analysis on h-harmonics and dunkl transforms

Analysis on h -Harmonics and Dunkl Transforms

Feng DaiYuan Xu

Advanced Courses in Mathematics CRM Barcelona

Advanced Courses in MathematicsCRM Barcelona

Centre de Recerca Matemàtica

Managing Editor:Carles Casacuberta

ttp://www.springer.com/series/5038More information about this series at h

http://www.springer.com/series/5038

Feng Dai • Yuan Xu

and Dunkl Transforms

Editor for this volume:

Analysis on h-Harmonics

Sergey Tikhonov, ICREA and CRM, Barcelona

Library of Congress Control Number: Mathematics Subject Classification (2010):

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Department of Mathematics Department of MathematicsUniversity of Oregon

Primary: 41A10, 42B15; Secondary: 42B25, 42B08, 41A17

Feng Dai Yuan Xu

Eugene, OR, USA

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and Statistical SciencesUniversity of Alberta

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DOI 10.1007/978-3-0348-0887- 3

ISSN 2297-0304 ISSN 2297-0312 (electronic)

ISBN 978-3-0348-0886 ISBN 978-3-0348-0887- (eBook) -6 3Advanced Courses in Mathematics - CRM Barcelona

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Contents

Preface vii

1 Introduction: Spherical Harmonics and Fourier Transform 1

1.1 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Dunkl Operators Associated with Reflection Groups 7

2.1 Weight functions invariant under a reflection group . . . . . . . . . . . . 72.2 Dunkl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Intertwining operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Notes and further results . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 h-Harmonics and Analysis on the Sphere 15

3.1 Dunkl h-harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Projection operator and intertwining operator . . . . . . . . . . . . . . . 203.3 Convolution operators and orthogonal expansions . . . . . . . . . . . . . 233.4 Maximal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Convolution and maximal function . . . . . . . . . . . . . . . . . . . . . 313.6 Notes and further results . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Littlewood–Paley Theory and the Multiplier Theorem 35

4.1 Vector-valued inequalities for self-adjoint operators . . . . . . . . . . . . 354.2 The Littlewood–Paley–Stein function . . . . . . . . . . . . . . . . . . . 374.3 The Littlewood–Paley theory on the sphere . . . . . . . . . . . . . . . . 39

4.3.1 A crucial lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.2 Proof of Theorem 4.3.3 . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 The Marcinkiewicz type multiplier theorem . . . . . . . . . . . . . . . . 454.5 A Littlewood–Paley inequality . . . . . . . . . . . . . . . . . . . . . . . 474.6 Notes and further results . . . . . . . . . . . . . . . . . . . . . . . . . . 50

v

vi Contents

5 Sharp Jackson and Sharp Marchaud Inequalities 51

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Moduli of smoothness and best approximation . . . . . . . . . . . . . . . 525.3 Weighted Sobolev spaces and K-functionals . . . . . . . . . . . . . . . . 545.4 The sharp Marchaud inequality . . . . . . . . . . . . . . . . . . . . . . . 565.5 The sharp Jackson inequality . . . . . . . . . . . . . . . . . . . . . . . . 595.6 Optimality of the power in the Marchaud inequality . . . . . . . . . . . . 615.7 Notes and further results . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Dunkl Transform 65

6.1 Dunkl transform: L2 theory . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Dunkl transform: L1 theory . . . . . . . . . . . . . . . . . . . . . . . . . 726.3 Generalized translation operator . . . . . . . . . . . . . . . . . . . . . . 76

6.3.1 Translation operator on radial functions . . . . . . . . . . . . . . 776.3.2 Translation operator for G = Zd

2 . . . . . . . . . . . . . . . . . . 806.4 Generalized convolution and summability . . . . . . . . . . . . . . . . . 82

6.4.1 Convolution with radial functions . . . . . . . . . . . . . . . . . 826.4.2 Summability of the inverse Dunkl transform . . . . . . . . . . . . 846.4.3 Convolution operator for Zd

2 . . . . . . . . . . . . . . . . . . . . 866.5 Maximal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.5.1 Boundedness of maximal function . . . . . . . . . . . . . . . . . 876.5.2 Convolution versus maximal function for Zd

2 . . . . . . . . . . . 906.6 Notes and further results . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7 Multiplier Theorems for the Dunkl Transform 95

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.2 Proof of Theorem 7.1.1: part I . . . . . . . . . . . . . . . . . . . . . . . 967.3 Proof of Theorem 7.1.1: part II . . . . . . . . . . . . . . . . . . . . . . . 1017.4 Proof of Theorem 7.1.1: part III . . . . . . . . . . . . . . . . . . . . . . 1057.5 Hormander’s multiplier theorem and the Littlewood–Paley inequality . . . 1067.6 Convergence of the Bochner–Riesz means . . . . . . . . . . . . . . . . . 1087.7 Notes and further results . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography 111

Index 117

Preface

These lecture notes were written as an introduction to Dunkl harmonics and Dunkl trans-forms, which are extensions of ordinary spherical harmonics and Fourier transforms withthe usual Lebesgue measure replaced by weighted measures.

The theory was initiated by C. Dunkl and subsequently developed by many authorsin the past two decades. In this theory, the role of orthogonal groups, which provide theunderline structure for the ordinary Fourier analysis, is played by a finite reflection group,the partial derivatives are replaced by the Dunkl operators, which are a family of commut-ing first order differential and difference operators, and the Lebesgue measure is replacedby a weighted measure with the weight function hκ invariant under the reflection group,where κ is a parameter. The theory has a rich structure parallel to that of Fourier analysis,which allows us to extend many classical results to the weighted setting, especially in thecase of h-harmonics, which are the analogues of ordinary spherical harmonics. There arestill many problems to be solved and the theory is still at its infancy, especially in the caseof Dunkl transform. Our goal is to give an introduction to what has been developed so far.

The present notes were written for people working in analysis. Prerequisites onreflection groups are kept to a bare minimum. In fact, even assuming the group is Zd

2,which requires essentially no prior knowledge of reflection groups, a reader can still gainaccess to the essence of the theory and to many highly non-trivial results, where theweight function hκ is simply

hκ(x) =d

∏i=1

|xi|κi , κi ≥ 0, 1 ≤ i ≤ d,

the surface measure dσ on the sphere Sd−1 is replaced by h2κ dσ , and the Lebesgue mea-

sure dx on Rd is replaced by h2κ dx.

To motivate the weighted results, we give a brief recount of basics of ordinary spher-ical harmonics and the Fourier transform in the first chapter, which can be skipped alto-gether. The Dunkl operators and the intertwining operator between partial derivatives andthe Dunkl operators, are introduced and discussed in the second chapter. The intertwin-ing operator plays a key role in the theory as it appears in the concise formula for thereproducing kernel of the h-spherical harmonics and in the definition of the Dunkl trans-form. The next three chapters are devoted to analysis on the sphere. The third chapter isan introduction to h-harmonics and essential results on harmonic analysis in the weighted

vii

viii Preface

space. The Littlewood–Paley theory on the sphere is developed in the fourth chapter, andis used to establish a Marcinkiewicz type multiplier theorem on the weighted sphere.As an application, two inequalities, the sharp Jackson and sharp Marchaud inequalities,are established in the fifth chapter, which are useful for approximation theory and in theembedding theory of function spaces. The final two chapters are devoted to the Dunkltransform. The sixth chapter is an introduction to Dunkl transforms, where the basic re-sults are developed in detail. The Littlewood–Paley theory and a multiplier theorem areestablished in the seventh chapter, using a transference between h-harmonic expansionson the sphere and the Dunkl transform in Rd .

The topics reflect the authors’ choice. There are many results for Dunkl transformson the real line (where the measure is |x|κ dx) that we did not discuss, since the setting onthe real line is closely related to the Hankel transforms and often cannot even be extendedto the Zd

2 case in Rd . There are also results on partial differential-difference equations,in analogy to PDE, that we did not discuss. Because of the explicit formula for the inter-twining operator, the case Zd

2 has seen far more, and deeper, results, especially in the caseof analysis on the sphere such as those for Cesaro means. We chose the Littlewood–Paleytheory and the multiplier theorem, as this part is relatively complete and the results arerelated in the two settings, the sphere and the Euclidean space.

These lecture notes were written for the advanced courses in the program Approxi-mation Theory and Fourier Analysis at the Centre de Recerca Matematica, Barcelona. Weare grateful to the CRM for the warm hospitality during our two months stay, to the partic-ipants in our lectures, and thank especially the organizer of the program, Sergey Tikhonovfrom CRM, for his great help. We gratefully acknowledge the support received fromNSERC Canada under grant RGPIN 311678-2010 (F.D.), from National Science Foun-dation under grant DMS-1106113 (Y.X.), and from the Simons Foundation (# 209057 toY. X.).

Edmonton, Alberta, and Eugene, Oregon Feng DaiSeptember, 2014 Yuan Xu

Chapter 1

Introduction:

Spherical Harmonics and

Fourier Transform

The purpose of these lecture notes is to provide an introduction to two related topics:h-harmonics and the Dunkl transform. These are extensions of the classical sphericalharmonics and the Fourier transform, in which the underlying rotation group is replacedby a finite reflection group. This chapter serves as an introduction, in which we brieflyrecall classical results on the spherical harmonics and the Fourier transform. Since allresults are classical, no proof will be given.

1.1 Spherical harmonics

First we introduce several notations that will be used throughout these lecture notes.For x ∈ Rd , we write x = (x1, . . . ,xd). The inner product of x,y ∈ Rd is denoted by

〈x,y〉 := ∑di=1 xiyi and the norm of x is denoted by ‖x‖ :=

√〈x,x〉. Let Sd−1 := {x ∈ Rd :‖x‖= 1} denote the unit sphere of Rd , and let N0 denote the set of nonnegative integers.For α = (α1, . . . ,αd)∈Nd

0, a monomial xα is a product xα = xα11 · · ·xαd

d , which has degree|α| := α1 + · · ·+αd .

A homogeneous polynomial P of degree n is a linear combination of monomialsof degree n, that is, P(x) = ∑|α|=n cα xα , where cα are either real or complex numbers.A polynomial of (total) degree at most n is of the form P(x) = ∑|α|≤n cα xα . Let Pd

n

denote the space of real homogeneous polynomials of degree n and Πdn the space of real

polynomials of degree at most n. Counting the cardinalities of {α ∈ Nd0 : |α| = n} and

{α ∈ Nd0 : |α| ≤ n} shows that

dimPdn =

(n+d −1

n

)and dimΠd

n =

(n+d

n

).

1


in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-08 -3_1, , Advanced Courses

5F. Dai, Y. Xu Analysis on h-Harmonics and Dunkl Transforms

87

2 Chapter 1. Introduction: Spherical Harmonics and Fourier Transform

A harmonic polynomial is a homogeneous polynomial that satisfies the Laplaceequation. Let ∂i be the partial derivative in the i-th variable and Δ the Laplacian operator

Δ = ∂ 21 + · · ·+∂ 2

d .

Definition 1.1.1. For n = 0,1,2, . . ., let H dn be the linear space of real harmonic polyno-

mials, homogeneous of degree n, on Rd , that is,

H dn =

{P ∈ Pd

n : ΔP = 0}.

Spherical harmonics are the restrictions of elements in H dn on the unit sphere Sd−1.

