L = h'R7r*f, EB hR(7rjp)*(jp)*f, EEl hR(7ri)*(i*f,®r<).

Soit 2h(Re1r*L, hR7r*L) E pE / pE.O la classe de Bott-Chern [BGSl] telle que

~~ ch(Re1r*L, hR7r*L) = ch( R(1rjp)*(jp)*~, hR(7r jP )*(jP)*E,) (2.3)

-ch(Re1r*~, h,R7r*f;) - ch( R(1ri)*(i*~ ® h:), hR(7ri)*(i*f;®K»).

On rappelle que T(wZ, hf" h,R7r*E,) E pE / pE.O est definie par Ie Theoreme

1.3, et que T(wZ, hf;) et T(wL" hi*f;®K) sont definies a §lb). On pose

T'(w z , hf;, h,R7r*f,) = T(wZ, hf;) - T(wL" hi*f,®,,) - ch(Re1r*L, hR7r*L). (2.4)

L'objet de cet article est de donner une formule pour

T'(w z ,hf"h,R7r*f.) _T(wz ,hE.,h,R7r*f.) E pE/pE.O. (2.5)

Soit ((s) la fonction zeta de Riemann. Soit R(x) la serie de Gillet-Soule [GS]

( ('(-n) n 1) xn R(x) = '" 2-- + '" -:- ((-n)-. ~ ((-n) ~J n!

n2':O. n impair j=l (2.6)

Si f(x) est une serie entiere, on pose f(x) = 'E-k>of[klxk. On definit Sex), T(x) par

Sex) = L t (-l)jC~((_n)x~, n2':l. n impair j=l J n.

T(x)= '" 1 [(l+x)n+llOg(l+x)][nlxn. ~ . (n + 3)! 1 + 2x

n2':2. n palr

(2.7)

On identifie R(x), Sex), T(x) aux genres additifs correspondants. On pose

1 2 1 Q(x ,x ) = 1 2

X - X

l x2 Td( _Xl )Td( _x2 ) - Td( -s)Td( _(Xl + x 2 - s)) x ds.

xl Td( -s)(s - Xl )(s - x 2 )

(2.8)

Alors Q(Xl, x 2 ) est une fonction symetrique de (Xl, x 2 ). On peut considerer Q comme une fonction ad-invariante des matrices 2 x 2. On identifie Q au genre correspondant. De plus Q(x, -x) et Td(x)Q(x, 0) sont des fonctions paires.

On rappelle que TY sur VI: \ P(NI:/v) c::::: V \ ~ s'etend en un sous-fibre v(TVjB x S) de p*TV sur VI: [B3, Theoreme l.1]. Soit gv(TV/BXS) la met rique induite par gTV sur v(TVj B x S). Soit U Ie fibre en droite universel sur P(NI:/v). Soit PI: : P(NI:/v) ~ ~ la projection canonique. Sur P(NI:/v), on a la suite exacte [B2, (0.10)]

o ~ pfT~jB x ~ ~ v(TVjB x S) ---+ U Q9pfK, ~ O. (2.9)

Soit gNE/\/ une metrique sur N E/V telle que Ie scindement local NI:/v =

NE/Z' + NI:/Z2 (ou NI:/Zi represente Ie fibre normal dans l'une des deux branches locales de Y) soit orthogonal, et que l'isomorphisme canonique A 2 N L/ v Q9 pfK, ~ 7f2[~] soit une isometrie [B2, §6.3 et §9.4]. Par [BGSl, §If)], on peut associer une classe de Bott-Chern

a la suite exacte de fibres holomorphes hermitiens (2.9) telle que

2fHJ Td(v(TVjB x S),gv(TV/BXS),gU@P'i;K)

Z7f = Td(v(TVj B x S), gv(TV/BXS))

-pfTd(T~jB x ~,gTL/BX.6.)Td(U Q9PfK"gU®P~K).

Theoreme 2.1. On a la for-mule suivante dans p E j p B .O,

T'(w Z , h<;, h,R7r.<;) - T(w z , h<;, h,R7r.<;)

= 2 r ch(';,g<;) r. Td(v(TVjB x S),gV(TV/RXS),gU®P'i;K) jLB j P(NE / F )

(2.10)

- r Td(T~B)(Td(NL/V)T(N2;/V)+2Q(N2;/V))ch(O (2.11) jLB

+ ~ Td(Tf;B)(Td(x)(R:S + Q(x, 0))) (N1:/i)ch(O. hB Remarque 2.2.

i) Si B est un point, Ie Theoreme 2.1 est [B2, Theoreme 0.2].

ii) Dans Ie Theoreme 2.1, si Vest projective, par l'argument de resolution acyclique, on peut supposer seulement que

et

sont localement libres sur B x S. B x ~ et B x ~.

404 Xiaonan Ma

3. Preuve du Theoreme 2.1: Ie cas oil Ia fibre exceptionnelle est scindee

Cette section est organisee de la fa<;on suivante. Dans a), on introduit notre hypothese sur Z. Dans b), on compare des formes de torsion analytique. Dans c), on montre Ie Theoreme 2.1 sous notre hypothese sur Z.

a) Dne hypothese simple sur Z

Dans la suite, on fixe toujours So E Do. So it Z = Zso. On suppose qu'il existe deux diviseurs lisses Zl et Z2 dans V intersectes

transversalement Ie long de ~ tels que

Alors Ie fibre Ii sur ~ est trivial. Soit i : ~ -+ V, jk : Zk -+ V, j : Z -+ VIes immersions naturelles. Pour k = 1,2, soit O"[Zk] la section canonique de [Zk]. Alors la section canonique O"[Z] de [Z] est donnee par

(3.2)

Par (3.2), on identifie A2([_Zl] EB [_z2]) a [-Z]. Les complexes de Koszul (A([_Zk], O"[Zkl), (A[-Z], O"[ZI), (A([_Zl] EB [_Z2]),

O"[ZII EB 0"[Z21) sont des resolutions des Ov-faisceaux j~C,j*C,i*C sur V. On a Ie complexe de faisceaux sur V

0 0 0 T j j

0-+ i*C----+ 0-+ 0-+ 0 j j j

0-+ C----+ C-+ 0-+ 0 iU[Zlj EEl ( -iUjZ2j H ej j (3.3)

0-+ [_Zl] EB [_Z2([Zl.l!:[Z2 j CEf)C-+ j;C EB j';C-+ 0 iU[Zlj EEl ( -iU[Z2j)T ej eT

0-+ [-Z]~ C-+ j*C-+ 0 j j j

0 0 0

Dans (3.3), e: C -+ CEBC est Ie complexe diagonal f -+ (j,j) et e: CEBC -+ C est donne par (j,g) -+ (j - g), et j;CEBj;C = (jp)*C, e: j*C -+ j;CEBj;C = (jp)*C est induit par Ie morphisme de Oz-faisceaux sur Z, C -+ p*C considere dans (2.1).

Dans (3.3), les trois premieres lignes (a partir du bas) sont des suites exactes de faisceaux. Les deux premieres lignes correspondent aux complexes de Koszul pour les diviseurs Z, Zl et Z2 dans V. Les deux premieres colonnes dans (3.3) sont aussi des suites exactes de faisceaux. La premiere colonne correspond au complexe de Koszul associee a ~ = Zl n Z2.

Par un argument de suite spectrale, la cohomologie totale du bicomplexe (3.3) est exactement i.C.


Bien sur, tous les complexes de Oz-faisceaux peuvent etre tensorises par ~. Done on a un bicomplexe de fibres holomorphes sur V,

0 0 I I

0--> ~---+ ~--> 0 III III

0--> ([_Zl] EB [_Z2]) ® ~~ ~ED~--> 0 (3.4) III v III

0--> [-Z] ®~---+ ~--> 0 I I

° 0

Dans (3.4), on numerote les lignes de haut en bas par (~!)(i = 0,1,2) et les colonnes de droite a gauche par (G)(i = 0,1). Par (1.8), on sait que

(~;,v) = (1J,v)Wx{so}' (3.5)

b) Comparaison des formes de torsion analytique

Pour k = 1,2, soit g[Zk] une met rique hermitienne sur [Zk]. On munit [Z] de la met rique g[Z] = g[Zl] ®g[Z2]. On munit aussi [Zl] ED [Z2] de la met rique g[Zl]EIl[Z2] =

g[Zl] EDg[Z2]. Alors A([_Zk]), A([-Z]), A([_Zl] ED [_Z2]) sont munis naturels des metriques hermitiennes gA([-ZkJ) , gA([-Z]) , gA([-Z']EIl[-Z2]).

Soit H(X, ~:Ix) l'hypercohomologie de (Ox(~:lx), jL + v). Soient H(X, ~!Ix)

et H(X,~7Ix) les hypercohomologies de (Ox(~!lx)'v) et (Ox(~~x),jL). Comme dans [B3, §3.2]' par la theorie de Hodge, on peut equiper des metriques £2, hH(x·E.:IX) hH(X·E.~lx) et hH(x.E.7Ix) (i = ° 1 2) sur H(X C· ) H(X ci ) et , , , '''.Ix ' '''.Ix H(X, ~7Ix) sur B associees a gA([_Zk]), gA([-Zll, gA([-Z']EIl[- Z2 ll, hE. et gTX. D'apres

(3.3), sur B, on a

H(X, ~~Ix) = H(X, ~~Ix) = 0, HP(X'~~lx) = RP(7rjp)*(jp)*O, HP(X'~;lx) = HP(X,1Jlx) = RP7r*~, HP-l(X'~:lx) = HP(X'~~lx) = RP(7ri)*i*~, hH(x'E.~lx) = hH(x·E.:lx), h H(X·E.;lx) = hH(x·1)lx).

(3.6)

Par [B3, §3.2]' on peut construire des formes de torsion analytique T(wV,hE.:), T(w v, hE.7) et T(w v, hE.~) E pE, telles que

28.0 T(wv, hE.:) =ch(H(X'~:lx),hH(x.E.:lx))- r Td(TX,gTX)ch(~:,hE.:). (3.7) Z7r Jx

et les formes T(w v, hE.;), T(w v, hE.~) verifient des equations analogues. Remarque

aussi ch(~!, hE.~) = L:( -l)jch(~j, hE.j). Alors, par (1.9), on a

i* T(n*wv hrl) = T(w v hE.;) So ' ,. (3.8)

406 Xiaonan Ma

On rappelle que Ie complexe Lest defini en (2.1). Soit h'R1f*L = hH(XJ';lx) ffi

h H(X'<:;IX) ffihH(X'<::lx) la metrique sur R e 7r *L = R7r *E,ffiR( 7rj p)* (j p)* E,ffiR( 7ri)* i* E,. Comme en (2.3), so it ch(Re 7r*L, h,R1f*L) E pB I pB.O la classe de Bott-Chern pour Ie complexe (2.2).

Theoreme 3.1. On a l'identite suivante dans pB I pB.a, 2

-T(wV, h<:~) = 2) -1)iT(wV, h<:~) + ch(R e 7r*L, h,R1f*L). (3.9) i=l

Preuve. Pour Ie bicomplexe E = (O(X,E,:),ax +V,IL), si on Ie filtre par IL, alors Ie premier terme de la suite spectrale associee est Ef'i = HP(X, E,!lx)' D'ou Ie suite spectrale (End,,) degenere a E2 . D'apres l'argument de [Mal, Theoreme 11.1], [B3, Theoreme 0.2], pour Ie complexe (3.4), dans pB I pB,a, on a

T(wV,h<::) = L;=()(-l)iT(wV,h<:~) +ch(Re 7r*L,h,R1f*L), T(wV, h<::) = L~=a( -l)iT(wV, h<:7).