If Y ∈ H dn , then Y (x) = ‖x‖nY (x′), where x = ‖x‖x′ and x′ ∈ Sd−1. We will also call H d

nthe space of spherical harmonics.

Spherical harmonics of different degrees are orthogonal with respect to the innerproduct

〈 f ,g〉Sd−1 :=1

ωd

∫Sd−1

f (x)g(x)dσ(x),

where dσ is the surface area measure on Sd−1, and ωd denotes the surface area of Sd−1,

ωd :=∫Sd−1

dσ =2πd/2

Γ(d/2).

Theorem 1.1.2. If Yn ∈ H dn , Ym ∈ H d

m , and n = m, then 〈Yn,Ym〉Sd−1 = 0. For n =0,1,2, . . ., Pd

n admits the decomposition

Pdn =

⊕0≤ j≤n/2

‖x‖2 jH dn−2 j.

In other words, for each P ∈ Pdn , there is a unique decomposition

P(x) = ∑0≤ j≤n/2

‖x‖2 jPn−2 j(x) with Pn−2 j ∈ H dn−2 j.

From the orthogonal decomposition, one immediately deduces the following:

Corollary 1.1.3. For n = 0,1,2, . . .

dimH dn = dimPd

n −dimPdn−2 =

(n+d −1

n

)−

(n+d −3

n−2

),

with the convention dimPdn−2 = 0 for n = 0,1.

In the spherical-polar coordinates x = rξ , r > 0, ξ ∈ Sd−1, the Laplace operator iswritten

Δ =∂ 2

∂ r2 +d −1

r∂∂ r

+1r2 Δ0,

1.1. Spherical harmonics 3

where Δ0, called the Laplace–Beltrami operator, can be given explicitly by

Δ0 =d−1

∑i=1

∂ 2

∂ξ 2i−

d−1

∑i=1

d−1

∑j=1

ξiξ j∂ 2

∂ξi∂ξ j− (d −1)

d−1

∑i=1

ξi∂

∂ξi.

Using this expression of Δ, ΔY = 0 for Y being a homogeneous polynomial leads to thefollowing result.

Theorem 1.1.4. The spherical harmonics are eigenfunctions of Δ0:

Δ0Y (ξ ) =−n(n+d −2)Y (ξ ), ∀Y ∈ H dn , ξ ∈ Sd−1.

In spherical coordinates, an orthogonal basis of H dn can be given explicitly. Let

{Yα} be an orthonormal basis of H dn , that is, 〈Yα ,Yβ 〉Sd−1 = δα,β . A function f in L2(Sd−1)

can be expanded in a Fourier series

f (x) = ∑cαYα(x), where cα =1

ωd

∫Sd−1

f (y)Yα(y)dσ(y).

It is often more convenient to consider the orthogonal expansions in terms of the spacesH d

n . Collecting terms of spherical harmonics of the same degree, the Fourier series takesthe form

f (x) =∞

∑n=0

projn f (x),

where projn f is the orthogonal projection of f onto the space H dn and satisfies

projn f (x) =1

ωd

∫Sd−1

f (y)Zn(x,y)dσ(y), x ∈ Sd−1,

in which Zn(·, ·), called the reproducing kernel of H dn , is given by

Zn(x,y) =dimH d

n

∑k=1

Yk(x)Yk(y), x,y ∈ Sd−1.

Since the space of spherical polynomials is dense in C(Sd−1) by the Weierstrasstheorem and, as a consequence, dense in L2(Sd−1), the following theorem is a standardHilbert space result for L2(Sd−1):

Theorem 1.1.5. The set of spherical harmonics is dense in L2(Sd−1) and

L2(Sd−1) =∞

∑n=0

H dn , f =

∞

∑n=0

projn f

in the sense that limn→∞ ‖ f −Sn f‖2 = 0 for any f ∈ L2(Sd−1), where Sn f := ∑nj=0 proj j f .

In particular, for f ∈ L2(Sd−1), the Parseval identity holds:

‖ f‖22 =

∞

∑n=0

‖projn f‖22.

Much of the analysis on the sphere beyond the L2 setting depends on the knowledgeof the kernel Zn. It is known that this kernel is uniquely determined by its reproducingproperty

1ωd

∫Sd−1

Zn(x,y)p(y)dσ(y) = p(x), ∀p ∈ H dn , x ∈ Sd−1

and the requirement that Zn(x, ·) is an element of H dn for each fixed x. In particular, Zn

is independent of the particular choice of bases of H dn . The space H d

n is invariant underthe action of the orthogonal group O(d) and the surface measure is also invariant, so thatthe kernel Zn(·, ·) satisfies Zn(x,y) = Zn(xg,yg) for all g ∈ O(d). This implies that Zn(x,y)depends only on the distance between x and y, where the distance is the geodesic distanced(x,y) = arccos〈x,y〉. Hence, Pn(x,y) = Fn(〈x,y〉), which is often called a zonal harmonicas it is harmonic and depends only on 〈x,y〉. It turns out that the function Fn has a conciseformula in terms of the Gegenbauer polynomial Cλ

n of degree n, defined by

Cλn (x) :=

(λ )n2n

n!xn

2F1

( − n2 ,

1−n2

1−n−λ ;1x2

), (1.1.1)

for λ > 0 and n ∈ N0, where 2F1 is the hypergeometric function.

Theorem 1.1.6. For n ∈ N0 and x,y ∈ Sd−1, d ≥ 3,

Zn(x,y) =n+λ

λCλ

n (〈x,y〉), λ =d −2

2. (1.1.2)

The Gegenbauer polynomials are also called ultra-spherical polynomials. They areorthogonal with respect to the weight function

wλ (t) := (1− t2)λ− 12 , t ∈ [−1,1].

Let cλ be the normalization constant of wλ , cλ = 1/∫ 1−1 wλ (t)dt. Then

cλ

∫ 1

−1Cλ

n (x)Cλm(x)wλ (x)dx =

λ(n+λ )

(2λ )n

n!δn,m.

These classical polynomials have been extensively studied. For their essential properties,see [53]. In particular, there is a generating function

1(1−2rt + r2)λ+1 =

∞

∑n=0

n+λλ

Cλn (t)r

n, 0 ≤ r < 1, λ > 0.

The concise formula for Zn(x,y) is one of the most useful ingredients for analysison the sphere. For example, it leads to the following definition of a convolution on thesphere: for f ∈ L1(Sd−1) and g ∈ L1(wλ , [−1,1]) with λ = d−2

2 ,

( f ∗g)(x) :=1

ωd

∫Sd−1

f (y)g(〈x,y〉)dσ(y), x ∈ Sd−1. (1.1.3)

1.2. Fourier transform 5

The generating function of the Gegenbauer polynomials leads to the following def-inition: for f ∈ L1(Sd−1), the Poisson integral of f is

Pr f (ξ ) := ( f ∗Pr)(ξ ), ξ ∈ Sd−1,

where the kernel Pr(t) is given by

Pr(t) :=1− r2

(1−2rt + r2)d/2 , t ∈ [−1,1],

for 0 ≤ r < 1.The Poisson kernel and Poisson integral satisfy the following properties:

(1) for x,y ∈ Sd−1, Pr(〈x,y〉) = ∑∞n=0 Zn(x,y)rn;

(2) Pr f = ∑∞n=0 rn projn f ;

(3) Pr(〈x,y〉)≥ 0 and ω−1d

∫Sd−1

Pr(〈x,y〉)dσ(y) = 1.

Using these properties, it is easy to prove the following well-known theorem.

Theorem 1.1.7. Let f be a continuous function on Sd−1. For 0 ≤ r < 1, u(rξ ) := Pr f (ξ )is a harmonic function in x = rξ , and limr→1− u(rξ ) = f (ξ ), ∀ξ ∈ Sd−1.

Spherical harmonics appear in many disciplines and in many different branches ofmathematics. We outlined the essential structure for analysis on the sphere. For proofsand further results we refer to [16, 40, 52] and the discussion at the end of Chapter 1in [16].

1.2 Fourier transform

For f ∈ L1(Rd), the Fourier transform of f is (well) defined by

f (x) =1

(2π)d/2

∫Rd

f (y)e−i〈x,y〉dy, x ∈ Rd .

For f ∈ L1(Rd), f ∈ C0(Rd). The basic properties of the Fourier transform are summa-

rized in the following theorem:

Theorem 1.2.1. (i) If f ∈ L1(Rd) and f ∈ L1(Rd), then the inversion formula,

f (y) =1

(2π)d/2

∫Rd

f (x)e−i〈x,y〉dx,

holds for almost every y ∈ Rd.

(ii) The Fourier transform extends uniquely to an isometric isomorphism on L2(Rd):‖ f‖2 = ‖ f‖2 for all f ∈ L2(Rd).


(iii) If f ,g ∈ L2(Rd), then ∫Rd

f (x)g(x)dx =∫Rd

f (x)g(x)dx.

(iv) If f (x) = f0(‖x‖) is radial, then f (x) = Hd−22

f0(‖x‖) is again a radial function,where Hα denotes the Hankel transform defined by

Hα g(s) =1

Γ(α +1)

∫ ∞

0g(r)

Jα(rs)(rs)α r2α+1 dr,

in which Jα denotes the Bessel function of the first kind.

The usual proof of (i) uses convolution defined by

f ∗g(x) =1

(2π)d/2

∫Rd

f (y)g(x− y)dy,

for f ,g ∈ L1(Rd). It is easy to see that if f ,g ∈ L1(Rd), then

f ∗g(x) = f (x)g(x).

Let Φ be a nice function, say Φ(x) = e−‖x‖ or e−‖x‖2/2, and let φ := Φ; normalize Φ sothat

∫Rd φ(x) = 1. For ε > 0, define φε(x) := ε−dφ(x/ε). It is easy to see that

( f ∗φε)(x) =1

(2π)d/2

∫Rd

Φ(εy) f (y)ei〈x,y〉dy.

Thus, the proof of (i) comes down to showing that f ∗φε(x)→ f (x) as ε → 0.The eigenfunctions of the Fourier transform can be given in terms of spherical har-

monics. Let Y ∈ H dn . Define

φm(Y ;x) = Ln+ d−2

2m (‖x‖2)Y (x)e−‖x‖2/2, x ∈ Rd ,

where Lαn denotes the Laguerre polynomial of degree n with index α , normalized so that

1Γ(α +1)

∫ ∞

0Lα

n (x)Lαm(x)x

α e−xdx =(

n+αα

)δm,n.

If {Yk,n : 1 ≤ k ≤ dimH dn } denotes an orthonormal basis of H d

n , then it is easy to verify,using spherical polar coordinates and the orthogonality of Lα

n , that {φm(Yk,n;x) : m,n ≥0,1 ≤ k ≤ dimH d

n } is an orthogonal basis of L2(Rd).

Theorem 1.2.2. For m,n = 0,1,2, . . ., Y ∈ H dn and x ∈ Rd,

φm(Y )(x) = (−i)n+2mφm(Y ;x).

There are many books on Fourier transforms. For the basics that we need here andthe proofs, we refer to [31, 46, 52].

Chapter 2

Dunkl Operators Associated with

Reflection Groups

In this chapter we introduce the essential ingredient in the Dunkl theory of harmonicanalysis. Since our purpose is to study harmonic analysis in weighted spaces, we startwith the definition of a family of weight functions invariant under a reflection groupin Section 2.1. Dunkl operators are a family of commuting first-order differential anddifference operators associated with a reflection group, and are introduced in Section2.2. The intertwining operator between the Dunkl operators and ordinary derivatives isdiscussed in Section 2.3.

For readers who are primarily interested in analysis, the prerequisites on reflectiongroups are reduced to a minimum. In fact, all essential ideas are presented in the case ofG = Zd

2, which requires no prior knowledge of reflection groups.

2.1 Weight functions invariant under a reflection group

The simplest family of weight functions in d variables that we consider is defined by

hκ(x) :=d

∏i=1

|xi|κi , x ∈ Rd , (2.1.1)

for κi ≥ 0, 1 ≤ i ≤ d, and x = (x1, . . . ,xd). Obviously, they are invariant under signchanges, that is, invariant under the group Zd

2. This is a special case of weight functionsinvariant under reflection groups. To define the general weight functions, we first need torecall basic facts on reflection groups. Readers who are not interested in reflection groupscan skip to the end of the section and keep in mind the functions hκ in (2.1.1) and Zd

2 inthe rest of these lecture notes.

For x ∈ Rd , let 〈x,y〉 denote the usual Euclidean inner product and ‖x‖ :=√〈x,x〉

the Euclidean norm of x. For a nonzero vector v ∈ Rd , let σv denote the reflection with


in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-08 -3_, , Advanced Courses


87

7

2

8 Chapter 2. Dunkl Operators Associated with Reflection Groups

respect to the hyperplane v⊥ perpendicular to v,

xσv := x−2(〈x,v〉/‖v‖2)v, x ∈ Rd .

A root system is a finite set R of nonzero vectors in Rd such that u,v ∈ R implies uσv ∈ R.If, in addition, u,v ∈ R and u = cv for some scalar c implies that c =±1, then R is calledreduced. The set {u⊥ : u ∈ R} is a finite set of hyperplanes, hence, there exists u0 ∈ Rd

such that 〈u,u0〉 = 0 for all u ∈ R. With respect to u0 define the set of positive rootsR+ := {v ∈ R : 〈v,u0〉> 0}. If u ∈ R, then −u = uσu ∈ R, so that R = R+∪ (−R+).

The finite reflection group G generated by the root system R is the subgroup of O(d)generated by {σu : u ∈ R}. If R is reduced, then the set of reflections contained in G isexactly {σu : u∈R+}. For a given root system R, a multiplicity function v �→ κv : R→R≥0is a nonnegative function defined on R such that κv = κu whenever σu is conjugate to σv,that is, there exists g ∈ G such that ug = v.

Given a reduced root system R on Rd and a multiplicity function κv on R, we definea weight function hκ by

hκ(x) := ∏v∈R+

|〈x,v〉|κv , x ∈ Rd . (2.1.2)

Then hκ is invariant under the reflection group G generated by R. It is a homogeneousfunction of degree

γκ := ∑v∈R+

κv. (2.1.3)

For hκ in (2.1.1) associated with Zd2, γκ = |κ|= κ1 + · · ·+κd .

Let us give two examples beyond Zd2. Let e1, . . . ,ed be the standard Euclidean basis,

that is, the i-th component of ei is 1 and all other components are 0.

Symmetric group. The root system is R = {ei−e j : 1 ≤ i = j ≤ d}. Choosing u0 = (d,d−1, . . . ,1), one has R+ = {ei−e j : 1 ≤ i < j ≤ d}. There is only one conjugacy class in thisgroup, so that the weight function is

hκ(x) = ∏1≤i< j≤d

|xi − x j|κ , κ ≥ 0, x ∈ Rd . (2.1.4)

This reflection group is of the type Ad−1 and it is the same as the symmetric, or permuta-tion, group of d objects. Evidently, hκ is symmetric under permutations of x1, . . . ,xd .

Octahedral group. The positive root system is R+ = {ei−e j,ei+e j : 1≤ i = j ≤ d}∪{ei :1 ≤ i ≤ d}. There are two conjugacy classes in this group, so that the weight function is

hκ(x) =d

∏i=1

|xi|κ0 ∏1≤i< j≤d

|x2i − x2

j |κ1 , κ0,κ1 ≥ 0, x ∈ Rd . (2.1.5)

This reflection group is of the type Bd and it is the symmetric group of the octahedron{±e1, . . . ,±ed} of Rd , or the cube in Rd . Obviously, hκ is symmetric under permutationsof x1, . . . ,xd and sign changes.

2.1. Weight functions invariant under a reflection group 9

The analysis in these lecture notes is in the setting of weighted Lp spaces with thesereflection invariant weight functions on the unit sphere and on Rd .

Let dσ denote the surface measure on the unit sphere Sd−1. Let ωd denote thesurface area and ωκ

d denote the normalization constant of hκ :

ωd :=∫Sd−1

dσ =2πd/2

Γ(d/2)and ωκ

d :=∫Sd−1

h2κ(y)dσ .

The closed form of ωκd is known for every reflection group (cf. [26]). For 1 ≤ p ≤ ∞ we

denote by Lp(h2κ) the space of functions defined on Sd−1 with finite norm

‖ f‖κ,p :=( 1

ωκd

∫Sd−1

| f (y)|ph2κ(y)dσ(y)

)1/p, 1 ≤ p < ∞,

and for p = ∞ we assume that L∞ is replaced by C(Sd−1), the space of continuous func-tions on Sd−1 with the usual uniform norm ‖ f‖∞.

Let Pdn denote the space of homogeneous polynomials of degree n in d variables.

Consider the measure on Rd defined by

dμ := h2κ(x)e

−‖x‖2/2dx, x ∈ Rd ,

and let ch denote the normalization constant

ch :=(

1(2π)d/2

∫Rd

h2κ(x)e

−‖x‖2/2dx)−1

. (2.1.6)

The inner product with respect to dμ is closely related to the one with respect to dxwhen we restrict to the space of homogeneous polynomials. For n ∈ N0 and a ∈ R, let(a)n := a(a+1) · · ·(a+n−1) denote the shifted factorial of a.

Proposition 2.1.1. For p,q ∈ Pdn ,

ch

∫Rd

p(x)q(x)h2κ(x)e

−‖x‖2/2dx = (2π)d/22n(λκ +1)n1

ωκd

∫Sd−1

p(ξ )q(ξ )h2κ(ξ )dσ(ξ ),

where λκ = d−22 + γκ .

Proof. If g is a homogeneous polynomial of degree 2n, then using spherical polar coor-dinates x = rξ , r > 0 and ξ ∈ Sd−1, we have∫

Rdg(x)h2

κ(x)e−‖x‖2/2dx =

∫ ∞

0r2γκ+2n+d−1e−r2/2dr

∫Sd−1

g(ξ )h2κ(ξ )dσ(ξ ) (2.1.7)

= 2n+λκ Γ(λκ +n+1)∫Sd−1

g(ξ )h2κ(ξ )dσ(ξ ),

from which the result follows. �It is worth noting that, setting n = 0 in (2.1.7),

c−1h =

1(2π)d/2 2λκ Γ(λκ +1)ωκ

d = 2λκ Γ(λκ +1)Γ(d/2)

ωκd

ωd. (2.1.8)


2.2 Dunkl operators

The main ingredient of the theory of h-harmonics is a family of first-order differential-difference operators, Di, called the Dunkl operators.

Definition 2.2.1. Let R+ be a positive root system and κv be a multiplicity function fromR+ to R≥0. For 1 ≤ i ≤ d, define

Di f (x) := ∂i f (x)+ ∑v∈R+

κvf (x)− f (xσv)

〈x,v〉〈v,ei〉, 1 ≤ i ≤ d. (2.2.1)

It is easy to verify that DiPdn ⊂Pd

n−1, so that the Di are indeed first-order differential-difference operators. In the case of Zd

2, the Dunkl operators take on the form

Di f (x) = ∂i f (x)+κif (x)− f (xσi)

xi(2.2.2)

where xσi = (x1, . . . ,xi−1,−xi,xi+1, . . . ,xd).The most important property of these operators is that they commute.

Theorem 2.2.2. The Dunkl operators commute:

DiD j = D jDi, 1 ≤ i, j ≤ d.

Proof. The proof of the general case is rather involved. We give the proof only for thecase of Zd

2, for which a straightforward computation shows that, for i = j,

DiD j f (x) = ∂i∂ j f (x)+κi

xi(∂ j f (x)−∂ j f (xσi))+

κ j

x j(∂i f (x)−∂i f (xσ j))

+κiκ j

xix j( f (x)− f (xσ j)− f (xσi)+ f (xσ jσi)),

from which DiD j = D jDi follows immediately. �The Dunkl operators are akin to the partial derivatives and they can be used to define

an analog of the Laplace operator, denoted by Δh:

Δh := D21 + · · ·+D2

d . (2.2.3)

This is a second-order differential-difference operator and it reduces to the usual Lapla-cian Δ when all κi = 0. It has the following explicit formula related to the weight functionhκ in (2.1.2):

Proposition 2.2.3. The Dunkl Laplacian Δh can be written as Δh = Dh +Eh, with

Dh f :=Δ( f hk)− f Δhk

hκand Eh f := −2 ∑

v∈R+

κvf (x)− f (xσv)

〈v,x〉2 ‖v‖2,

and both Dh and Eh commute with the action of the reflection group.

2.2. Dunkl operators 11

Proof. Again, we give the proof only for the case of Zd2. Let

E j f (x) :=f (x)− f (xσ j)

x j, 1 ≤ j ≤ d,

so that D j = ∂ j +κ jE j in the case of G = Zd2. A straightforward computation shows that

E2j = 0 and

D2j = ∂ 2

j f +κ j∂ jE j +κ jE j∂ j = ∂ 2j +2

κ j

x j∂ j − κ j

x jE j.

Summing over j we obtain

Dh = Δ f +2d

∑j=1

κ j

x j∂ j f −

d

∑j=1

κ j

x jE j.

In this case the sum over R+ means the sum over 1 ≤ j ≤ d, so that the sum over E j givesEh and the differential part is Dh, which can be written in the stated expression in termsof hκ by a simple verification. �

Later we will need to perform an integration by parts for the Dunkl operator, atleast over the space of polynomials. For this to make sense, we consider the integral withrespect to the measure

dμ := h2κ(x)e

−‖x‖2/2dx, x ∈ Rd .

Theorem 2.2.4. The adjoint D∗i acting on L2(Rd ;dμ) is given by

D∗i p(x) = xi p(x)−Di p(x), p ∈ Πd .

Proof. Assume κv ≥ 1. Analytic continuation can be used to extend the range of validityto κv ≥ 0. Let p and q be two polynomials. Integrating by parts shows that∫

Rd

(∂i p(x)

)q(x)h2

κ(x)e−‖x‖2/2dx =−

∫Rd

p(x)(∂iq(x)

)h2

κ(x)e−‖x‖2/2dx

+∫Rd

p(x)q(x)[−2hκ(x)∂ihκ(x)+h2

κ(x)xi]e−‖x‖2/2dx.