On veri fie facilement qu'on a

T(wV,h<:~)=O,

Par (3.10), (3.11), on a (3.9).

c) Preuve du Theoreme 2.1

Soit

(3.10)

(3.11)

o

(3.12)

h.' T(wv, h<:~) - T(w E , h<:) + ch( H(X, E,~lx), hH(X'<:~IX), hR(1fi)*(i*O) ,

h.,,, T(w v, h<:;) - T(wZ, h<:) + Zh(H(X, E,~lx), hH(X'<:;IX), hR(1fjp)* (jp)* <:).

Alors en utilisant [B3, Theoreme 0.1] et en procedant comme en [B2, Theoreme 6.1], on sait que dans pB I pB,O, 1;", Iv' sont egaux formellement aux memes formules que les termes a droite de [B2, (6.11)].

On rappelle que h.. est defini en (1.17). Par [BGSI, Theoreme 1.20], [Mal, (10.7)], dans pB I pB.D, on a

ch(Re 7r*L, h,R1f*L) _ ch(Re 7r*L, hR1f*L) (3.13)

= Ch( R(7ri)*i*E" hH(X'<:~IX), hR(1fi)*(i*<:))

-Zh ( R( 7rj p)* (j p)* E" hH(X'<:;IX), hR(1fjp)* (jp)* <:)

+Zh(R e 7r*E" hH(x·t,;IX), h,R1f*t,).

Par (1.10), (1.15), (3.7), (3.9) et (3.13), dans pB IpB.O, on a

T'(w z , h<:, h,R1f*t,) - T(wz, ht" h'R1f*t,) = h. + h." - I),,,. (3.14)

Par (3.14), en procedant comme dans [B2, §6 et §7], on a Ie Theoreme 2.1. 0

4. Preuve du Theoreme 2.1: Ie cas general

Dans cette section, on montre Ie Theoreme 2.1 dans Ie cas general. On ne suppose donc plus que Z = Zl UL; Z2. L'idee principale de la preuve est d'utiliser la technique de la deformation du cone normal [BaFM], [BGS4]. Elle reduit notre probleme a un probleme analytique naturellement localise sur P(NL;/V ffi 1), auquel la technique de [B2, §4 et §6] peut etre appliquee.

Cette section est organisee de la fa<;on suivante. Dans a), soit W l'eclate de V x pI Ie long de ~ x {oo}, on releve Ie diviseur Z a un diviseur D dans W. Dans b), on calcule l'asymptotique des formes de torsion analytique sur B x pl. Dans c), en utilisant les resultats de la Section 3, on sait que Ie probleme peut etre localise sur P(NL;/V ffi1). Et on etablit un resultat analogue de [B2, Theoreme 9.16]. Dans d), on montre Ie Theoreme 2.1 dans Ie cas general.

Pour simplifier les notations, on suppose toujours que .6. = {so}. On utilise les meme notations qu'aux Sections 1 et 2.

a) Diviseur D

Soit W l'eclate de V x pI Ie long de ~ x {CXl}. Soit p, q les projections p : W -+ V, q: W -+ pl. Pour Z E pI, on pose

( 4.1)

Si Z "I 00, l'application p : Wz -+ Vest une identification de varietes complexes. Alors p* Z est un diviseur sur W.

Definition 4.1. Soit D Ie diviseur sur W

D = p* Z - 2P(NL;/V ffi 1). (4.2)

Par [B3, Proposition 9.2], D est un diviseur effectif dans W, transversaux avec W=, et a croisements normaux Ie long de ~ x pl. On verifie facilement que D est exactement I' eclate de Z x P I Ie long de ~ x {oo}. En particulier , Woo n D est l'union de la normalisation Z = VL; n D de Z et de P(NL;/V ffi 1) n D qui est un diviseur aux croisem:nts normaux P(Nf;/z ffi1) dans P(NL;/V ffi1), et ils sont

transversaux Ie long de ~. Soit io l'immersion D -+ W. So it Po,qo les restrictions de p,q a D. Alors

po(D) c Z. On note Zi = (HI 0 p, q) : W -+ B X pI, Zio = Zilo. Pour s E pI, on

note aussi D z = pr/(z). Soit DB la fibre de la fibration HOp: D -+ B x {so}. Soit H(ZB,j*t,lzB) la cohomologie de t, Ie long des fibres ZB, alors H(ZB,j*t,lzB) RH*t,IBX{so}. Par [B2, Proposition 9.3], on a

Proposition 4.2. L'application lineaire ClO E H(ZB,j*t,lzB) -+ PDClO E H(DB' (jPO)*t,IOB) induit une trivialisation de R-Zio*(jpo)*t, sur pl.

408 Xiaonan Ma

b) L'asymptotique des formes de torsion analytique sur pi

Soit gTS une met rique sur T8. Soit O"[ool la section canonique de [00] sur pi, soit g[ool la met rique canonique sur [00] = 0(-1).

On rappelle que par (1.5), [Z]lz = 7r2T8. So it g[Zl une met rique hermitienne sur [Z] telle que

(4.3)

Evidemment

[P(NE/V EB l)]IExP' = q*[oo]IEXPl. ( 4.4)

Soit g[P(NE l v EB l)l la met rique sur [P(NE/V EB 1)] sur W telle que

( [P(NE/V EB 1)]'9[P(NElv EB l)l) = q*([oo],g[ool). IExP'

(4.5)

Par (4.2), on a

[D] = q*[Z] Q9 [P(NE/ V EB 1)]-2. (4.6)

Soit g[Dl la met rique sur [D] induite par g[Zl et g[P(NElv EBl)l. Par (4.2)-(4.6), on a

([D],g[Dl)IEXP' = ((7r2P)*T8 Q9 q*[-2oo], (7r2p)*gTS Q9 q*g[-2ool ) . (4.7) IExP'

On rappelle aussi que Ie complexe

(4.8)

est une resolution canonique de ilhOD((jPD )*E;,). Soit H(X, E;,Dlx) l'hypercohomologie de (OX (E;,D) , v), alors par (4.8), H(X,E;,Dlx) c:::: R"ijD*(jPD)*E;,.

Comme Vest Kahlerienne, West aussi Kahlerienne. Soit gTW une met rique Kahlerienne sur TW. Alors gTW induit une met rique Kahlerienne gTX sur TX, et on munit les metriques hermitiennes associees sur les sous-varietes de W. Soit hc' une met rique hermitienne sur t;,.

Soit 7f : W --+ B x 8 X pi (resp. 1f: W --+ B X pi) definie par

7f = (7rlP, 7r2P, q) (resp. 1f = (7rlP, q)). (4.9)

Plus precisement, 7f : W \ (Woo U Z X pi) --+ B x (8 \ ~) x (pi \ {oo}) est une submersion de fibre compacte Y. Soit PBxS : B x 8 1 X pi --+ B x 8 la projection naturelle. Par la Proposition 4.2, R-7f*E;, est localement libre sur B x 8 X pI, et R-7f *E;, = Ps xsR7r *E;,. Soit hR1i .C, la met rique L2 sur R7f *E;, sur B x (8\~) x (pl\ {oo}) associee a hC" gTW. Soit h,R1i.C, une metrique hermitienne sur R7f*E;, sur B x 8 X pl.

Soit hH(x'c'Dlx) la metrique L2 sur H(X, E;,Dlx) associee aux metriques hC" h[Dl, gTX definie par [B3, §3.2] sur B x (pl \ {oo}). En rempla<;ant formellement V par W, 7r par 7f, E;, par p*E;, a la Section 1, soient

T(w w, hc') E pBx (5\6) x (P'\{ oo}) / pBx (5\6) x (P' \{oo} ),0,

T(wW, hc'D) E pBX(P'\{oo}) / pBX(P'\{oo}),O


les formes de torsion analytique associees a (1f, wW , h~), ('if, wW , h~D) pour (4.8) [BKJ, [B3, §3.2]' en particulier,

2~f) T(ww, h~D) = ch(H(X'~Dlx), hH(X'~DIX)) - f Td(TX,gTx)ch(~D' h~D). m b

et Ch(~D' h~D) = ch(p*~, hP'~) - ch(p*~ ® [-D], hP·~I8>[-D]).

Pour Z E pI, S E S, soient Ts,z : B ~ B x S X pI et Tz : B ~ B X pI les applications de y E B a (y,s,z) E B x S X pI et (y,z) E B X pl.

Definition 4.3. Pour Z E pI \ {oo}, Sa E ~, dans pB / pB,a, on pose

Tz(wD,h~,h'R1i.~) = \ lim {T;,z[T(wW,h~) +clJ.(R·1f*~,hR1i·~,h'R1i*~)] S Ll.3s-+sQ

+~ log(lla[Ll.]II;lla[oo]II;4) f Td(T~B)E(N~/v )Ch(~)}. 2 J~B

(4.10)

En effet, pour Z E pI \ {oo}, la Definition 4.3 correspond exactement au TMoreme 1.3, dans lequel Vest remplace par W z . Comme en (1.17), [.>.,z est bien defini, et dans pB / pB,a, on a

[, = T*T(ww h~D) - T (w D h~ h'R1i.~) - T*clJ.(R1f (: h'R1i.~ hH(X'~DIX)) /\,z z , z,' z *":", . (4.11)

En procedant comme dans [B2, TMoreme 9.9]' on a

Theoreme 4.4. La fonction Z E pI \ {oo} ~ [.>.,z E pB / pB,a se prolonge en une fonction continue sur pl.

Remarque 4.5. Si on note l.>.,oo = limz-+oo [.>.,z E pB / pB,a, alors ['>',00 a une formule analogue que Ie terme a droite de [B2, (9.37)].

On rappelle que U est Ie fibre en droite universel sur P(N~/v).

Theoreme 4.6. Quand Z ~ 00

z~~ {Tz(w D, h~, h'R1i.~) -log Iz12[ ~ Td(Ti:B)E(U)ch(~)]} hB (4.12)

a une limite dans pB/pB,a. On note cette limite comme Too(wD,h~,h'R1i.~) E pB/pB,a.

Preuve. On rappelle que la fibration W ~ B X pI verifie les hypotheses de la Section 1. Comme H(X, ~Dlx) est localement libre sur B x pI, par les Theoremes 1.2 et 1.3 et [B2, (9.47)], pour la fibration q : W ----t B x pI, on a

lim {T;T(WW , h~D) - T;clJ.(R1f*~, h'R1i.~, hH(X'~DIX)) z-+oo

- log Iz12[ ~ Td(Ti:B)E(U)ch(~)]} hB existe dans pB / pB,a. Par Ie TMoreme 4.4, on a Ie TMoreme 4.6. o

410 Xiaonan Ma

Pour la fibration 7r : W \ Woo ---+ B x S X (pI \ {oo}), Z X (pI \ {oo}) sont les fibres singulieres, comme en (2.4), sur B x pI \ {oo}, on peut definir

T'(w D, hI;, h,R7i.I;) E pBX(pl\{oo}) / p BX (P 1 \{00}),0.

Theoreme 4.7. Quand z ---+ 00

}~~ {T;T'(w D,hl;,h'R7i.I;) -loglzI2[ ~ Td(TI;B)E(U)ch(~)]} hE (4.13)

a une limite dans pB / pB ,0, qu 'on le note comme T!xo (w D, hI;, h,R7i.I;) E pB / pB,O.

Preuve. Comme en [B2, (9.40)],

Nfj=/D = Np(NE/V)/wlfj= = (U EEl U-I)If;·

Par les Theoremes 1.2 et 1.3, on a Ie Theoreme 4.7. o Remarque 4.8. Pour (T;T' - T z )(wD ,hl;,h'R7i.I;), on a aussi un result at analogue que [B2, Theoreme 9.12].

c) Localisation du calcul it 00 sur P(N"E/v EEl 1)

On a P(N"E/v) C P(N"E/v EEl 1) C Woo, et I;oo = D n P(N"E/v). Soit

D' = D n P(N"E/v EEl 1). (4.14)

Alors D' est un diviseur a croisements normaux Ie long de ~ dans P(N"E/v EEl 1), et ~ C D' est l'ensemble des points singuliers dans D'. Soit j' l'immersion D' ---+

P(N"E/v EEl 1). Soit ji : D' ---+ Doo = Woo n D l'immersion naturelle.

Par §4a), I;oo s'identifie a I; dans §la), et p : I;oo ---+ ~ est un revetement double et K = det p10fj. Soit P"E : P(N"E/v EEl 1) ---+ ~ la projection canonique. Sur ~, on a une suite exacte de faisceaux

(4.15)

En (4.15), e est l'application diagonale x ---+ (x, x), fest l'application de difference (x, y) ---+ x - y. Rappelle que i : ~ ---+ Vest l'immersion naturelle. Par (4.15), on a un complexe acyclique de fibres holomorphes sur B

(R(7ri)*K, v) :- .. ---+ Hj(I;B, (7rip)*~) ---+ Hj (~B' i*U?) K) ---+ Hj+I(~B' i*~)---+ ...•

(4.16) Soit k l'immersion ~ ---+ D', soit ijt la normalisation de D'. Alors D' est

l'eclate de D' Ie long de ~. Soit p' : D' ---+ D' la projection canonique. Alors on a une suite exacte de faisceaux

0---+ OD' ---+ p:Ofj, ---+ k*O"E(K) ---+ O. (4.17)

Sur P(N"E/v EEl 1) c Woo, Ie complexe

(~D"V): 0 ---+ Op(NE/VEBI) ([-D'] ® (ip)*~) D"i:';l Op(NE/VEB1)((iP)*~) ---+ O. (4.18)

est une resolution canonique de j~OD'((ipj')*f,) sur P(N"E/v EEl 1), et (~D"V) est exactement la restriction du complexe (~D'V) en (4.8) a P(N"E/V EEl 1).


On note D~, D~, P(NE/V EB I)B' P(NE/V )B, VE1B les fibres des fibrations de D',IY, P(NE/V EB 1), P(NE/V), VE a B. Par un argument de suite spectrale, on sait que [B3, §ge)]

H(D~, (ipj')*~) = H(L,B, i*~),

H(D~, (ipj' p')*~) = H(f,B' (iP')*~), H(P(NE/V EB I)B' [-D'] ® (ip)*~) = 0,

H(P(NE/V EB I)B, (ip)*~) = H(L,B, i*~).

(4.19)

Plus precisement, on verifie facilement que, SOliS l'isomorphisme canonique de (4.19), les suites exactes longues de cohomologie associees a (4.15), (4.17) sont (4.16).

On rappelle que L, et P(NE / v EBl) ont ete equipes naturellement des metriques induites par gTW comme dans §4a). La met rique gTW induit aussi une met rique sur TD'. De meme, tous les fibres holomorphes consideres (ip)*~, [-D']' ... sont deja equipes des metriques hermitiennes induites, par exemple, gTP(NE/v(f)l)B, gTfj~ sont les metriquess induites sur les fibres tangents relatifs T P(NE/V EB I)B' T D~.

So it H(P(NE/V EB I)B' ~D/) l'hypercohomologie de (Op(NE/v(f)l)B (~D/)' v). Par (4.18), on sait que

H(P(NE/V EB I)E' ~D/) = H(D~, (ipj') *0 = H(L,B, i*~). (4.20)

So it hH(P(NE/V(f)l)B,I;D/), hH(fj~.(ipj'pl)*EJ les metriques L2 sur H(L,B,i*~), H(f,E' (ip')*~) sur B induites par (gP(NE/V(f)l)B,ht;D), (gTfj~,hl;) [B3, §3.2]. Rappelle que h R (7ri)*(i*t;0 K ) est la met rique L2 sur R(7ri)*(i*~ ® K,) definie a §lb). On munit Re(7ri).K de la met rique h R (7ri)*K = hH(P(NE/v(f)l)B,t;D/ ) EBhH(fj~.(ipjlpl)*O EO

h R (7ri)*(i*t;0 K ). Alors par (4.16), il existe une classe de Bott-Chern ch(ReCrri).K, h R (7ri)*K) E p B / pB,G [BGS1, §If)] telle que

28.8 ch(Re (7ri).K, h R (7ri)* K) =ch(H(f,B, (ip')*~), h H(fj~ ,(ipj' p')* t;)) (4.21) ~7r

- ch(H(L,B, i*~), hH(P(NE/V(f)l)B.t;D/ »)

- ch(Re(7ri).(i'~ ® K,), h R (7ri)*(i*1',0 K »).

So it T(wP(NE/V(f)l) , ht;D / ) E p B / pB,G les formes de torsion analytique as

sociees au complexe (4.18) et (7riPE, hl',D) [B3, §3.2]. So it T(Wfj/, ht;) E p B / p E •G

les formes de torsion analytique associees a (ipj' p' : D' --> B x {so}, Wfj/, hl',). Soit

h(ipjl) = _T(wP(NE/V(f)l) , ht;D / ) + T(Wfj/, ht;)

-T(wE, h i *1',0 K ) - clJ.(Re(rri).K, h R (7ri)*K). (4.22)

Rappelle que VE est l'eclate de V Ie long de L,. Par la Section 4a) et (4.8), Ie complexe

(4.23)

412 Xiaonan Ma

est une resolution canonique de 0z(P*~) sur VI;. Le complexe

(~DIP(NE/V)' v) : 0 -+ (p*~ @ [-D])IP(NE/V) a~l P*~IP(NE/V) -+ O.

est une resolution canonique de 0t(P*O sur P(NI;/V). Soient H(VI;IB'~D) et H(P(NI;/V)B, ~D) les hypercohomologies de

(OVE1B (~DIVE)' v), (Op(NE/V)B (~DIP(NE/V))' v).

Par (4.23) et (4.24), on a

H(VI;IB, ~D) = H(ZB' (jp)*~) = R(7rjp)*(jp)*~, H(P(NI;/V )B, ~D) = H(f-B' (ip)*~).

(4.24)

(4.25)

Soient hH(VEB,I;D), hH(P(NE/V)B,I;D) les metriques L2 associees sur H(ZB' (jp)*~),

H(f-B , (ip)*~) sur B [B3, §3.2].

On rappelle que Ie complexe (Re 7r*L, v) est defini en (2.2). On munit R-7r*L de la met rique

h,Rn*L = ffih ,R1i*1; ffi hH(VE1B,I;D) ffi hR(7ri).(i*I;®><).

Soit clt(Re7r*L, h'Rn•L ) E pB / pB,a la classe de Bott-Chern [BGSl] telle qu'elle verifie une equation analogue de (2.3).

On munit R-(7ri)*K de la met rique

h,R(ni).K = ffihH(P(NE/vfJ)l)B,I;DI) ffi hH(VEIB,I;D) ffi hR(ni).(i·I;®><).

Comme en (4.21), on l'associe une classe de Bott-Chern clt(Re (7ri).K, h'R(7ri).K) E p B / pB,a. On a aussi [B2, (9.35)]

(4.26)

Soit W P(NE/V) l'eclate de W Ie long de P(NI;/V) c Woo et PP(NE/V) : W P(NE/ V ) -+

W la projection correspondante. Alors, TV 00 la normalisation de Woo est un diviseur lisse dans W P(NE / V )' qui sont transversaux Ie long de P(U ffi U-1 ).

Soient 15,1500 les normalisations de D, Doo. So it PD : 15 -+ D, p: 1500 -+ Doo les projections naturelles. Soit i 1 : ~ -+ Doo l'immersion naturelle.

Sur D oo , on a un bicomplexe de faisceaux

0 0 0 1 1 1

0-+ OWoolDoo v, ° V2 ihPD*Ot-+ 0 --+PD* DIDoo-+ I"d I"d I"d

0-+ P P(NE/V)* Ow 00 IDoo v, ~ ° V2 ihPD*Ot-+ 0 --+P* Doo-+

1"21 1"21 1 0-+ ihP*Otoo

v,. ° 0-+ 0 --+ZhP* t oo -+ 1 1 1 0 0 0

Dans (4.27), on a

PP(NE/V)*OWooIDoo = ODoonvE ffi OD',

{i*ODoo = ODoonVE ffi P:ODI'

(4.27)

(4.28)


Les VI, /11 sont des morphismes naturelles, V2, /12 sont des morphismes de difference. De plus, comme Woo = VE Up(NE / V ) P(NE/V EEl 1), les VI, /11 des troisiemes lignes et colonnes sont des identites.

Apres avoir tensorise (4.27) par p* f" on note les lignes de haut en bas par L~ (i = 0,1,2), et les colonnes de gauche a droite par Li (i = 0,1,2). On note aussi R7f*L~, R7r*Li (i = 0,1,2) les suites exactes longues de la cohomologie de L:IDooIB' L;IDooIB' Par (4.27), on a un bicomplexe de fibres vectoriels sur B

0 0 0 1 1 1

Hi(Doo1B,P*f,) v,

Xi V2

Hi(L,IB, i*f, ® K,) ••• ----+ ----4 ----4 ----4 •••

,",1 1 ,",1 1 ,",1 1 Hi(DooIB,PP(NE/V)*Owoo ® p*f,)

VI 1';

V2 Hi(L,IB,i*f, ® K,) ••• ----4 ----4 ----4 ----4.' •

,",2 1 ,",2 1 ,",2 1 Hi(L,IB, i*f, ® K,)

v, Zi

V2 0 .•• ---+ ----4 ----4 ----4 •••

(4.29) (avec Xi = Hi(DooIB' PD*Ofj ®p*f,), Yi = Hi(Doo1B,()*Ofjoo ®p*f,), Zi = Hi(L,IB, i* f, ® K)) dont les lignes sont exactes.

Soit h'R1r.Lj (i,j = 0,1,2) une metrique hermitienne sur R-7f*Lj sur B.

Alors elle induit des metriques h'R1r.L~ et h'R1r.Lj sur R-7f*L~ et R-7f*Lj. Soit

ch(R-7f *L~, h' R1r.L~) et ;;h(R-7f*Lj, h' R1r.Lj) E pE / pB,O les classes de Bott-Chern associees [BGSI, §If)] qui verifient des equations analogues a (4.21). Par (4.27), (4.29), d'apres un argument de suite spectrale, en utilisant [Mal, Theoreme 10.7], dans pB / pB.O, on a

2 2

2:) -l)i;;h(R-7f*L:, h' R1r.L~) = 2) -ljJch(R-7f*Lj, h'R1r*Lj). (4.30) i=O j=O

Rappelle que PE : P(NE/V EEl 1) ----4 L, est la projection naturelle. Soient T(wVE,h!;D), T(wP(NE/V),h!;D) E pB/pB.O les formes de torsion analytique de (4.23), (4.24) associees a (7fllJ, h!;D), (7fiPEIP(NE/v), h!;D) [B3, §3.2]. On pose

h" =T(wVE,h!;D) -T(wZ,h!;)

-;;h( H(ZB' (jp)*f,), hR(1r jp ).(jp)'E., hH(XEB'!;D») ,

1;.'2 = T(WP(NE/v ), h!;D) - T(w E , hi*c,0"')

-;;h( H(f:,B' (ip)*f,), hR(1ri).(i'c,0",) , hH(P(NE/V)B,!;D»).