For a fixed root v,∫Rd

p(x)− p(xσv)

〈x,v〉 q(x)dμ =∫Rd

p(x)q(x)〈x,v〉 dμ −

∫Rd

p(xσv)q(x)〈x,v〉 dμ

=∫Rd

p(x)q(x)〈x,v〉 dμ +

∫Rd

p(x)q(xσv)

〈x,v〉 dμ,

where in the second integral we have replaced x by xσv which changes 〈x,v〉 to 〈xσv,v〉=−〈x,v〉 and leaves h2

κ invariant. Note also that

hκ(x)∂ihκ(x) = ∑v∈R+

κvvi

〈x,v〉h2κ(x).

Combining these ingredients, we obtain∫Rd

Di p(x)q(x)dμ =∫Rd

[p(x)

(xiq(x)−∂iq(x)

)+ ∑

v∈R+

(κvvi p(x)

(−2q(x)+q(x)+q(xσv))/〈x,v〉)]dμ,

where the term inside the square brackets is exactly p(x)(xiq(x)−Diq(x)

). �

2.3 Intertwining operator

There is a linear operator that intertwines between the Dunkl operators and the partialderivatives, which plays an important role in harmonic analysis.

Definition 2.3.1. Let Di be the Dunkl operators associated with a given positive rootsystem and a multiplicity function κ . A linear operator Vκ on the space Πd of algebraicpolynomials on Rd is called an intertwining operator if it satisfies

DiVκ =Vκ ∂i, 1 ≤ i ≤ d, Vκ 1 = 1, VκPn ⊂ Pn, n ∈ N0. (2.3.1)

Strictly speaking, (2.3.1) is not the definition of Vκ , but rather the property thatwe most often use. Indeed, the existence of such a Vκ is by no means automatic. Theoperator Vκ was introduced in [26], where it was defined inductively on homogeneouspolynomials. The definition is extended from homogeneous polynomials to the spaceA(Bd) defined below.

For f ∈ Πd , let ‖ · ‖A := ∑∞n=0 ‖ fn‖S, where f = ∑∞

n=0 fn with fn ∈ Pdn and ‖ f‖S =

supx∈Sd−1 | f (x)|. Let A(Bd) be the closure of Πd in A-norm. Then A(Bd) is a commutativeBanach algebra under the pointwise operations and it is contained in C(Bd)∩C∞({x :‖x‖< 1}), where Bd = {x : ‖x‖≤ 1} is the unit ball of Rd . Then the following propositionholds (see [26]):

Proposition 2.3.2. For f ∈ Πd and x ∈ Bd,

|Vκ f (x)| ≤ ‖ f‖A and ‖Vκ f‖A ≤ ‖ f‖A.

The existence of the operator Vκ for a generic reflection group satisfying (2.3.1)requires substantial knowledge of reflection groups and considerable efforts [26]. For ourpurpose, however, it is not necessary to know the proof.

In the case of Zd2, the intertwining operator Vκ has an explicit expression as an

integral operator.

Theorem 2.3.3. Let κi ≥ 0. The intertwining operator for Zd2 is given by

Vκ f (x) = cκ

∫[−1,1]d

f (x1t1, . . . ,xdtd)d

∏i=1

(1+ ti)(1− t2i )

κi−1dti, (2.3.2)

2.3. Intertwining operator 13

where cκ = cκ1 · · ·cκd with cμ = Γ(μ + 1/2)/(√

πΓ(μ)), and if any one of κi = 0, thenthe formula holds under the limit

limμ→0

cμ

∫ 1

−1f (t)(1− t2)μ−1dt =

f (1)+ f (−1)2

. (2.3.3)

Proof. The integrals are normalized so that Vκ 1 = 1. Recall that D j = ∂ j +κ jE j. Takingderivatives we get

∂ jVκ f (x) = cκ

∫[−1,1]d

∂ j f (x1t1, . . . ,xdtd)t j

d

∏i=1

(1+ ti)(1− t2i )

κi−1dti.

Taking into account the parity of the integrand, integration by parts shows that

κ jE jVκ f (x) =κ j

x jcκ

∫[−1,1]d

f (x1t1, . . . ,xdtd)2t j

(∏i = j

(1+ ti))( d

∏i=1

(1− t2i )

κi−1)

dti

=cκ

∫[−1,1]d

∂ j f (x1t1, . . . ,xdtd)(1− t j)d

∏i=1

(1+ ti)(1− t2i )

κi−1dti.

Adding the last two equations gives D jVκ =Vκ ∂ j for 1 ≤ j ≤ d. �

Apart from partial results for the symmetric group on three variables and the dihe-dral group D4, an explicit formula for Vκ is not known. In general, however, we have thefollowing theorem of Rosler, which, in particular, asserts that Vκ is nonnegative.

Theorem 2.3.4. For each x ∈ Rd, there exists a unique probability measure μκx on the

Borel σ -algebra of Rd such that for all algebraic polynomials f on Rd,

Vκ f (x) =∫Rd

f (ξ )dμκx (ξ ). (2.3.4)

Furthermore, the measures μκx are compactly supported in the convex hull C(x) :=

conv{xg : g ∈ G} of the orbit of x under G, and satisfy

μκrx(E) = μκ

x (r−1E), and μκ

xg(E) = μκx (Eg−1) (2.3.5)

for all r > 0, g ∈ G and each Borel subset E of Rd.

Remark 2.3.5. This theorem was proved in [43, Th. 1.2 and Cor. 5.3]. Note that themeasure μκ

x depends on x. The most useful part of the result is that Vκ is a nonnegativeoperator. By means of (2.3.4), Vκ can be extended to an operator in the space C(Rd) ofcontinuous functions on Rd , which we will denote by Vκ again.

Definition 2.3.6. For x,y ∈ Rd , define

E(x,y) :=V (x)κ

(e〈x,y〉

)


and, for n = 0,1,2, . . ., define

En(x,y) :=1n!

V (x)κ (〈x,y〉n),

where the superscript x means that Vκ acts on the x variable.

Definition 2.3.7. For p,q ∈ Pdn , define 〈p,q〉D := p(D)q(x).

Proposition 2.3.8. The kernel En satisfies the following properties

(1) En is symmetric, En(x,y) = En(y,x);

(2) En(xg,yg) = En(x,y), g ∈ G;

(3) En is the reproducing kernel of 〈·, ·〉D , that is,

〈En(x, ·), p〉D = p(x), ∀p ∈ Pdn .

Proof. The first two properties follow from the inductive definition of Vκ , see [26]. Forthe third property, we note that if p ∈Pd

n , then p(x) = (〈x,∂ (y)〉n/n!)p(y). Applying V (x)κ

leads to V (x)κ p(x) = En(x,∂ (y))p(y). The left-hand side is independent of y so, applying

V (y) to both sides, we get V (x)κ p(x) = En(x,D (y))V (y)

κ p(y). Thus, the desired identity holdsfor all Vκ p with p ∈ Pd

n , which completes the proof, since Vκ is one-to-one. �

2.4 Notes and further results

The Dunkl operators were introduced in [25] and the intertwining operators and the innerproducts in Section 2.3 were studied in [26]. For a complete proof of the existence of theintertwining operator and its basic properties, see [29, Chapter 6]. The positivity of theintertwining operator was proved in [43]. The explicit formula for Vκ in the case of Zd

2was given in [69]. For the symmetric group S3 with hκ(x) = |(x1−x2)(x2−x3)(x3−x1)|κfor x ∈ S2 and the dihedral group I(4) with hκ(x) = |x1x2|κ0 |x2

1 − x22|κ1 , some explicit

integral formulas for Vκ are given in [28] and [70], respectively. But neither of them isin a strong enough form for carrying out the harmonic analysis that will be developed inlatter chapters.

Chapter 3

h-Harmonics and Analysis on the

Sphere

Dunkl h-harmonics are defined as homogeneous polynomials satisfying the Dunkl Lapla-cian equation. They are defined and studied in Section 3.1. Projection operators and or-thogonal expansions in spherical h-harmonics are studied in Section 3.2, which includesa concise expression for the reproducing kernel of the spherical h-harmonics. This ex-pression is an analog of the zonal harmonics, which suggests a definition of a convolutionoperator, defined in Section 3.3 and it helps us to study various summability methodsfor spherical h-harmonic expansions. Maximal functions are introduced in Section 3.4and proved to be of strong type (p, p) and weak type (1,1). Finally, the relation betweenconvolution and maximal functions is discussed in Section 3.5.

3.1 Dunkl h-harmonics

We are now in a position to define h-harmonics.

Definition 3.1.1. Let Y ∈ Pdn be a homogeneous polynomial of degree n. If ΔhY = 0,

then Y is called an h-harmonic polynomial of degree n.

For n = 0,1,2, . . ., let H dn (h2

κ) denote the linear space of h-harmonic polynomialsof degree n. Elements of H d

n (h2κ) are homogeneous polynomials so that they are uniquely

determined by their restrictions to the unit sphere Sd−1. The restrictions of h-harmonicsto the sphere are spherical h-harmonics, analogues to spherical harmonics. We shall notdistinguish between Y h

n ∈ H dn (h2

κ) and its restriction to the sphere.Let hκ be the weight function defined in (2.1.2). The inner product in L2(h2

κ ,Sd−1)

is denoted by

〈 f ,g〉κ :=1

ωκd

∫Sd−1

f (x)g(x)h2κ(x)dσ(x). (3.1.1)




87

15

3

16 Chapter 3. h-Harmonics and Analysis on the Sphere

Theorem 3.1.2. With respect to 〈·, ·〉κ , spherical h-harmonics of different degree areorthogonal. More precisely, if f ∈ H d

n (h2κ), g ∈ H d

m (h2κ) and n = m, then 〈 f ,g〉κ = 0.

Proof. As in the classical proof for ordinary harmonics, this follows from an analog ofGreen’s formula stated for the differentiation part Dh of Δh:∫

Sd−1

∂ f∂n

gh2κ dσ =

∫Bd(gDh f + 〈∇ f ,∇g〉)h2

κ dx,

where ∂ f/∂n denotes the normal derivative of f . Consequently, since ∂ f∂n = n f for f

homogeneous of degree n, and Δh f = 0, Δhg = 0,

(n−m)∫Sd−1

f gh2κ dσ =

∫Bd(gDh f − f Dhg)h2

κ dx =−∫Bd(gEh f − f Ehg)h2

κ dx

=−∫ 1

0r2γκ+n+m+d−5dr

∫Sd−1

(gEh f − f Ehg)h2κ dσ ,

using the spherical polar coordinates x = rξ , r > 0 and ξ ∈ Sd−1. The last integral is zerosince the difference part Eh of Δh is self-adjoint with respect to 〈·, ·〉κ . �

Theorem 3.1.3. For n = 0,1,2, . . ., Pdn admits the decomposition

Pdn =

⊕0≤ j≤n/2

‖x‖2 jH dn−2 j(h

2κ). (3.1.2)

Furthermore, for n = 0,1,2, . . .,

dimH dn (h2

κ) = dimPdn −dimPd

n−2 =

(n+d −1

d −1

)−

(n+d −3

d −1

). (3.1.3)

Proof. Briefly, the proof follows by induction, using the orthogonality of H dn (h2

κ) andthe fact that Δh maps Pd

n onto Pdn−2. �

From (2.3.1) it follows immediately that ΔhVκ = Vκ Δ and, consequently, if P is anordinary harmonic polynomial, then Vκ P is an h-harmonic.