(4.31)

Pour les formes de torsion analytique T(wW, h!;D) associees a q : W ----4

B X pI et le complexe (f,D, V) (cf. (4.8)), comme q est singuliere sur B x {oo}, en procedant com me dans les Sections 1 et 2, on definit T(wWoo, hC,D, h,R7r*!;) et T'(w Woo , h!;D, h'R7r*!;) E pB / pE,O. Comme on a fait apres (2.2), on definit

hR1r.L~ = h'R1r.c, EEl hH(DooIB,PP(NE/V)*Owoo0P'O EEl hH(E1B,P*OEoc,rSJi*O

414 Xiaonan Ma

la met rique L2 sur R e1f*Lo induite par hc'D ,gTW. Comme Woo = VE Up(NE/V) P(NE/V EB 1), par (2.4), on a

T'(wWoo,hC,D,h,R7i.c,) = T(wVE,hc'D) +T(wP(NE/V!fJI),hc'D) _T(wP(NE/V), hc'D) - Zh(Re 1f*Lo, hR7r.L~). (4.32)

On rappelle aussi que fj est la normalisation de D et P' : fj ---+ D est la projection canonique. Par la Proposition 4.2, R(qDP')*(jPDP')*~ est localement libre sur B x pl. Comme dans les Sections 1 et 2, on peut definir T(w Doo , hI:" h,R7i.I:,) et T'(w Doo , hI:" h,R7i.c,) E pB / pB,O sur B pour la fibration PDP' : i5 ---+ B{ so} X pl.

On a une equation analogue que [B2, (9.92)].

Proposition 4.9. On a l'equation suivante dans pB / pB,O,

h..,oo - i)" + 1;..2 - (T' - T)(w Doo , hI:" h,R7i.I:,)

+(T' - T)(wwoo, hC,D, h,R7i.c,)

= (T~ - Too )(wD ,hl:"h,R7i.I:,) - h..(ipjf).

(4.33)

Preuve. Par [Mal, (10.7)]' Remarque 4.5, (4.11)-(4.13), (4.22), (4.30)-(4.32), on a (4.33). 0

Theoreme 4.10. Le terme (T/x, -Too )(w D,hl:"h,R7i.c,) -I)"(ipjf) E pB /pB,O est egal exactement le meme terme a droite de [B2, (9.69)].

Preuve. En effet, Ie terme log(:: ::;«(;PD)'O )2(CXl) dans [B2, (9.69)] cor res pend a A«JPD)'!;)

(T/x, -Too), Ie terme log(:: ::~WPjf)'O)2 dans [B2, (9.69)] correspend a I)..(ipjf). Par A«,pjf)*O

(4.33), et en procedant com me en [B2, Theoreme 9.16], on a Ie Theoreme. 0

d) Preuve du Theoreme 2.1

En procedant comme en [B2, §9f) et §9g)], et en remplac;ant les Quillen par les formes de torsion analytique, on a Ie Theoreme 2.1.

5. Fonctorialite de la renormalisation de formes de torsion analytique

metriques de o

Dans cette section, on explique la compatibilite du Theoreme 2.1 et [BerB, Theoreme 3.1]. Comme c;a concerne des fibres singulieres, la verification n' est pas toujours triviale.

Cette section est organisee de la maniere suivante. Dans a), on etudie des classes de Bott-Chern pres des fibres singulieres. Dans b), on donne une version de [BerB, Theoreme 3.1] aux fibres singulieres.

Dans cette section, on fait les memes hypotheses et utilise toujours les memes notations qu'aux Sections 1 et 2.


a) Classes de Bott-Chern pres des fibres singulieres

On considere la suite exacte de fibres holomorphes hermitiens sur V \ ~

0----+ TY ----+ TZ ----+ 7r;TB ----+ O. (5.1 )

Soit Td(TZ,TB,gTZ,gTB) E pV\'E/pV\'E.o la classe de Bott-Chern de [BGSl] telle que

Par [B2, Theoreme 1.1], Ie fibre verctoriel TZ sur V'E \ P(N'E/v) c:::: V \ ~ s'etend en un sous-fibre v(TV / S) de p*TV sur V'E, de plus, Ie fibre v(TV/ B x S) defini a §le) est un sous-fibre de v(TV/ S). Sur Z, on a

TZ = v(TV/S). (5.3)

Soit gTZ,gv(TV/S) les metriques sur TZ,v(TV/S) induites par gTV. Par (2.9), on a un bicomplexe de fibres vectoriels holomorphes sur P(N'E/v),

Soient

o 1

o ----+7rf T~/ B x ~----+ 1

o ----+7rf T~/S----+ 1

o ----+7rf TB----+ 1 o

o 1

v( TV/B x S)----+ 1

v( TV/S) ----+ 1

7rf TB----+ 1 o

o 1

U@pf"'----+O 1

U@pf"'----+O 1 o ----+ 0 1 o

Td(v(TV/ S), gv(TV/S) , gU@PE K ), Td(v(TV/ S), T B, gv(TV/S), gTB) E pP(NE/ V ) / pP(NE/V ),0

(5.4)

les classes de Bott-Chern associees aux deuxiemes ligne et colo nne de (5.4). So it Td(T~/S,TB,gT'E/s,gTB) E p'E/p'E,O la classe de Bott-Chern associee a la premiere colonne de (5.4). Par (5.4), et l'unicite de la classe de Bott-Chern [BGSl, Theoreme 1.29], dans pP(NE / V ) / PP(NE/V ),0, on a ([B2, (9.72)])

Td( v(TV/ S), gv(TV/S) , gU@PE K) (5.5)

-Td(v(TV/B x S),gv(TV/BXS),gU@PEK)Td(TB,gTB)

= Td(v(TV/S), TB, gv(TV/S), gTB)

-Td(T~/ S, TB,gT'E/S,gTB)Td(U @ pf""gU@PE K ).

416 Xiaonan Ma

Proposition 5.1. Pour J1 E pV fixe, quand S E S \ ~ ---7 So E ~, on a

r J1Td(TZ,TB,gTZ,gTB) = CJ1Td(TZ,TB,gTZ,gTB) (5.6) }~ h

+2 r J1Td(v(TVjS),TB,gv(Tv/S),gTB)+O(s-so). }P(NE/V)

Preuve. So it J11 E pVE, par l'argument dans [B2, Theoreme 2.1], quand s E S\~ ---7

So E~, on a

r J11 = C J11 + 2 r J11 + O(s - so). }zs h }P(NE / V )

(5.7)

On a aussi

Td(TZ, T B, gTZ, gTB)BX{s} = Td(v(TVj S), T B, gv(TV/S) , gTB)BX{s}. (5.8)

Par (5.7) et (5.8), on a la Proposition. 0

b) Fonctorialite des formes de torsion analytique

On rappelle que h'R-rr.c, est une met rique Coo sur R-7r*c, sur B x S. Soit '\(c,) et ,\(R-7r*c,) les fibres en droite sur S qui sont les inverses des

determinants de H(Z,C,lz) et H(B,R-7r*C,IB)' Alors par [KM], Ie fibre en droite ,\(c,) ® ,\-1 (R-7r*c,) a une section canonique non nulle 0"1 sur S. Soit II 11~(Ro7r'c')

la metrique de Quillen sur ,\(R-7r*c,) associees aux metriques gTB, h'R7r .C, sur TB, R7r*c,. Soit II 11.x(O,C>. la met rique renormalisee definie dans [B2, (0.7)]. Soit

II 11.x(c,)IC>.<SI.x- 1 (R°7r.c,) la met rique sur ,\(c,) 11>. ® ,\-1 (R-7r*c,) induite par II Ih(c'),C>.

et II 11~(Ro7r'c')' Theoreme 5.2. On a l'identite

log(IIO"lll~(c')IC>.<SI.x-l(R07r'c')) (5.9)

= L Td(T B, gTB)T(w z , hC" h,R7r.C,)

- kTd(TZ,TB,gTZ,gTB)Ch(c"hc')

-2 r Td(v(TVjS),TB,gv(Tv/S),gTB)ch(c"hC,). }P(NE / V )

Preuve. Par (1.16), Theoreme 1.3, Proposition 5.1, on a Ie TMoreme 5.2. 0

Soient 0"2 et 0"3 les sections canoniques non-nulles de

Soient '1'1 et '1'2 les sections canoniques non-nulles des inverses du determinant de cohomologie associees a (2.1) et (2.2). Pour simplifier, on note IITill, II00ili les metriques de Quillen associees. Par un argument de suite spectrale [KM], on a

(5.10)

Par [BGS2, TMoreme 1.23], on a

log IIT2W = l Td(TB, gTB)ch(Re7r*L, hR-rr.L). (5.11)

Si on applique [BerB, TMoreme 3.1] a 0"2,0"3, [B2, TMoreme 0.2] (ou TMoreme 2.1) a la section Tl, par le TMoreme 2.1, (5.5), (5.9) et (5.11), a partir de l'equation des metriques de Quillen pou (5.10), on ramene finalement l'equation suivante

Mais par [B2, (6.64)]' on a

r Td(U ®p~""gU®P~I<) = _~. J P(NE / V ),:: 2

(5.13)

Done le TMoreme 2.1 et [BerB, TMoreme 3.1] sont compatibles.

On peut interpreter le TMoreme 2.1 et la compatibilite de TMoreme 2.1 et [BerB, Theoreme 3.1] comme "la renormalisation commute a la limite adiabatique pour la met rique de Quillen".

References

[BaFM] Baum P., Fulton W., MacPherson R., Riemann-Roch for singular varieties, Publ. Math. IHES 45 (1975), 106-146.

[BerB] Berthomieu A., Bismut J.-M., Quillen metric and higher analytic torsion forms, J. Reine Angew. Math 457, 1994, 85-184.

[BBos] Bismut J.-M., Bost J.B., Fibres determinants, metriques de Quillen et degenerecence des courbes, Acta Math. 165 (1990), 1-103.

[B2] Bismut J.-M., Quillen metrics and singular fibres in arbitrary relative dimension, J. of. Alg. Geom. 6 (1997), 19-149.

[B3] Bismut J.-M., Families of immersions, and higher analytic torsion, Asterisque 244, 1997.

[BGS1] Bismut J.-M., Gillet H., Soule c., Analytic torsion and holomorphic determinant bundles. I, Comm. Math. Phys. 115 (1988), 49-78.

[BGS2] Bismut J.-M., Gillet H., Soule c., Analytic torsion and holomorphic determinant bundles.III, Comm. Math. Phys. 115 (1988), 301-35l.

[BGS4] Bismut J.-M., Gillet H., Soule c., Complex immersions and Arakelov geometry, Progress in Math., no. 86, Birkhiiuser, Boston 1990, 249-33l.

[BK] Bismut J.-M., Kohler K., Higher analytic torsion forms for direct images and anomaly formulas, J. of Alg. Geom. 1 (1992), 647-684.

[Bo] Bott R., Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231-244.

[GrH] Griffiths P., Harris J., Principles of Algebraic Geometry, New York, Wiley 1978.

418 Xiaonan Ma

[GS] Gillet H., Soule C., Analytic torsion and the arithmetic Todd genus, Topology 30 (1991), 21-54.

[KM] Knudsen P.F., Mumford D., The projectivity ofthe moduli space of stable curves, I, Preliminaires on "det" and "div", Math. Scand. 39 (1976), 19-55.

[Mal] Ma Xiaonan., Formes de torsion analytique et familles de submersions. I, Bull. Soc. Math. France 127 (1999), 541-621.

[Ma2] Ma Xiaonan., Formes de torsion analytique et familles de submersions. II, Asian J. of Math., 4 (2000), 633-668.

[RS] Ray D.B., Singer I.M., Analytic torsion for complex manifolds, Ann. of Math. 98 (1973), 154-177.

Xiaonan Ma Humboldt-Universitat zu Berlin Institut fur Mathematik Rudower Chaussee 25 D-12489 Berlin, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 145, 419~437 © 2003 Birkhiiuser Verlag Basel/Switzerland

Regularisation of Secondary Characteristic Classes and Unusual Index Formulas for Operator-valued Symbols

Grigori Rozenblum

Abstract. The usual expression for the secondary characteristic classes is regularized so that it fits for the case of Hilbert bundles. This gives index formulas for pseudodifferential operators with operator-valued symbols.

1. Introduction

The present article continues the paper [14] where analytical index formulas for pseudodifferential operators in the Euclidean space with operator-valued symbols were obtained. The special feature of these formulas was that they look different compared with usual index formulas, well known for the case of scalar or matrix symbols: the latter simply do not make sense in the operator-valued situation since they contain the trace of non-trace-class operators. New formulas, involving certain regularisation based on the suspension homomorphism in cyclic co-homologies, can be naturally called 'unusual index formulas'. In [14], the operators in the Euclidean space were considered. Now we turn our attention to operators acting on arbitrary compact manifolds and obtain unusual formulas for this case. Again, as in [14], there are two main ingredients in our work. First, is the analysis of a special class of operator-valued symbols, possessing the property of improvement under differentiation in co-variables, this improvement expressed in the terms of Shatten classes. The second ingredient is the regularisation of classical Chern-Weyl formulas for the characteristic classes involved in the index formulas, by means of cyclic co-homology methods.

The co-homological form of the classical Atiyah-Singer index formula can be expressed in two versions. The more usual one involves the Chern character, the even co-homological class on the co-ball bundle of the manifold constructed from the symbol of the operator and connections on the start and target vector bundles.

2000 Mathematics Subject Classification. Primary 35P15; Secondary 47 AlD, 47B38, 47B65, 47D06. Key words and phrases. Pseudodifferential operators, index, characteristic classes, cyclic cohomology. This work was completed with the support of grant from the Swedish Royal Academy of Sciences.

420 G. Rozenblum

Another version, applicable only for the case when these latter bundles are isomorphic, involves the Chern-Simons character, an odd-dimensional co-homology class constructed from the symbol of the operator.

Consider the most simple case of an elliptic zero-order operator in ]R.m with symbol a(x,~), invertible outside some ball BR (with boundary SR) in ]R.2m. The index of the pseudodifferential operator Ops(a) depends only on the values of the symbol a on the sphere SR, or, to be more exact, on the class of this symbol in the group KI(SR)' Here, the two types of formulas mentioned above are

ind Ops(a) = J eh2m(a), (1.1)

BR

and

ind Ops(a) = J eS2m-l(a). (1.2)

SR

The analytical expression for the Chern character, or, to be more precise, its highest degree component eh2m(a) (which is the only one involved in (1.1)) is given by

eh2m(a) = em tr (((dr + r(da)r)dar), (1.3)

where r is a regularizer for a, i.e., symbol coinciding with a-I outside the ball B R , and the 2m - I-degree component of the Chern-Simons character eS2m-r(a) equals

(1.4)

Note, that (1.2), (1.4) involve only those values of the symbol on which the index actually depends, while (1.1), (1.3) contain values of the symbol inside the ball, although the index does not depend on these values. This circumstance favors odd index formulas, admitting a more natural interpretation for the case of more general symbol algebras, not being algebras of (matrix- or operator-valued) functions on the co-spheric bundle (see [10],[3]).

When we pass to the operator-valued case (to emphasize this, we boldface operator symbols), the analogy of (1.1), (1.3) requires the symbol r to be a very good regularizer of a: the operators ar - 1, ra - 1 must belong to the trace class. Thus, although the index (for a proper class of symbols) still depends only on the values of the symbol on the sphere, a special continuation of the symbol inside the ball is required, admitting this nice regularizer. On the other hand, (1.2), (1.4) still express the index in the terms of the 'essential' values of the symbol a, however, a new condition arises: the operator under the trace sign in (1.4) must belong to the trace class - but, in typical applications, it does not. This situation is dealt with in [14], where the expression (1.4) for the Chern-Simons character was modified, or, in other words, regularized, by algebraic methods, so that it makes sense for wider (and, in fact, arising in applications) classes of symbols and, substituted into (1.2), gives a correct expression for the index of the pseudodifferential operator.

Regularisation of Secondary Characteristic Classes 421

The approach in [14] originates in [13], where a similar regularisation procedure was used for Toeplitz operators on the line, with operator-valued symbols.

On a general manifold X, co-homological index formulas have a more complicated structure; they contain the full Chern (Chern-Simons) character as well as the Todd class reflecting the topology of the manifold. The analytical description of these formulas can be found, for example, in [6]. Analogies of (1.1)-(1.4) are

where 1

ind Ops(a) = J T(X)ch(a), (1.5)

T*X

ind Ops(a) = J T(X)sc(a),

S*X

ch(a) = Tr (exp((dr + rdar)da) - 1),

1 11 (a-1da)2t(1 - t) cs(a) = ----; Tr(a-1daexp(- . ))dt,

m 0 2m

(1.6)

(1. 7)

(1.8)

and T(X) = LT4j is the complete Todd class of the manifold X. Again, in the operator-valued case, the operators under the trace sign in (1.5), (1.7) belong to the trace class provided a proper continuation of the symbol a from the cospheric bundle S* X to the co-ball bundle B* X is constructed, admitting an upto-trace-class-error regularizer r. Formulas (1.6), (1.8), on the other hand, do not make sense unless the operator-valued form a-Ida belongs to some Shatten ideal, depending on the dimension and topology of the manifold (generally, the need for trace class property may arise). It is not reasonable to restrict the consideration to such operator symbols. In fact, as it is well known, pseudo differential operators acting in the space of functions with values in infinite-dimensional spaces arise in a natural way in the analysis on singular manifolds (see, e.g., [4], [15]). The properties of the symbols depend, to begin with, on the dimension of the manifold and on the form of the singularity, therefore, they can't be postulated arbitrarily. Thus, some regularisation of odd index formulas (1.6), (1.8), and, in particular, of the Chern-Simons character, is needed. An additional feature of our problem is that, unlike the matrix case, homogeneous symbols are not the interesting ones. The symbols arising in applications are quite far from being homogeneous, and therefore even the notion of the principal symbol does not make immediate sense.

In the present paper we extend analysis of classes of operator-valued symbols, introduced in [14], to the case of manifolds. We explain what is the natural notion of the principal symbol for this class, and it is in the terms of this principal symbol that the index of the operator must be expressed. Similar to [14], we define an index as a homomorphism from the K1-group of an algebra of symbols to Z. To find the

lCompared with the usual formula, the unit operator is subtracted in the expression (1.7). This does not effect the index since T(X) contains only dx differentials, so the result of integration in (1.5) does not change. On the other hand, in the infinite-dimensional case, it is necessary to subtract the identity operator, otherwise trace never makes sense.

422 G. Rozenblum

analytical description of this homomorphism, we construct regularisations of the Chern-Simons character (1.8) related to these symbols. As a result, we obtain new index formulas for elliptic pseudodifferential operators with operator-valued symbols. Note that, unlike the finite-dimensional case, we do not lose in generality when passing to the odd-dimensional index formulas: infinite-dimensional Hilbert bundles are always trivial.

To make the exposition more independent of [14], we describe here the general scheme of regularisation.

Let 6 be a Banach *-algebra with unit, M(6) be the set of matrices over 6. The groups Kj(6),j = 0,1 are the usual K-groups in the theory of Banach *algebras, in particular, Kl (6) consists of equivalence classes of invertible matrices in M(6), i.e., elements in GL(6). If 6 does not have a unit, one attaches it and thus replaces M(6) by M(6)+ in the latter definition. (We use boldface K in order to distinguish operator algebras K -groups from topological ones.) The notion of these K-groups carries over to local *-sub-algebras of 6, i.e., sub-algebras closed with respect to the holomorphic functional calculus in 6. Important here is the fact that the K-groups of a dense local sub-algebra in 6 are isomorphic to the ones of 6 (see, e.g., [1]).

For a normed *-algebra 6, the group C~(6) of cyclic co chains consists of (k+ 1)- linear continuous functionals ep(ao, aI, ... ,ak), which are cyclic in the sense ep(ao, aI, ... ,ak) = (-I)kep(al' a2, ... ,ao). The Hochschild co-boundary operator b: C~(6) ---+ C~+1(6) generates, in a usual way, co-homology groups HC~(6).

There is a coupling of HC;v-l(6) and Kl(6) (see [3], Ch. III.3):

[ep] Xv [a] = "'(v(ep 0tr)(a- 1 -1,a -1,a-1 -1,a -1, ... ,a-1 -1,a -1), (1.9)

where tr is the matrix trace and "'(v is tilt ~!'Jfmalisation constant, chosen in [2] to be equal to (2i)-1/22- 2v+1 f(v+!), for functoriality reasons. The sense of constructing index formulas is to find a proper cyclic co cycle ep, for which the coupling (1.9) gives the index of an operator with symbol a under a certain 'quantisation', i.e., the rule associating an operator to the symbol.

The suspension (or periodicity) homomorphism in cyclic co-homologies is S : HC;v-l(6) ---+ HC~V+l(6). This operation is not, in general, an isomorphism. In fact, it is a monomorphism, with range isomorphic to the kernel of the homomorphism I associating to every cyclic co cycle representing a class in H C~v+ 1 (6), the class of the same co cycle in the Hochschild co-homology group H2v+l(6) (see [3], III.1."'(). The suspension homomorphism is consistent with the coupling (1.9):

(1.10)

Based upon (1.10), the regularisation goes in the following manner. Suppose that for a certain 'nice' sub-algebra 6 0 in 6, we have a functional

on GL(60 ) being a sum of expressions of the form (1.9)

<J>(a) = 2:)epj] X Vj [a], j

(1.11)

where 'Pj are some cyclic co cycles of degree 2vj - 1. For a wider *-algebra 6, the co cycles 'Pj may be not extendible by continuity. However, when we apply sufficiently many times the suspension S to functionals 'Pj in (1.11), it may (and in fact, it often does) turn out that the suspended functionals admit such extension by continuity. In particular, when the original functional describes the index of the operator with symbol a under a certain sanitation, the suspension produces unusual index formulas on 6.

In [14], this strategy was implemented for pseudodifferential operators in the Euclidean space, where the original expression (1.11) for the index involves only one term. For general manifolds, (1.6)-(1.8) contains, generally, the sum of several such terms, see (1.6). So we take as the integral of the product of the ChernSimons character of the symbol a with an arbitrary scalar closed form 7jJ. After applying the iterated suspension, we arrive at an expression of the functional, being the integral of 7jJ multiplied by a new form, which is natural to consider as the regularized Chern-Simons form, since it defines the same co-homology class. Substituting this latter form into (1.6) we get unusual index formulas.

The research was partly supported by grant from the Royal Swedish Academy of Science. The author is grateful to Professors B. Fedosov, B.-W. Schulze and B. Sternin for helpful discussions.

2. Operator-valued symbols

The calculus of pseudodifferential operators with operator-valued symbols, acting on functions defined in a domain of the Euclidean space, was developed in [14]. The main feature of this calculus was the choice of the notion of negligible operator. In the classical calculus, one usually considers infinitely smoothing operators as negligible, and it is in the terms of this notion, that the main facts of calculus are expressed, such as pseudo-locality, composition formula, change of variables etc. In the operator-valued case, infinitely smoothing operators are not sufficiently nice. In [14], we proposed to consider the trace class operators as negligible for the needs of calculus.