In terms of the spherical polar coordinates x = rξ , the Dunkl Laplacian Δh admits adecomposition as in the case of ordinary Laplace operator. Let us define

λk := γk +d −2

2= ∑

v∈R+

κv +d −2

2. (3.1.4)

Lemma 3.1.4. In the spherical-polar coordinates x = rξ , r > 0, ξ ∈ Sd−1, the DunklLaplace operator can be expressed as

Δh =d2

dr2 +2λκ +1

rddr

+1r2 Δh,0, (3.1.5)

3.1. Dunkl h-harmonics 17

where

Δh,0 f =1

hκ[Δ0( f hκ)− f Δ0hκ ]−E(ξ )

h f , (3.1.6)

Δ0 denotes the usual Laplace–Beltrami operator, and E(ξ )h means that Eh is acting on the

ξ variable.

Proof. By the decomposition Δh = Dh +Eh, we can apply the decomposition of the or-dinary Laplacian Δ to the differential part Dh, which gives the part of Δh,0 expressed interms of the classical Laplace–Beltrami operator Δ0. The difference part follows readilyfrom the definition of Eh. �

The operator Δh,0 is the analogue of the Laplace–Beltrami operator on the sphere,which, in particular, has spherical h-harmonics as eigenfunctions.

Theorem 3.1.5. The spherical h-harmonics are eigenfunctions of Δh,0:

Δh,0Y hn (ξ ) =−n(n+2λκ)Y h

n (ξ ), ∀Y hn ∈ H d

n (h2κ), ξ ∈ Sd−1. (3.1.7)

Proof. Since Y hn is a homogeneous polynomial of degree n, Y h

n (x) = rnY hn (ξ ). Apply-

ing (3.1.5) to Y hn , equation (3.1.7) follows from ΔhY h

n = 0. �

The following theorem gives an orthogonal basis for H dn (h2

κ).

Theorem 3.1.6. For α ∈ Nd0 , n = |α|, define

pα(x) :=(−1)n

2n(λκ)n‖x‖2|α|+2λκ Dα{‖x‖}−2λk , (3.1.8)

where Dα := Dαdd · · ·Dα1

1 . Then

1. pα ∈ H dn (h2

κ) and pα is a monic spherical h-harmonic of the form

pα(x) = xα +‖x‖2qα(x), qα ∈ Pdn−2; (3.1.9)

2. pα satisfies the recurrence relation

pα+ei(x) = xi pα(x)− 12n+2λκ

‖x‖2Di pα(x);

3. {pα : |α|= n,αd = 0 or 1} is a basis of H dn (h2

κ).

The proof of this theorem is more or less a straightforward computation. Indeed, forg ∈ Pd

n and ρ ∈ R, the explicit expressions for Di and Δh can be used to show that

Di(‖x‖ρ g) = ρxi‖x‖ρ−2g+‖x‖ρDig, (3.1.10)

Δh(‖x‖ρ g) = ρ(2n+2λk +ρ)‖x‖ρ−2g+‖x‖ρ Δhg. (3.1.11)


The recurrence relation follows immediately from (3.1.10), which shows, by induction,that pα is homogeneous of degree n. Using (3.1.11), a quick computation shows thatΔh pα(x) = 0, so that pα ∈ H d

n (h2κ).

However, for a generic reflection group, it is not clear how to evaluate the norm ofpα , that is, an explicit formula for the norm of pα is not known. As a consequence, wecannot provide an explicit orthonormal basis from {pα}. In fact, no orthonormal basis forH d

n (h2κ) is explicitly known beyond the case of the group Zd

2. For Zd2, the norm of pα can

be evaluated so that an orthonormal basis can be derived via the Gram–Schmidt process.In fact, for Zd

2, an orthonormal basis can be explicitly given in spherical coordinates, ash2

κ in this case is a simple product.Let projκn denote the orthogonal projection operator

projκn : L2(Sd−1;h2κ) �→ H d

n (h2κ).

By the orthogonal decomposition of homogeneous polynomials, p ∈ Pdn can be written

as p(x) = pn +‖x‖2qn, where pn ∈ H dn (h2

κ) and qn ∈ Pdn−2; we have, by definition, that

pn = projκn p. In this regard, by (3.1.9), the polynomial pα in (3.1.8) is the orthogonalprojection of xα ,

pα(x) = projκn qα(x), qα(x) = xα with |α|= n. (3.1.12)

Proposition 3.1.7. Let p ∈ Pdn . Then

projκn p =�n/2�∑j=0

14 j j!(1−n−λκ) j

‖x‖2 jΔ jh p. (3.1.13)

Proof. It suffices to prove equation (3.1.13) for p = qα . Since projκn qα = pα , we need toshow that pα equals the right-hand side of (3.1.13) with p= qα . This can be established byinduction on the degree |α| of qα . Indeed, the case |α|= 1 is obvious. Suppose equation(3.1.13) has been proved for all α such that |α|= n, which gives

Dα{‖x‖−2λ

}= (−1)n2n(λ )n‖x‖−2λ−2n

[n/2]

∑j=0

14 j j!(−λ −n+1) j

‖x‖2 jΔ jhqα(x),

where λ = λκ . Applying here Di and using the first identity in (3.1.10) with g = Δ jh{xα},

we conclude that

DiDα{‖x‖−2λ

}= (−1)n2n(λ )n(−2λ −2n)‖x‖−2λ−2n−2

×[(n+1)/2]

∑j=0

14 j j!(−λ −n) j

‖x‖2 j[xiΔ j

h{xα}+2 jΔ j−1h Di{xα}

], (3.1.14)

since Di commutes with Δh. From (3.1.10) and (3.1.11) it is easy to see, by induction onj, that

Δ jh{xi f (x)}= xiΔ j

h f (x)+2 jDiΔ j−1h f (x), j = 1,2,3, . . . .

3.1. Dunkl h-harmonics 19

Thus, by the definition of pα and (3.1.12), the left-hand side of (3.1.14) is a constantmultiple of the projection of xiqα(x), and the right-hand side is a constant multiple of theright-hand side of (3.1.13) with p(x) = xiqα(x), which completes the induction. �

There is another inner product on the space of homogeneous polynomials Pdn that

will be useful in our study below.

Theorem 3.1.8. For p,q ∈ Pdn ,

〈p,q〉D = En(D(x),D (y))p(x)q(y) = 〈q, p〉D .

Proof. By Proposition 2.3.8, p(x) = En(x,D (y))p(y). The operators D (x) and D (y) com-mute and thus

〈p,q〉D = En(D(x),D (y))p(y)q(x) = En(D

(y),D (x))p(y)q(x).

The last expression equals 〈q, p〉D . �

The pairing 〈·, ·〉D is related to 〈·, ·〉κ when p ∈ Pdn and q ∈ H d

n (h2κ).

Theorem 3.1.9. If p ∈ Pdn and q ∈ H d

n (h2κ), then

〈p,q〉D = 2n(λκ +1)n〈p,q〉κ . (3.1.15)

Proof. Since p(D)q(x) is a constant, it follows from Theorem 2.2.4 that

〈p,q〉h = ch

∫Rd

p(D)q(x)h2κ(x)e

−‖x‖2/2dx

= ch

∫Rd

q(x)(

p(D∗)1)h2

κ(x)e−‖x‖2/2dx.

Repeatedly applying the formula for the adjoint operator D∗i g(x) = xig(x)−Dig(x), and

using the fact that the degree of Dig is lower than that of g, we see that p(D∗)1 = p(x)+s(x) with a polynomial s of degree less than n. Since q ∈ H d

n (h2κ), it follows from the

spherical polar integral that∫Rd q(x)s(x)h2

κ(x)e−‖x‖2/2dx = 0. Consequently, we conclude

that〈p,q〉h = ch

∫Rd

q(x)p(x)h2κ(x)e

−‖x‖2/2dx.

Now, (3.1.15) follows from Proposition 2.1.1. �

In the case where both p,q ∈ Pdn , the inner product 〈p,q〉D also has an integral

expression:

Theorem 3.1.10. For p,q ∈ Pdn ,

〈p,q〉D = ch

∫Rd

(e−Δh/2 p(x)

)(e−Δh/2q(x)

)h2

κ(x)e−‖x‖2/2dx. (3.1.16)

Proof. First of all, decomposing p ∈ Pdn as p(x) = ∑0≤ j≤n/2 ‖x‖2 j pn−2 j, where pn−2 j ∈

H dn−2 j(h

2κ), and decomposing q similarly, by the definition of Δh and 〈pn−2 j,qn−2 j〉h, we

conclude that

〈p,q〉h =�n/2�∑j=0

�n/2�∑i=0

〈pn−2 j,qn−2 j〉h =�n/2�∑j=0

�n/2�∑i=0

Δih pn−2i(D)

(‖x‖2 jqn−2 j(x)).

By the identity (3.1.11),

Δih(‖x‖2 jqn−2 j(x)

)= 4i(− j)i(−n−λ + j)i‖x‖2 j−2iqn−2 j(x),

which is zero if i > j. If i < j, then 〈‖x‖2 jqn−2 j(x),‖x‖2i pn−2i(x)〉h = 0 by the sameargument and the fact that 〈p,q〉= 〈q, p〉. Hence, the only remaining terms are those withj = i, which are given by 4 j j!(−n−λ + j) j pn−2 j(D)qn−2 j(x). Therefore,

〈p,q〉h =�n/2�∑j=0

4 j j!(n−2 j+λκ +1) j〈pn−2 j,qn−2 j〉h.

Thus, we only need to prove (3.1.16) for polynomials of the form p(x) = ‖x‖2 j pm(x) andq(x) = ‖x‖2 jqm(x) with pm,qm ∈ H d

n−2 j(h2κ). For such p,q, by (3.1.11) and (3.1.15),

〈p,q〉D = 4 j j!(m+λκ +1) j〈pm,qm〉D= 4 j j!(m+λκ +1) j2m(λκ +1) j〈pm,qm〉κ

= 2m+2 j j!(λκ +1)m+ j〈pm,qm〉κ .

On the other hand, a straightforward computation using (3.1.11) shows that

e−Δh/2 [‖x‖2 j pm(x)]= (−1) j j!2 jLn+λκ

j (‖x‖2/2)pm(x), (3.1.17)

where Lαj is the standard Laguerre polynomial. As a consequence, the right-hand side of

the stated formula becomes, using spherical polar coordinates,∫Rd[Lm+λκ

j (‖x‖2/2)]2 pm(x)qm(x)h2κ(x)e

−‖x‖2/2dx

= 2m+λκ Γ(m+ j+λκ +1)j!

∫Sd−1

pm(ξ )qm(ξ )h2κ(ξ )dσ .

Putting these together we get (3.1.16). �

3.2 Projection operator and intertwining operator

Let {Yν ,n : 1 ≤ ν ≤ adn}, ad

n := dimH dn (h2

κ), be an orthonormal basis of H dn (h2

κ). Forf ∈ L2(Sd−1,h2

κ), the usual Hilbert space theory shows that f can be expanded in sphericalh-harmonics as

f =∞

∑n=0

adn

∑ν=0

fν ,nYν ,n, fν ,n := 〈 f ,Yν ,n〉κ ,

3.2. Projection operator and intertwining operator 21

where the convergence holds in L2(Sd−1;h2κ) norm. In terms of the orthogonal projection

operator from L2(Sd−1,h2κ) onto H d

n (h2κ),

projκn : L2(Sd−1;h2κ) �→ H d

n (h2κ),

the expansion in h-harmonics can be rewritten as

f =∞

∑n=0

projκn f .