In the present section we establish further properties of this operator-valued calculus, having in mind the study of operators on compact manifolds. We start with the definition of the classes SJ consisting of symbols, being bounded operators in some separable Hilbert space R. These classes involve a certain sufficiently large integer N, concrete choice of which depends on particular applications. By I . Ip we denote the norm (quasi-norm for p < 1) in the Shatten ideal sp.

Our symbols satisfy the following conditions.

Definition 2.1. Let 'Y :s; 0, q > O. The class SJ = SJ (jRrn X jRrn') consists of

functions2 a(y, TJ), (y, TJ) E jRrn x jRrn', such that for any (y, TJ), a(y, TJ) is a bounded

2Not following the unwritten tradition, we denote the variables by (y, 1]), and not by (x, ';). This somehow reflects the fact that in the most usual situation, when considering pseudodifferential

424 G. Rozenblum

operator in .st and, moreover,

IID~D~a(y, 1])11 ::; C a ,/3(l + 11]1)-la l+"I, (2.1)

ID~D~a(y,1])I_-Y~IQI ::; Ca ,/3, lad::; I'+q, (2.2)

ID~D~a(y,1])ll ::; Ca ,/3(l + 11]I)Q+"I-la l, lal > I' + q (2.3) for lal, 1131 ::; N (where all derivatives are understood in the strong sense).

Remark 1. In [14] the estimate (2.2) was required for alllal, 1131 ::; N. Our definition here introduces a somewhat larger class, which follows from the obvious inequality lall ::; lal p ilail l - p , p < 1. In fact, it is (2.3) that is actually needed in the proofs of all the facts in the pseudodifferential calculus (this is mentioned in [14] and one can even see this in the formulation of the key Proposition 2.2 below), while it is easier to check (2.3) in concrete situations.

Having a symbol a(y, y', 1]) E SJ (1R2m X IRm), we define the pseudo differential operator with this symbol as

(OP S(a)u)(y) =(a(y, y', Dy)u)(y) = (21f)-m J J ei(Y-Y')'l)a(y, y', ry)u(y')d1]dy',

(2.4) where u(y) is a function on IRm with values in .st. In particular, if a does not depend on y', this is the usual formula involving the Fourier transform:

a(y, Dy)u = OPS(a) = F- l a(y, 1])Fu.

The following proposition proved in [14] gives a sufficient condition for a pseudodifferential operator to belong to trace class.

Proposition 2.2. Let the operator-valued symbol a(y, y', 1]) in 1R2m X IRm be smooth

with respect to y, y', let all y, y'-derivatives D~ D~; a up to some (sufficiently high) order N be trace class operators with trace class norm bounded uniformly in y, y' . Suppose that g(y), h(y) = O((1+lyl)-2m). Then the operator ha(y, y', Dy)g belongs to 51(L2(lRm ; .st)).

All usual properties of the pseudodifferential calculus hold for our operatorvalued version, with trace class operators acting as negligible. In particular, the composition rule takes the following form (see Theorem 3.6 in [14]).

Proposition 2.3. If a E SJl, b E SJ2 ,1'1,1'2 ::; 0, then, for N large enough,

where

h(y)Ops(a)Ops(b) - h(y)Ops(a ON b) E 51,

aONb= 2:: (a!)-lD a a8/3b lal:SN

and h(y) is a function decaying as (1 + Iyl)-m-l.

(2.5)

operators on a manifold X with singularities lying on a submanifold Y, it is the variables on T*Y which the operator symbol depend on.


The property of pseudo-locality established in [14] (Theorem 3.5), has the following form.

Proposition 2.4. Let the symbol a(y,y',ry) belong to SJ(ffi2n X ]Rn) for some q > 0, ry s: 0, let 'P, 1/J be bounded functions with disjoint supports, at least one of them being compactly supported. Then (for N large enough) the operator 'Pa(y, y', D)1/J belongs to 51 (L2(]Rm;.fl.)), moreover, the 51 -norm of this operator is controlled by Leo-norms of 'P, 1/J, the distance between their supports, and constants C a ,{3 in (2.1), (2.2), (2.3).

In order to consider an operator on manifolds, we need this property in a somewhat stronger form.

Theorem 2.5. Let the symbol a(y, y', ry) belong to SJ(]R2m X ]Rm) for some q > 0, ry s: 0, have compact support in one of variables y, y' and, moreover, a(y, y', ry) = bey, y', ry)ly - y'I 2N for Iy - y'l s: 6, for some fixed positive 6 > 0, large enough N and a symbol b E Sr Then the operator a(y, y', D) belongs to the trace class, with trace class norm estimated to be the constants Ca .{3 in (2.1), (2.2), (2.3) and similar constants for b.

Proof. The proof follows the lines of the reasoning in [14] leading to Proposition 2.4. The first-order partial differential operator L = L(D1)) = -ilY - y'I- 2(y -y')D1) has the property Le i 1)(Y-Y') = ei1)(Y-Y'), and since Iy - y'I- N a does not have singularities, we can insert LN into the expression for any N. After integration by parts (first formal, but then justified in the usual way), we obtain that (2.4) equals

(21T) -rn J e i 1)(Y-Y') 'P' (y) Iy - y'1-2N ((y - y')Dr/)N a(y, y', ry )1/J' (y')u(y')dy' dry. (2.6)

Now, provided N is large enough, we arrive at the conditions of Proposition 2.2. 0

We note an important particular case of Theorem 2.5.

Corollary 2.6. If the symbol a(y, y', ry) in SJ vanishes in a neighborhood of the diagonal y = y', then the operator (2.4) belongs to the trace class.

Now we are able to establish for our classes the analogies of two important results of the scalar calculus (see, e.g., [8] ).

Theorem 2.7. An operator with the symbol a(y, y' , ry) E SJ which has compact support in one of the variables y, y' can, up to a trace class error, be represented as an operator with some other symbol a', not depending on y', moreover

a'(y, ry) = L (0'!)-1 D~,8~a(y, y', ry)ly,=y' lal~N

(2.7)

The proof literally follows the (rather long one) in [8]. One considers the starting section of the Taylor expansion of the symbol a near the diagonal y' = y in y - y'. These terms give the expression in (2.7). The remainder term has zero

426 G. Rozenblum

of high order on the diagonal, and therefore gives a trace class operator, according to Theorem 2.5. D

The second important fact, formulated but not proved in [14], concerns the behavior of pseudo differential operators under the change of variables.

Theorem 2.8. Let the symbol a(y, 7]) belong to SJ and have compact support 0 in y. Let z = ",(y) be a diffeomorphism defined on a neighborhood oj O. Then, up to a trace class error, the operator

B = '" 0 Ops(a) 0 ",-1 (2.8)

is a pseudodifferential operator with the usual expression Jor the symbol b(z, () oj B, in particular,

(2.9)

Again, the prooJfollows the pattern in [8]. First, cut off the complement to a neighborhood of the diagonal y = y', which, according to Corollary 2.6 gives a trace class perturbation to the operator. Then, after the change of variables, the operator B = '" 0 Ops(a) 0 ",-1 takes the form

Bv(z) = J J ei(K-'(z)-K-'(z'»7)a(",-1(z),7])u(",-1(z))18yj8zldzd7].

Now, the Kuranishi trick, the linear change of variables 7] = G(y, y')( near the diagonal y' = y, transforms B to the usual pseudodifferential form, with the phase function (z - z')( and with a certain symbol depending on z, z', (, which, by means of Theorem 2.7 and Corollary 2.6 reduces to a symbol not depending on z'. D

This latter result enables one to define pseudodifferential operators on compact manifolds.

Let Y be a compact manifold of dimension m. We consider the space L 2 (Y) = L2 (Y, Ji), where Ji is an infinite-dimensional Hilbert space. (There is no need to consider Hilbert bundles over Y, since all such bundles, unlike the finite-dimensional case, are trivial.)

Definition 2.9. An operator A in L2 (Y) is called pseudodifferential in the class LJ (Y) if for some co-ordinate covering Y by open sets U, with co-ordinate mappings "'L to domains in ]Rm, a partition of unity 'PL subordinated to U, and another system 'l/JL of cut-off functions with supports in UL such that 'l/J,'PL = 'PL, the operators

"'L'l/J,A'PL",-;l (2.10)

are pseudodifferential operators with symbols in SJ, plus some trace class operator.

According to Theorem 2.8, the property described by this definition does not depend on the choice of the covering, cut-offs and mappings. Moreover, the symbols at a point y E Y obtained by different choices are related by the formula of the type (2.9).


Definition 2.10. A pseudo differential operator A E .c~ is called elliptic if for any point Y E UL and cut-off functions lP"1/J,, such that lPL(Y) =I- 0, the symbol bL(z,() of operator (2.10) is invertible for z in a neighborhood of "'L(Y) and for sufficiently large 1(1, with norm-bounded inverse.

Theorem 2.8 implies that the property of the operator to be elliptic does not depend on the choices made in the definition.

In the usual, finite-dimensional, pseudo differential calculus, one easily associates canonically to a pseudo differential operator on a manifold its leading symbol, being a matrix defined on the co-spheric bundle on the manifold; there, the crucial role is played by the homogeneity property of symbols. In the operatorvalued context, homogeneous symbols are not the ones which are interesting for applications (see, e.g., [4], [15], [14]). In what follows we are going to associate an invertible operator family on the co-spheric bundle to an elliptic operator. This association, although non-canonical, generates a homomorphism from the group of elliptic operators to the Kl-grouP of a proper algebra of operator-valued functions. We follow the pattern of the construction in [9]. For the given elliptic operator A, fix some co-ordinate covering, cut-off functions and co-ordinate mappings, as in Definition 2.9. Fix also some Riemannian metric on Y. For each neighborhood UL , let b L (z, () be the symbol of the corresponding to UL pseudodifferential operator in a domain in the Euclidean space. Define then the operator function on T*UL as aL(y, ry) = bL("'L(Y), TDK,L(y)-lry). This symbol is invertible for Iryl large enough (the length is understood in the sense of the Riemannian metric). Symbols obtained in this way over the same point in the intersection of co-ordinate neighborhoods differ by a term in S;; 1; this implies that by gluing such symbols together by means of a partition of unity, we get an operator function a on T*Y, invertible for large Iryl. Let R be so large, that this a is invertible for Iryl 2: R. We associate to our operator the invertible operator-function c(y, ry) = a(y, Rry), for Iryl = 1. One easily checks that this operator function is differentiable in y, and it improves its properties under ry-differentiation: if Tv are vector fields on S*Y which are tangent to S*Yly at every point Y E Y then

TVJTV2 ... TVkc(y, ry) E 5* (.It), k ~ q. (2.11)

Denote the class of such symbols, satisfying (2.11) for k = 1 by 6 q (S*Y). Then, as it follows from our construction, different choices of coverings, cut-offs etc. lead to homotopic elements in GL (6 q (S*Y)). Thus, we have associated to any elliptic operator A a class [a(A)] in the group K 1(6 q (S*Y)). It is clear that this is a homomorphism.

As is usual in the elliptic theory, the index homomorphism factors through the above symbol homomorphism.

Theorem 2.11. Elliptic operators in .c~(Y) are Fredholm, and the index depends only on its class [a(A)] in K 1(6 q (S*Y)).

Proof. The reasoning goes along the standard lines, following the proof in the matrix case, say, as in [5],[14]. We explain the main issues, emphasizing the operator

428 G. Rozenblum

specifics. First, the Fredholm property is proved by constructing the regularizer R for the elliptic operator A. In our case, it is important that it regularizes A up to a trace class term:

(2.12)

This is performed by the usual construction, first locally, then gluing together local representations. One takes a symbol ro E S~ which equals a-I for large 17]1, as the initial approximation for r. Then one adds 'lower order terms' to ro, to compensate the extra terms in the composition formula (2.5), and as a result, the operator version of the composition rule gives (2.12). For large 17]1, the terms in rON a, a ° N r contain products of a, a-I and their derivatives. Let rON al M denote the sum of those terms in rON a where the sum of orders of derivatives does not exceed 2M. This symbol, for N ~ M, does not depend on N, and, for M large enough,

Ops(r ON aiM) - 1, Ops(a ON riM) - 1 E 51. (2.13)

Moreover, the symbols ro N aiM -I, ao N riM -1 have compact support and belong to the trace class.

The next step is to obtain a rough analytical index formula. It has almost the same form as the analytical index formula (0.6) in [5] (the additional parameter was present there, for bookkeeping the orders of derivatives):

ind OPS(a) = (27T)-m J Tr (a ON riM - r ON aiM )dyd7]. (2.14)

T*Y

One interprets (2.14) in the following way. Of course, the concrete expression for the integrand depends on the co-ordinate system and is not a density on T*Y. However, as it is shown in [5], after integration in 7], the integrand in (2.14) becomes a density on Y, thus it can be integrated over Y. The proof of (2.14) is the same as for the analogous formula for the Euclidean space in [14] (see Proposition 4.1 there).

In fact, the particular form of the expression in (2.14) does not matter. The only thing which is important, is that (2.14) involves integration of some expression containing the symbol and its derivatives over a compact set in T*Y. This enables us to repeat the reasoning in [14]. First, it follows from (2.14) that if two elliptic symbols al and a2, invertible for 17]1 ~ R, coincide in the co-ball bundle B'RY = {17]1 ::; R}, then the indices of operators coincide: in fact, the integrands in (2.14) for al and a2 coincide inside B'RY and are zero outside it. Further, if two such symbols are equal only on the co-spheric bundle S'RY = aB'RY, then their indices still coincide since they can be made equal inside BR Y by means of a homotopy of elliptic symbols (in fact, the homotopy acting inside BRY only). Finally, if the elliptic symbols aI, a2 are invertible on and outside <SRI Y, <SR2 Y and their values on these co-spherical bundles are homotopic in the class of invertible symbols, they can be deformed by a homotopy in the class of elliptic symbols to the previous situation. We refer to Section 4 in [14] for the details of the construction. 0


Thus it makes sense, for a given invertible element a E 6 q , to denote by ind ([a]) the index of any elliptic pseudodifferential operator A on Y, for which a(A) = [a].

Remark 2. A result similar to Theorem 2.11 was formulated in [12]; the proof there was given for homogeneous symbols, while for general, 'hypo-elliptic' symbols it was only declared that the approach in [9] can be used. This may be correct, but not automatically. In the approach by Hormander, the case of non-homogeneous symbols is reduced to the case of homogeneous ones by means of an approximation procedure, where the given operator and its adjoint are approximated by operators with homogeneous symbols in the strong sense. This works fine for the usual matrix-valued symbols. However, in the infinite-dimensional case, strong convergence behaves badly under the operation of taking adjoint, and this surely would require some extra conditions to be imposed on the symbols, as well as cause some other complications.

3. Regularisation

Now we will follow the general scheme of [14] in finding the regularisation of the Chern-Simons character of an automorphism of a Hilbert bundle.

Let M be a compact manifold of dimension nand jt be an infinite-dimensional Hilbert space. By 6 0 = 6 0 (M, jt) we denote the algebra of functions on M with values being finite rank operators in jt. Thus GL(60 ) is the set of invertible operator-valued functions on M having form l+k with k E 6 0 . For any q E (0,00), the algebra 6~ consists of functions having values in Sq(jt). This algebra produces GL(6~). We always suppose that differentiation preserves the Shatten class properties of the operator-functions in question. In natural topologies, GL(6~) is dense in GL(60 ).

For a E GL(60 ), the Chern Simons character is given by the formula

1

cs(a) = --. tr(a- 1daexp(--.t(1- t)(a- 1 da)2))dt 1 J 1 27TZ -2m

o

= I: Pv tr ((a-1da?v-l), (3.1) v=l

where 1 v (v - I)!

PV=-(27Ti) (2v-l)!' (3.2)

This expression makes sense for a E GL(60 ), and, moreover, for a E GL(6~), since the expression under the trace sign contains at least one trace class operator. Our task is to find some other expression, valid for a E GL(6~), for a given q, and defining the same co-homology class as (3.1).

430 G. Rozenblum

We consider a single term in (3.1), with a fixed v. Take an arbitrary closed (scalar-valued) form 'ljJ of degree n - (2v - 1) and consider

'ljJ[a] = ('ljJ, cs(a)) = J'ljJcs(a) (3.3)

M

for a E GL(6~), so (3.3) makes sense. We can rewrite (3.3) as

'ljJ[a] =!3v J 'ljJtr «a- l da)2v-l)

M

= (-lr- l !3v J 'ljJtr «a- l - 1)d(a - 1)d(a- l - 1) ... d(a - 1)). (3.4)

M

Consider the multilinear functional, cochain, on 6~, corresponding to (3.4):

<p,p(ao, al,···, ak) = (-lr- l !3v J 'ljJ tr (aOdal .,. dak)i ao,···, ak E 6 1 , k = 2v-1.

M

Due to the cyclicity property of the trace and anti-commutativity of the multiplication of the forms, this cochain is cyclic:

<p,p(al,a2, ... ,ak,ao) = (-lr- l !3v J'ljJtr(alda2 ... dao)

M

= (-It- l !3v J tr (dao'ljJal ... dak) = (-lr- l ,8v J 'ljJ tr (d(aoaI) ... dak)

M M

-( -It- l !3v J 'ljJ tr (aodal ... dak) = -<p,p(ao, ... , ak).

M

Here we used the fact that the scalar form Tr (d(aoaI) ... dak)) is exact. A similar calculation shows that the cochain <p,p is closed with respect to the Hochschild co-boundary operation,

(b<p,p)(ao, ... ,ak+I) = <p,p(aoal, ... ,ak+d - <p,p (ao, ala2, ... ,ak+d + ...

+<p,p(ak+lao, ... , ak) = O. Thus <p,p is a cyclic cocycle and it defines a class in the cyclic co-homology

group HCk(6~). Now we apply the iterated suspension homomorphism S to the cocycle <p,p. In the calculations below we follow the reasoning in [14], with modifications needed to deal with our new situation. We use the representation of S in the terms of the universal differential graded algebra for 6~ as in [3], [10].

In general, for a given algebra 6, denote by 6 the algebra obtained by adjoining the unit 1. For each kEN, let Ok be the linear space

Ok = Ok(6) = 606 6®ki 0 = E90k.

The differential d : OJ.! ---+ OJ.!+l is given by

d«ao + .Xl) ® al 0··· 0 ak) =.Xl 0 ao 0···0 ak E Ok+l.


By construction, one has d2 = O. One defines a right 6-module structure on Oil by setting

k

Uio Q9 al Q9 .•. Q9 ak)a = 2:) -l)k-iao Q9 •.. Q9 aiaHI Q9 .•. Q9 a. i=O

This right action of 6 extends in a natural way to the product in 0: - - k' w(bo Q9 b I Q9 ... Q9 b k ) = (wbo) Q9 b I Q9 •.. Q9 b k , W EO.

The product satisfies

aOdaI ... dak = ao Q9 al Q9 ... ® ak, ai E 6

and gives 0 the structure of a graded differential algebra. This algebra is universal in the sense that any homomorphism p of 6 into some differential graded algebra (O',d') extends to a homomorphism p* of (O,d) to (O',d') respecting the product of differentials:

The particular situation where we use this universality property is for 6 = 6~ and for the differential algebra 0' being the algebra of operator-valued differential forms on M,

w' = aOd'aI ... d'ak, ai E 6~

where in the product, for the terms of the form

d'a = 2:aidxi'

the usual product of operators and the exterior product of differentials is used. Thus in the degree zero, two differential algebras coincide, and we take identity as the homomorphism p involved in the definition of the universality property. We will omit the prime symbol in the sequel.

According to Proposition 4, Ch. III.1, in [3], any cyclic co cycle rp E C~(6) of dimension k can be represented as

rp(ao, ... ,ak) = cp(aOdaI ... dak),

where cp is a closed graded trace of dimension k on 0(6). In our particular case, for the cocycle rp7jJ of degree k = 2v - 1, this representation is generated by

cp7jJ(aOdaI .. . dak) = (-It-I;]" J 7,btr(aOdaI ... dak),

!vI

where b" are given in (3.2). In the terms of the above model, on the co cycle level, S admits the follow

ing description. For the algebra of complex numbers C, we consider the graded differential algebra 0(6) Q9 O(C), with elements having the form

(ao ® wO)d(aI Q9 WI)'" d(ak Q9 Wk),

432 G. Rozenblum

where Wo, .. . , Wk E <C, with differential

d(a Q9 w) = (da) Q9 W + a Q9 dw.

For a cyclic co cycle r.p E C~(6) and cyclic cocycle a E Cf(q, following [3], we define the cup product r.prta E C~+P(6 Q9 q = C~+P(6) by setting

r.prta(ao, ... , ak+p) = (cp Q9 G-)((ao Q9 e)d(a1 Q9 e) ... d(ak+p Q9 e)). (3.5)

Here, e is the unit in C, i.e., the element 1 + 01 E t, cp, G- are graded closed traces of degree, respectively, k and p in n(6), n(C) representing r.p, a and thus only terms ofbidegree (k,p) survive in (3.5). It is shown in [3] that r.prta is a cyclic cocycle. In particular, take 'l/J = a1 E C~(C), a1(e,e,e) = 1. Cup product with a1 generates the homomorphism S in cyclic co-homologies.

Now we consider iterations of S. For an even integer p = 2l, we set az = a~l where rtl denotes taking to the power l in the sense of rt operation. According to Corollary 13, Ch. IIL1 in [3], az(e, e, ... , e) = lL To the cocycle az, there corresponds the graded closed trace G-z of degree p on n*(C), moreover G-z(ede ... de) = lL Cup multiplication with the cocycle az generates the iterated homomorphism SZ in cyclic cohomologies of the algebra 6. We will study the structure of SZr.p E

C~+P(6), for r.p E C~(6). According to (3.5),

SZr.p(ao, a1 ... ak+p) = Slr.p((ao Q9 e)d(a1 Q9 e) ... d(ak+p Q9 e))

= cp Q9 G-((ao Q9 e)(da1 Q9 e + a1 Q9 de) ... (dak+p Q9 e + ak+p Q9 de)). (3.6)

Since cp is a graded trace of degree k and G-z is a graded trace of degree p = 2l, only the terms ofbidegree (k,p) contribute to (3.6). There are a lot of such terms, and each of them involves the value of cp on a certain ordered product of ai and dai, where exactly k + 1 factors are of the form ai, including ao, as well as the value of G-z on the corresponding ordered product of the elements e and de, with k + 1 factors e, including the first one, and p factors de. Quite a lot of these terms vanish. In fact, since e is an idempotent, e = e2 , we have

de = e(de) + (de)e, e(de)e = 0, e(de)(de) = (de)(de)e. (3.7)

Therefore, if some term in the expansion of (3.6) contains the product of an odd number of consecutive de surrounded bye, it vanishes. Thus only those terms survive where each group of consecutive de in the product contains an even number of de. For any such product, using (3.7), we can rearrange the factors e, de and arrive at the expression G-z (ede ... de) which equals lL This leaves us with the contribution to (3.6) involving ai and dai. In these terms, the variables ai enter in a very special way. Since, for i # 0, ai stand on the places where we had de in (3.6), and dai on the places where we had e, only those terms survive in (3.6), where an even number of variables ai stand in succession, not counting ao. This gives us the following characterisation of SZr.p.