In particular, the projection operator can be expressed as an integral,

projκn f (x) =1

ωκd

∫Sd−1

f (y)Zκn (x,y)h

2κ(y)dσ(y), x ∈ Sd−1, (3.2.1)

where Zκn (·, ·) is the kernel function defined by

Zκn (x,y) =

adn

∑ν=0

Yν ,n(x)Yν ,n(y), x,y ∈ Sd−1.

This kernel, however, is independent of the choice of particular basis of H dn (h2

κ). Indeed,it is the reproducing kernel of H d

n (h2κ), i.e.,

1ωd

∫Sd−1

Zκn (x,y)p(y)h2

κ(y)dσ(y) = p(x), ∀p ∈ H dn (h2

κ), x ∈ Sd−1. (3.2.2)

In terms of the intertwining operator Vκ , the reproducing kernel Zκn has a concise

expression given in terms of the Gegenbauer polynomial Cλn :

Theorem 3.2.1. Let λκ = γk +d−2

2 . For ‖y‖ ≤ ‖x‖= 1,

Zκn (x,y) = ‖y‖n n+λκ

λκVκ

[Cλκ

n

(⟨·, y‖y‖

⟩)](x). (3.2.3)

Proof. From Proposition 2.3.8 (ii), it follows that for every p ∈ H dn (h2

κ),

p(x) = 〈En(x, ·), p〉D = 〈projκn (En(x, ·)), p〉D= 2n(λk +1)n〈projκn (En(x, ·)), p〉κ .

Since Zκn (·, ·) is uniquely determined by the reproducing property, this shows that

Zκn (x, ·) = 2n(λk +1)n projκn (En(x, ·)).

Using the intertwining property of Vκ and the definition of En(·, ·), it follows from (3.1.13)that

Zκn (x,y) = ∑

0≤ j≤n/2

(λκ +1

)n2n−2 j(

1−n−λκ) j j!‖x‖2 j‖y‖2 jEn−2 j(x,y).

When ‖x‖ = 1, we can write the right-hand side as Vκ(Gn(〈·,y/‖y‖〉))(x), where Gn is ahypergeometric function 2F1, which turns out to be a constant multiple of the Gegenbauerpolynomial. �


The identity (3.2.3) also indicates that in the theory of h-harmonics zonal functions,which depend only on 〈x,y〉, should be replaced by functions of the form Vκ [ f (〈·,y〉)](x).Indeed, we have an analogue of the Funk–Hecke formula.

Theorem 3.2.2. Let f be a continuous function on [−1,1]. Then for any Y hn ∈ H d

n (h2κ),

1ωκ

d

∫Sd−1

Vκ [ f (〈x, ·〉)](y)Y hn (y)h

2κ(y)dσ(y) = Λn( f )Y h

n (x), x ∈ Sd−1, (3.2.4)

where Λn( f ) is a constant defined by

Λn( f ) = cλκ

∫ 1

−1f (t)

Cλκn (t)

Cλκn (1)

(1− t2)λk− 12 dt,

and where cλ = Γ(λ +1)/√

πΓ(λ +1/2) and Λ0(1) = 1.

Proof. If f is a polynomial of degree m, then we can expand f in terms of the Gegenbauerpolynomials

f (t) =m

∑k=0

Λkk+λκ

λκCλκ

k (t),

where Λκ are determined by the orthogonality of Gegenbauer polynomials,

Λk =cλκ

Cλκk (1)

∫ 1

−1f (t)Cλκ

k (t)(1− t2)λκ− 12 dt,

and c−1λ =

∫ 1−1(1− t2)λ− 1

2 dt. Using (3.2.3) and the reproducing property of Zκn (x,y) it

follows that, for n ≤ m,

1ωκ

d

∫Sd−1

Vκ [ f (〈x, ·〉)](y)Y hn (y)h

2κ(y)dσ(y) = ΛnY h

n (x), x ∈ Sd−1.

Since Λn/ωκd =Λn( f ) by definition, we have established the Funk–Hecke formula (3.2.4)

for polynomials, and hence, by the Weierstrass theorem, for continuous functions. �Theorem 3.2.3. Let f : Bd → R be a continuous function. Then

1ωκ

d

∫Sd−1

Vk f (y)h2κ(y)dσ(y) = aκ

∫Bd

f (x)(1−‖x‖2)|κ|−1dx. (3.2.5)

In particular, if f (y) = g(〈x,y〉) with g : R→ R, then

1ωκ

d

∫Sd−1

Vk[g(〈x, ·〉)](y)h2κ(y)dσ(y) = cλκ

∫ 1

−1g(‖x‖t)(1− t2)λκ− 1

2 dt. (3.2.6)

Proof. Applying the Funk–Hecke formula (3.2.4) with n= 0 to the function ξ �→ g(‖x‖ξ ),ξ ∈ Sd−1, gives (3.2.6). It is known that every polynomial f can be written as a linear sumof p j(〈x,ξ j〉), where p j : [−1,1]→ R and ξ j ∈ Sd−1, so that (3.2.5) follows from (3.2.6)for all polynomials. For general f we can then pass to the limit, since the right-hand sideof (3.2.5) is clearly closed under limits. �

3.3. Convolution operators and orthogonal expansions 23

Remark 3.2.4. Using Theorem 3.2.3 and the positivity of Vκ , for any g ∈C(Bd),

‖Vκ g‖L1(h2κ ;Sd−1) ≤ bκ

∫Bd

|g(x)|(1−‖x‖2)|κ|−1dx.

Since the right-hand side is a constant multiple of the norm of L1(W ;Bd), where W (x) :=(1−‖x‖2)|κ|−1, this allows us to extend Vκ to a positive, bounded operator from L1(W ;Bd)to L1(h2

κ ;Sd−1), so that (3.2.5) holds for all g ∈ L1(W ;Bd).For a generic reflection group, not very much is known on specifics of the intertwin-

ing operator Vκ . Property (3.2.5) of Vκ is highly non-trivial, as can be seen in the specialcase of Zd

2, where Vκ is given explicitly by formula (2.3.2), and it is highly useful, as thedevelopment below will show.

3.3 Convolution operators and orthogonal expansions

The expression of Zκn (·, ·) at (3.2.3) suggests the following definition of convolution on

the sphere.

Definition 3.3.1. For f ∈ L1(Sd−1;h2κ) and g ∈ L1(wλκ , [−1,1]),

( f ∗κ g)(x) :=1

ωκd

∫Sd−1

f (y)Vκ [g(〈·,y〉)](x)h2κ(y)dσ(y). (3.3.1)

Denote the norm of the space Lp(wλ ; [−1,1]) by ‖ · ‖λ ,p, and the norm of the spaceLp(hκ ,S

d−1) by ‖ · ‖κ,p for 1 ≤ p < ∞ and by C(Sd−1) for p = ∞. The convolution ∗κsatisfies Young’s inequality:

Theorem 3.3.2. Let p,q,r ≥ 1 and p−1 = r−1 + q−1 − 1. For f ∈ Lq(Sd−1;h2κ) and g ∈

Lr(wλκ ; [−1,1]),‖ f ∗κ g‖κ,p ≤ ‖ f‖κ,q‖g‖λκ ,r. (3.3.2)

In particular, for 1 ≤ p ≤ ∞,

‖ f ∗g‖κ,p ≤ ‖ f‖κ,p‖g‖λκ ,1 and ‖ f ∗g‖κ,p ≤ ‖ f‖κ,1‖g‖λκ ,p. (3.3.3)

Proof. The standard proof of Young’s inequality applies in this setting. By Minkowski’sinequality, it suffices to show that

‖G(x, ·)‖κ,r ≤ ‖g‖λκ ,r, where G(x,y) =Vκ [g(〈x, · 〉)](y).The proof uses the integral relation (3.2.6). Indeed, the positivity of Vκ implies |Vκ g| ≤Vκ [|g|], so that ‖G(x, ·)‖κ,∞ ≤ ‖g‖λκ ,∞ and we deduce by (3.2.6) that

‖G(x, ·)‖κ,1 ≤ 1ωκ

d

∫Sd−1

Vκ [|g(〈x, · 〉)|] (y)h2κ(y)dσ = cλκ

∫ 1

−1|g(t)|wλ (t)dt = ‖g‖λκ ,1.

The log-convexity of the Lp-norm implies then ‖G(x, ·)‖κ,r ≤ ‖g‖λκ ,r. �

By (3.2.1) and (3.2.3), the projection projκn is a convolution operator

projκn f = f ∗κ Zκn , Zκ

n (t) :=n+λκ

λκCλκ

n (t). (3.3.4)

The following theorem justifies calling ∗κ a convolution:

Theorem 3.3.3. For f ∈ L1(Sd−1;h2κ) and g ∈ L1(wλκ ; [−1,1]),

projκn ( f ∗κ g) = gλκn projκn f , n = 0,1,2 . . . , (3.3.5)

where gλκn is the Fourier coefficient of g in the Gegenbauer polynomial,

gλκn = cλκ

∫ 1

−1g(t)

Cλκn (t)

Cλκn (1)

(1− t2)λκ− 12 dt.

Proof. By (3.2.1) and the Funk–Hecke formula in Theorem 3.2.2,

projκn ( f ∗κ g)(x) =1

ωκd

∫Sd−1

( f ∗g)(ξ )Zκn (x,ξ )h

2κ(ξ )dσ(ξ )

=1

ωκd

∫Sd−1

f (y)(

1ωκ

d

∫Sd−1

g(〈ξ ,y〉)Zκn (x,ξ )h

2κ(ξ )dσ(ξ )

)h2

κ(y)dσ(y)

= gλκn

1ωκ

d

∫Sd−1

f (y)Zκn (x,y)h

2κ(y)dσ(y) = gλκ

n projκn f (x),

which is what we needed to prove. �Since the convergence of the h-harmonic series does not go beyond L2 norm in

general, it is necessary to consider summability methods. We consider the Cesaro meansof the h-harmonic series. First, we give the definition of the Cesaro means for a sequenceof complex numbers.

Definition 3.3.4. The Cesaro (C,δ )-means of a given sequence {an}∞n=0 of complex num-

bers are defined by

sδn :=

n

∑j=0

Aδn− j

Aδn

a j, n = 0,1, . . . , (3.3.6)

where the coefficients Aδj are defined by

(1− t)−1−δ =∞

∑n=0

Aδn tn, t ∈ (−1,1).

For convenience, we also define Aδj = 0 for j < 0. The following useful properties

follow easily from the definition:

Aδj −Aδ

j−1 = Aδ−1j ,

n

∑j=0

Aδj = Aδ+1

n ,n

∑j=0

Aδn− jA

αj = Aα+δ+1

n ,

|Aδj | ∼ ( j+1)δ , whenever j+δ +1 > 0.

3.3. Convolution operators and orthogonal expansions 25

Let us denote by Sδn(h2

κ ; f)

the Cesaro means of the h-harmonic series, that is,

Sδn(h2

κ ; f)

:=1

Aδn

n

∑j=0

Aδn− j projκj f , (3.3.7)

where S0n(h

2κ ; f ) is the n-th partial sum. By (3.3.4), the Cesaro means are convolution

operators,Sδ

n(h2

κ ; f)= f ∗κ Kδ

n(h2

κ),

where the kernel is defined by

Kδn

(h2

κ ; t)

:=1

Aδn

n

∑k=0

Aδn−k

k+λκ

λκCλκ

k (t) = kδn(wλκ ;1, t

), (3.3.8)

in which kδn (wλκ ; ·, ·) is the kernel of the (C,δ )-means of the Fourier orthogonal series in

the Gegenbauer polynomials.

Theorem 3.3.5. The Cesaro means of the spherical h-harmonic series satisfy:

1. if δ ≥ 2λk +1, then Sδn (h

2κ) is a nonnegative operator;

2. if δ > λκ , then

supn≥0

‖Sδn (h

2κ ;g)‖κ,p ≤ c‖g‖κ,p, 1 ≤ p ≤ ∞. (3.3.9)

In particular, Sδn (h

2κ ; f ) converges to f in Lp(h2

κ ;Sd−1) for 1 ≤ p ≤ ∞.

Proof. By (3.3.8), the non-negativity of Sδn (h

2κ) follows from that of the Gegenbauer ker-

nel kδn (wλ ;1, t), which is a classical result. In order to prove the convergence, it is suffi-

cient to show that ‖Sδn (h

2k)‖κ,p is bounded. By Young’s inequality (3.3.3),

‖Sδn (h

2κ , f )‖κ,p ≤ ‖ f‖κ,p‖kδ

n (wλκ )‖λκ ,1 ≤ c‖ f‖κ,p,

whenever δ > λk, where the last inequality follows from a classical result on the (C,δ )summability of the Gegenbauer series. �

A careful examination of the proof of convergence shows that the essential ingredi-ents are (3.2.6) and the positivity of Vκ (namely, Theorem 2.3.4). Equality (3.2.6) removesVκ and allows us to reduce the problem to the Gegenbauer series. This is similar to theresult for ordinary spherical harmonics. However, for the ordinary spherical harmonics,this leads to the sharp condition δ > (d −2)/2, while for the h-harmonic series, δ > λκis not sharp in general. In fact, taking the average of Vκ by (3.2.6) erases the informationon the reflection group.

For a generic reflection group, we know very little about Vκ . In the case of Zd2, how-

ever, Vκ is given by the explicit formula in (2.3.2), which allows us to obtain much deeperresults on the convergence of the Cesaro means. These studies require sharp pointwiseestimates of the kernel functions and, sometimes, asymptotics of the kernel functions,

which in turn require long estimations and detailed analysis. We shall state the resultswithout proof.

Recall that, for the hκ defined in (2.1.1) associated with Zd2,

λκ = |κ|+ d −22

.

Theorem 3.3.6. Let hκ be as in (2.1.1). Let δ >−1 and define

σκ := λk − min1≤i≤d

κi = |κ|+ d −22

− min1≤i≤d

κi.

Then for p = 1 and p = ∞,

‖projn(h2κ)‖κ,p ∼ nσκ and ‖Sδ

n (h2κ)‖κ,p ∼

⎧⎪⎨⎪⎩1, δ > σκ

logn, δ = σκ

n−δ+σκ , −1 < δ < σκ .

In particular, the (C,δ )-means Sδn (h

2κ ; f ) converge to f in L1(h2

κ ;Sd−1) or in C(Sd−1) ifand only if δ > σκ .

The zero set of the weight function hκ(x) serves as boundary on the sphere, awayfrom which we have better convergence behavior. Indeed, for hκ in (2.1.1), the zero setis the collection of great circles defined by the intersection of Sd−1 with the coordinateplanes. Let us define

Sd−1int := Sd−1 \

d⋃i=1

{x ∈ Sd−1 : xi = 0},

which is the interior region bounded by these great circles on Sd−1.

Theorem 3.3.7. Let hκ be as in (2.1.1). Let f be continuous on Sd−1. If δ > d−22 , then

Sδn (h

2κ ; f ) converges to f for every x ∈ Sd−1

int and the convergence is uniform over eachcompact subset of Sd−1

int .

By the Riesz interpolation theorem, Theorem 3.3.6 also implies that Sδn (h

2κ ; f ) con-

verges to f in the Lp norm if δ > σκ , for all 1 < p < ∞. However, for each fixed p, thisorder is not sharp. The sharp results for Lp convergence are given in the following twotheorems:

Theorem 3.3.8. Suppose that f ∈ Lp(Sd−1;h2κ), 1 ≤ p ≤ ∞, | 1

p − 12 | ≥ 1

2σκ+2 and

δ > δκ(p) := max{(2σκ +1)| 1p − 1

2 |− 12 ,0}. (3.3.10)

Then Sδn (h

2κ ; f ) converges to f in Lp(h2

κ ;Sd−1) and

supn∈N

‖Sδn (h

2κ ; f )‖κ,p ≤ c‖ f‖κ,p.

3.4. Maximal functions 27

Theorem 3.3.9. Assume 1 ≤ p ≤ ∞ and 0 < δ ≤ δκ(p). Then there exists a functionf ∈ Lp(Sd−1;h2

κ) such that Sδn (h

2κ ; f ) diverges in Lp(Sd−1;h2

κ).

The proofs of these two theorems are much more involved and require heavy ma-chinery, such as the Fefferman–Stein inequality and Stein’s analytic interpolation theo-rem.

3.4 Maximal functions

For x ∈ Sd−1 and 0 ≤ θ ≤ π , we define

b(x,θ) := {y ∈ Bd : 〈x,y〉 ≥ cosθ}.

Let χE denote the characteristic function of the set E.

Definition 3.4.1. For f ∈ L1(Sd−1;h2κ), define the maximal function

Mκ f (x) = sup0<θ≤π

∫Sd−1 | f (y)|Vκ [χb(x,θ)](y)h2

κ(y)dσ(y)∫Sd−1 Vκ [χb(x,θ)](y)h2

κ(y)dσ(y)(3.4.1)

= sup0<θ≤π

(| f | ∗κ χ[cosθ ,1])(x)

cλκ

∫ θ0 (sinφ)2λκ dφ

.

The second expression in this definition uses the identity∫Sd−1

Vκ [χ[cosθ ,1](〈x, ·〉)](y)h2κ(y)dσ(y) =

∫ θ

0(sinφ)2λκ dφ ∼ θ 2λκ+1, (3.4.2)

coming from (3.2.6). This maximal function can be used to study the h-harmonic ex-pansions, since we can often prove that |( f ∗κ g)(x)| ≤ cMκ f (x). We will show that itsatisfies the usual property of maximal functions, that is, it is of strong type (p, p) for1 < p < ∞, and of weak type (1,1). The proof of this last result relies on a general resultabout the following semi-groups of operators (see [49, p. 2] for more details):

Definition 3.4.2. Let (X ,μ) be a measure space with a positive measure μ . A family ofoperators {Tt}t≥0 is said to form a symmetric diffusion semi-group if

Tt1Tt2 = Tt1+t2 , T 0 = id,

and

(i) Tt are contractions on Lp(X ,μ), i.e., ‖Tt f‖p ≤ ‖ f‖p, 1 ≤ p ≤ ∞;

(ii) Tt are symmetric, i.e., each Tt is self-adjoint on L2(X ,dμ);

(iii) Tt are positivity preserving, i.e., Tt f ≥ 0 if f ≥ 0;

(iv) Tt f0 = f0 if f0(x) = 1.

The result that we shall need is given in [49, p. 48] and it is a special case of theHopf–Dunford–Schwartz ergodic theorem.

Theorem 3.4.3. Suppose that {Tt}t≥0 is a symmetric diffusion semi-group on a positivemeasure space (X ,μ). Then the function

M f (x) = sups≥0

(1s

∫ s

0Tt f (x)dt

)satisfies the inequalities

(a) ‖M f‖p ≤ cp‖ f‖p for each p with 1 α}) ≤ (c/α)‖ f‖1 for each α > 0 and f ∈ L1(X ,μ), where cis independent of f and α .

Our semi-group of operators is defined in terms of the Poisson integrals:

Definition 3.4.4. For f ∈ L1(Sd−1;h2κ), the Poisson integral of f is defined by

Pκr f (ξ ) :=

1ωκ

d

∫Sd−1

f (y)Pκr (ξ ,y)h

2κ(y)dσ(y), ξ ∈ Sd−1, (3.4.3)

where the kernel Pκr (x, ·) is given by

Pκr (x,y) :=Vκ

[1− r2

(1−2r〈·,y〉+ r2)λκ+1

](x), (3.4.4)

for 0 < r < 1.

Using the generating function of the Gegenbauer polynomials, the Poisson kernelis the generating function of the spherical h-harmonics, as seen in the first item of thefollowing lemma, from which the other two items follow directly.

Lemma 3.4.5. For 0 < r < 1, the Poisson kernel satisfies the following properties:

(1) for x,y ∈ Sd−1, Pκr (x,y) = ∑∞

n=0 rn n+λκλκ

Vκ

[Cλκ

n (〈x, ·〉)](y);

(2) Pκr f = ∑∞

n=0 rn projκn f ;

(3) Pκr (x,y)≥ 0 and 1

ωκd

∫Sd−1 Pκ

r (x,y)h2κ(y)dσ(y) = 1.

Put Tt = Pκr with r = e−t . Using the above lemma, it is easy to see that Tt is a

diffusion semi-group. We will need another semi-group, which is the discrete analog ofthe heat operator:

Hκt f := f ∗κ qκ

t , qκt (s) :=

∞

∑n=0

e−n(n+2λκ )t n+λκ

λκCλκ

n (s). (3.4.5)

Lemma 3.4.6. The family of operators {Hκt } is a symmetric diffusion semi-group.

3.4. Maximal functions 29

Proof. The kernel qκt is known to be nonnegative, from which it immediately follows

that Hκt are positive and that ‖qκ

t ‖λκ ,1 = 1, by the orthogonality of the Gegenbauer poly-nomials. Hence, by Young’s inequality, ‖Hκ

t f‖κ,p ≤ ‖ f‖k,p. The other requirements inDefinition 3.4.2 can be verified directly. �

Lemma 3.4.7. The Poisson and the heat semi-groups are connected by

Pκe−t f (x) =

∫ ∞

0φt(s)Hκ

s f (x)ds, (3.4.6)

where

φt(s) :=t

2√

πs−3/2e−

(t

2√

s−λκ√

s)2

.

Furthermore, if f (x)≥ 0 for all x, then

Pκ∗ f (x) := sup

0<r<1Pκ

r f (x)≤ csups>0

1s

∫ s

0Hκ

u f (x)du. (3.4.7)

Consequently, Pκ∗ f is bounded on Lp(h2κ ;Sd−1) for 1 0( 1

s∫ s

0 Hκu f (x)du

)is bounded on Lp(h2

κ ,Sd−1) for 1 0, (3.4.8)

with v = (n+λκ)t. Making the change of variable s = t2/4u, we obtain

e−nt = eλκ t 1√π

∫ ∞

0

e−u√

ue−

n(n+2λκ )t24u e−

λ2κ t24u du

=t

2√

π

∫ ∞

0e−n(n+2λκ )ss−3/2e−

(t

2√

s−λκ√

s)2

ds

=∫ ∞

0e−n(n+2λκ )sφt(s)ds.