Proposition 3.1. The image Slep in C~+P of a cyclic cocycle ep E C~ under the iterated homomorphism Sl equals

(3.8)

where the summation is performed over collections of Ui, Vi such that 1 = Vo ::;

Ul < VI < ... < Us ::; Vs = k + p + 1, Ui - Vi are even, Ai = aVi~l ... a ui -l,

Bi = daUi •.. davi-I, 2::(Vi - Ui) = k.

Getting an explicit analytical description of (3.8) for a general collection of ai might be quite troublesome. We, however, are interested only in the value of Slep on a very special collection of these variables. In fact, according to (3.2), (3.4), we are interested in evaluating ep,p(ao, ... ,ak), for a2i = a-I - 1, a2i+l = (a - 1), a E GL(6 1 ), so that ai E 6 1 . This enables us to simplify the expression (3.8).

Proposition 3.2. For a E GL(6?),

Slep,p(a- 1 - 1, a-I, ... , a-I - 1, a-I)

= (l!) -1,8" J 'lj! tr [( d~ )l(a -1 (1 - sc) -Ida)k 18=0] , M

Slep,p(a- 1 - 1, a-I, ... ,a-I - 1, a-I)

= ,8" tr J 'lj!(c + a-Ida)k+l, c = (a - 1)(a-1 - 1).

M

(3.9)

(3.10)

Proof. We have to show that for our specific choice of variables, the expression (3.8) takes the form (3.9), (3.10). We can see that in (3.8), each term Ai equals

(Ui -Vi_I)

C 2 • This means that all terms in (3.8) can be obtained in the following way. Write down the expression a- Idad(a- 1 ) •.• da, with v factors da and v-I factors d(a- 1 ). Before each da, d(a- 1 ) insert several factors c, so that there are l of them altogether. Summing all such products and taking into account that d(a~l) = -a- 1daa- 1 and that c and a commute, we come to the formula

(3.11)

To describe (3.11) more explicitly, introduce an extra variable s and consider the expression depending on s:

(3.12)

For s small enough, 1- sc is invertible, and therefore (3.12) can be rewritten as

(3.13)

434 G. Rozenblum

Now we can see that (3.11) equals the coefficient at sl in (3.13), and this gives us (3.9). As for (3.10), it is clear that the term of degree k = 2// - 1 in the integrand, the only one that has correct degree and thus survives under the integration, exactly equals the integrand in (3.11). 0

Now we are able to find the regularized expression for the functional 'ljJ[a] = IM'lj;cs(a).

Proposition 3.3. For a E GL(6?) and a closed n - (2// - 1) form ?/J,

'lj;[a] = iM 'lj;cs~I2-1 = iM(l!)-I(3v,I'lj;tr [(:s)l(a- l (I-SC)-l da)kl s =o], (3.14)

where -1 l!(// + l - I)!

(3v,1 = (3v'Yl/ 'Yv+l = (2// + 2l - I)!'

"Iv are constants in (1.9). The functional in (3.13) admits extension by continuity to a E GL(6~) for q < 2// + 2l - 1.

Proof. The expression (3.14) for the functional follows immediately from (3.9) and the periodicity relation (1.9). To prove the extension property, note that each summand in (3.9) contains 2//-1 factors da and l factors c; since for a E GL(6~), we have da E 5 q , C E 5~, and therefore the whole product belongs to 51. 0

Since the form 'lj; can be chosen arbitrarily, the relation (3.14) can be expressed in the following way.

Theorem 3.4. For a E GL(6d, the forms CS2v-l (a) and cs~I2-1 (a) define the same co-homology class in H 2v- l (M; C). The latter form can be extended to a E

GL(6~), q < 2// - 1 + 2l.

4. The full Chern-Simons character and unusual index formulas

In order to use the expression for the components of the Chern-Simons character found in Section 3 for calculating the index, we consider the full Chern-Simons character

(4.1) v

We are going to find for cs(l) (a) an expression similar to (3.1), therefore we perform a certain summation procedure for the terms in (3.14).

Consider the formal sum

We can transform it as

h(w) = ~ (// + l-I)! W2v-l. ~(2//+2l-1)!

(4.2)

Setting a = t(l - t)w2, we can write

hew) = w21 - 1 11 el(a)dt,

where el(() = exp(() - 1- (- (2/21 _ ... - (1-1 /(l-l)!. Simple calculations give

el(() = (l1)-1 1C, (( - (')1-1 exp((')d(' ,

thus eXP1(z) = (-lel(z) is the Riesz mean of order l of the exponential function. In this notation, (4.2) takes the form

hew) = 11 (t(l - t))I-lweXP1(t(1 - t)w2)dt.

We apply these formal calculations for w = (-27ri)-1/2(1 - sc)a-1da. The formally infinite series are in fact finite, since they cannot contain forms of order higher than n, and this gives us the regularisation for (4.1).

Theorem 4.1. Let a E GL(6~). Define 1 cs(l) (a) = --.

-27rz

x Tr(dd )1 [ t((l - sc)a- 1da)(t(1 - t))l-l exptCt(l - t)((l - sc)~-lda)2 )dt] 18=0. s 10 27rZ

(4.3) Then cs(l) (a) and cs(a) define the same co-homology class in Hodd(M), more

over cs(l) (a) extends by continuity to a E G L( 6g), q < 2l + 1.

Now we can apply our regularized Chern-Simons character for calculating the index of pseudo differential operators with operator-valued symbols.

For a given m-dimensional manifold Y, we set M = S*Y, n = 2m - 1. We consider an elliptic pseudodifferential operator A having the symbol in Sq. As it is shown in Section 2, the index of the operator A depends only on the stable homotopy class of its principal symbol a in Kl(6 q (S*Y)).

Suppose first that a E GL(6g(S*Y)). Recall that this means that the operator-valued function a is invertible and differs from the identity operator by an operator in the Shatten class Sq.

Theorem 4.2. If a E GL(6g(S*Y)), then

ind A = J T(Y)cs(l)(a), (4.4)

S'Y

where cs(l) (a) is defined in (4.3).

Proof. For a E GL(6~(S*Y)), the index formula (4.4) follows from the usual one and Theorem 4.1. Further, according to Theorem 2.11 and again Theorem 4.1, both parts in (4.4) are continuous in the topology of GL(6g(SY)). D

436 G. Rozenblum

Finally, we extend the index formula to principal symbols in 6 q (S*Y). Recall that these symbols, defined in Section 2, satisfy (2.11). Here the regularisation of the Chern-Simons classes given by (4.3) is not sufficient because for a symbol a E GL(6 q ), the symbol c = (a - 1)(a- 1 - 1) is, generally, only bounded and multiplication by it does not improve trace ideals properties. Therefore one more regularisation is needed.

Proposition 4.3. Let a E 6 q be an invertible symbol on S*Y. Then there exists an invertible operator-valued function ao (y) acting in the space JtEl:l <cd, for some finite d, differentiable on Y so that ao(y) - (a(y, 7]) EI:l1d) ESq.

Remark 3. In fact, this operator ao can be explicitly constructed, given a, but the particular choice is of no importance here.

Proof. Everything is very easy in the case of the manifold Y having vanishing Euler characteristic. For such manifolds, there exists a smooth section 7](y) of the co-spheric bundle of Y, and one can set ao(y) = a(y, 7](y)). For a manifold with a nonzero Euler characteristic, the construction is more complicated. Set a(y,O) = fs*ya(y,7])d7]. The definition of the class 6 q implies that for any 7] E S;(Y),

y

a(y, 0) differs from a(y, 7]), by an operator in Sq(Jt). Since a(y, 7]) is invertible, this implies that a(y,O) is a Fredholm operator in Jt with zero index. Having such a Fredholm family, one can, in a standard way, include a(y,O) as the upper left

corner in a matrix ao (y) = (a~(~ ~) ~~~ ~ ), with operator functions g : <cd ---+

Jt, h : Jt ---+ <cd and k : Cd ---+ <cd, so that this block-matrix is invertible (see e.g., in [7] the details of the construction.) Since the operator-functions g, h, k have finite rank, the difference (a(y,7]) EI:l1d) - ao(y) belongs to Sq. 0

Using this operator function ao(y), we can now establish the index formula. First, note that the symbols a and a = aEl:l1d represent the same class in the group Kl (6 q ), therefore they have the same index. The index of the symbol ao (y) equals zero, since the corresponding operator, multiplication by the operator-valued function ao (y), is an isomorphism. Therefore, the index of the symbol

b(Y,7]) = ao(y)-la(y, 7]) (4.5)

equals the index of the symbol a. But now, the symbol b belongs to the class 6~+, and Theorem 4.2 can be applied. This gives our final result.

Theorem 4.4. For an invertible symbol a E GL(6~), define the symbol b as in (4.5). Then

ind ([aJ) = r T(Y)cs(l)(b), Js*y

where the regularized Chern-Simons character is given by (4.3).


References

[1] B. Blackadar, K-theory for Operator Algebras. Springer-Verlag, New York et al. 1986.

[2] A. Connes, Noncommutative Differential geometry. Publ. Math. I.H.E.S 62 (1985), 257-360.

[3] A. Connes, Noncommutative Geometry. Academic Press, NY, 1994.

[4] C. Dorschfeldt, B.-W. Schulze, Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach. Ann. Global Anal. Geom. 12 (1994), 135-171.

[5] B. Fedosov, Analytic formulas for the index of elliptic operators. Trans. Moscow Math. Soc. 30 (1974), 159-240.

[6] B. Fedosov, Index theorems. Encyclop. Math. Sci. Vol. 65 (Partial Differential Equations, VIII) 155-251, Springer, Berlin, 1996.

[7] B. Fedosov, B.-W. Schulze, N. Tarkhanov, On the index of elliptic operators on a wedge. J. Funct. Anal. 157 (1998), 164-209.

[8] A. Grigis, J. Sjostrand, Microlocal analysis for differential operators. An Introduction. Cambridge Univ. Press, Cambridge, 1994.

[9] L. Hormander, On the index of pseudodifferential operators. Elliptische Differentialgleichungen, Band II, Akademie-Verlag, 127-146, 1969.

[10] M. Karoubi, Homologie Cyclique et K -theorie. Asterisque, 149, 1987.

[11] M. Karoubi, Theorie generale des classes caracteristiques secondaires. K-theory, 4 (1990), 55-88.

[12] G. Luke, Pseudodifferential operators on Hilbert bundles. Journal Diff. Equat. 12 (1972), 566-589.

[13] B. Plamenevsky, G. Rozenblum, Pseudodifferential operators with discontinuous symbols: K -theory and index formulas. (Russian) Funktsional. Anal. i Prilozhen. 26 (1992), 4, 45-56, (Translation in: Functional Anal. Appl. 26 (1993), 266-275).

[14] G. Rozenblum, On some analytical index formulas related to operator-valued symbols. Electr. J. Diff. Equat. (2002), 17, 31 pp.

[15] B.-W. Schulze, B. Sternin, V.Shatalov, Differential equations on singular manifolds: semiclassical theory and operator algebra. Wiley, Berlin, 1998.

Grigori Rozenblum Department of Mathematics Chalmers Univ. of Technology S-41296 Goteborg, Sweden e-mail: [email protected]

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