Multiplying by projκn f and summing up over n we obtain the integral relation (3.4.6).For the proof of (3.4.7), we use (3.4.6) and integration by parts to obtain

Pκe−t f (x) =−

∫ ∞

0

(∫ s

0Hκ

u f (x)du)

φ ′t (s)ds

≤ sups>0

(1s

∫ s

0Hκ

u f (x)du)∫ ∞

0s|φ ′

t (s)|ds,

where the derivative of φ ′t (s) is taken with respect to s. Furthermore, since Pκ

r f = f ∗k pκr

and |pκr (t)| ≤ c for 0 < r ≤ e−1, it follows that

sup0<r≤e−1

Pκr f (x)≤ c‖ f‖1,κ = c lim

s→∞

1s

∫ s

0Hκ

u (| f |)(x)du.

Therefore, to finish the proof of (3.4.7), it suffices to show that sup0<t≤1∫ ∞

0 s|φ ′t (s)|ds is

bounded by a constant. A quick computation shows that φ ′t (s)> 0 if s < αt and φ ′

t (s)< 0if s > αt , where

αt :=t2

3+√

9+4λ 2κ t2

∼ t2, 0 ≤ t ≤ 1.

Since the integral of φt(s) over [0,∞) is 1 and φt(s)≥ 0, integration by parts gives∫ ∞

0s|φ ′

t (s)|ds = 2αtφt(αt)−∫ αt

0φt(s)ds+

∫ ∞

αt

φt(s)ds

≤ 2αtφt(αt)+1 =t√παt

e−(t−2λκ αt )2

4αt +1 ≤ c,

as desired. �

We are now ready to prove the main result on the maximal function. To state theweak type inequality, we define, for any measurable subset E of Sd−1, the measure withrespect to h2

κ as

measκ E :=∫

Eh2

κ(y)dσ(y).

Theorem 3.4.8. If f ∈ L1(Sd−1;h2κ), then Mκ f satisfies

measκ{x ∈ Sd−1 : Mκ f (x)≥ α} ≤ c‖ f‖κ,1

α, ∀α > 0. (3.4.9)

Furthermore, if f ∈ Lp(Sd−1;h2κ) for 1 < p ≤ ∞, then ‖Mk f‖κ,p ≤ c‖ f‖κ,p.

Proof. From the definition of pκr in (3.4.4), if 1− r ∼ θ , then

pκr (cosθ) =

1− r2((1− r)2 +4r sin2 θ

2

)λκ+1

≥ c1− r2

((1− r)2 + rθ 2)λκ+1 ≥ c(1− r)−(2λκ+1).

For j ≥ 0 define r j := 1−2− jθ and set B j :={

y ∈ Bd : 2− j−1θ ≤ d(x,y)≤ 2− jθ}. The

lower bound of pκr proved above shows that

χB j(y)≤ c(2− jθ)2λk+1 pκr j(〈x,y〉),

3.5. Convolution and maximal function 31

which immediately implies that

χb(x,θ)(y)≤∞

∑j=0

χB j(y)≤ cθ 2λk+1∞

∑j=0

2− j(2λκ+1)pκr j(〈x,y〉).

Since Vκ is a positive linear operator, applying Vκ to the above inequality we get∫Sd−1

| f (y)|Vκ[χb(x,θ)

](y)h2

κ(y)dσ(y)

≤ cθ 2λκ+1∞

∑j=0

2− j(2λκ+1)∫

Sd−1| f (y)|Vκ

[pr j(〈x,y〉)

](y)h2

κ(y)dσ(y)

= cθ 2λκ+1∞

∑j=0

2− j(2λκ+1)Pκr j(| f |;x)

≤ cθ 2λκ+1 sup0<r<1

Pκr (| f |;x).

Dividing by θ 2λκ+1 and using the fact that

1ωκ

d

∫Sd−1

Vκ [χb(x,θ)](y)h2κ(y)dσ(y) = cλκ

∫ θ

0(sinφ)2λκ dφ ∼ θ 2λκ+1,

we have proved that Mκ f (x) ≤ cPκ∗ | f |(x). The desired result now follows from Lemma3.4.7. �

3.5 Convolution and maximal function

The ordinary convolution operator is often defined in terms of the translation operator.For f ∈ L2(Sd−1) and 0 ≤ θ ≤ π , the translation operator Tθ is defined by

Tθ f (x) :=1

ωd−1(sinθ)d−1

∫〈x,y〉=cosθ

f (y)d�x,θ (y),

where d�x,θ is the Lebesgue measure on the set {y∈ Sd−1 : 〈x,y〉= cosθ}. For the integralwith respect to the h2

κ dσ , we do not have an explicit extension of Tθ f , but we can definean extension as a multiplier operator.

Definition 3.5.1. For 0 ≤ θ ≤ π , the generalized translation operator T κθ is defined by

projκn (Tκ

θ f ) =Cλκ

n (cosθ)Cλκ

n (1)projκn f , n = 0,1, . . . . (3.5.1)

Since a function in L1(Sd−1;h2κ) is uniquely defined by its orthogonal projections

on H dn (h2

κ), the generalized translation operator is well defined.

Proposition 3.5.2. The operator T κθ is well defined for all f ∈ L1(Sd−1;h2

κ) and enjoysthe following properties:

(i) for f ∈ L2(h2κ ,S

d−1) and g ∈ L1(wλκ , [−1,1]),

( f ∗κ g)(x) = cλκ

∫ π

0T κ

θ f (x)g(cosθ)(sinθ)2λκ dθ ; (3.5.2)

(ii) T κθ preserves positivity, i.e., T κ

θ f ≥ 0 if f ≥ 0;

(iii) for f ∈ Lp(h2κ ,S

d−1) if 1 ≤ p < ∞, or f ∈C(Sd−1) if p = ∞,

‖T κθ f‖κ,p ≤ ‖ f‖κ,p and lim

θ→0‖T κ

θ f − f‖κ,p = 0; (3.5.3)

(iv) T κθ f (−x) = T κ

π−θ (x).

Proof. The equation (3.3.5) immediately implies

projκn ( f ∗κ g)(x) = cλκ

∫ π

0projκn (T

κθ f )(x)g(cosθ)(sinθ)2λκ dθ ,

which proves the identity (3.5.2), because f is uniquely determined by its orthogonalprojections. To prove (ii) and (iii), for fixed θ , we let Bn(θ) := cλκ

∫ θ+1/nθ−1/n (sin t)2λκ dt and

gn(cosφ) := 1/Bn(θ) if |φ −θ | ≤ 1/n, and gn(cosφ) := 0 otherwise. Then

( f ∗κ gn)(x) =1

Bn(θ)

∫ θ+1/n

θ−1/nT κ

φ f (x)(sinφ)2λκ dφ .

The proof comes down to showing that f ∗κ gn converges to f in Lp(h2κ ,S

d−1), which canbe established first for f being a polynomial and by choosing the polynomial to be, say,the (C,δ )-means of f for δ ≥ 2λκ +1. Finally, (iv) follows from a change of variable in(3.5.2). �

In terms of the generalized translation operator, the maximal function Mκ f can bewritten as

Mκ f (x) := sup0<θ≤π

∫ θ0 T κ

φ | f |(x)(sinφ)2λκ dφ∫ θ0 (sinφ)2λκ dφ

. (3.5.4)

This relation allows us to prove the following result on the convolution operator.

Theorem 3.5.3. Assume that g ∈ L1([−1,1],wλκ ) and |g(cosθ)| ≤ k(θ) for all θ , wherek(θ) is a continuous, nonnegative, and decreasing function on [0,π]. For f ∈L1(h2

κ ,Sd−1),

|( f ∗κ g)(x)| ≤ cMκ(| f |)(x), x ∈ Sd−1,

where c =∫ π

0 k(θ)(sinθ)2λκ dθ .

3.5. Convolution and maximal function 33

Proof. Let λ = λκ . Define

Λ(θ ,x) =∫ θ

0T κ

φ | f |(x)(sinφ)2λ dφ .

Then the relation (3.5.4) implies that

Λ(θ ,x)≤ Mκ f (x)∫ θ

0(sinφ)2λ dφ

for all x ∈ Sd−1. By Proposition 3.5.2,

| f ∗κ g(x)|=cλ

∣∣∣∣∫ π

0T κ

φ f (x)g(cosφ)(sinφ)2λ dφ∣∣∣∣

≤ cλ

∫ π

0T κ

φ | f |(x)k(φ)(sinφ)2λ dφ .

Integrating by parts, we obtain

| f ∗κ g(x)| ≤ cλ

[Λ(π,x)k(π)−

∫ π

0Λ(θ ,x)k′(θ)dθ

]≤ cλ Mκ f (x)

[k(π)

∫ π

0(sinφ)2λ dφ −

∫ π

0k′(θ)

∫ θ

0(sinφ)2λ dφdθ

],

since k′(cosθ)≤ 0. Integrating by parts again, we conclude that

| f ∗κ g(x)| ≤ Mκ f (x)cλ

∫ π

0k(θ)(sinθ)2λ dθ ≤ cMκ f (x).

This completes the proof. �Applying the above theorem on the Cesaro means gives the following:

Theorem 3.5.4. If δ > λκ and f ∈ L1(Sd−1;h2κ), then for every x ∈ Sd−1,

Sδ∗ (x)( f )(x) := sup

n≥0|Sδ

n (h2κ ; f ,x)| ≤ c [Mκ f (x)+Mκ f (−x)] . (3.5.5)

If, in addition, δ ≥ 2λk +1, then the term Mκ f (−x) in (3.5.5) can be dropped.

Proof. For the proof of the inequality, it suffices to consider the case λ < δ ≤ λ + 1,where λ = λκ , since it is well known that Sδ+τ∗ f (x)≤ Sδ∗ ( f )(x) for any τ > 0. Setting

Gδn,1(cosθ) :=nλ−δ (n−1 +θ)−(δ+λ+1)χ[0, π

2 ](θ),

Gδn,2(cosθ) :=nλ−δ (n−1 +θ)−λ χ[0, π

2 ](θ),

and using (3.3.8) and the pointwise estimate of the kernel kδn (wλ ; t,1) for the (C,δ )-means

of the Gegenbauer series in [53], we derive that, for λ < δ ≤ λ +1,

Kδn (h

2κ ; cosθ)≤ c

[Gδ

n,1(cosθ)+Gδn,2(cos(π −θ))

].

It is easy to see that g(t) = Gδn,i(t) satisfy the conditions of Theorem 3.5.3, which allows

us to conclude that

|Sδn f (x)| ≤

[(| f | ∗Gδ

n,1)(x)+(| f | ∗Gδn,2)(−x)

]≤ c [Mκ f (x)+Mκ f (−x)] .

Furthermore, if δ > 2λ +1, then the pointwise estimate of the kernel kδn (wλ ; t,1) shows

that |Kδn (h

2κ ; cosθ)| is bounded by a single term and the same proof yields |Sδ

n f (x)| ≤cM f (x). �


The first studies on h-harmonic expansions appeared in [69] (which contains (2.3.2) anda proof for the formula for Zκ

n , in the case of Zd2, by summing over a specific orthonor-

mal basis using special function identities), and [68] (which contains relations (3.2.3) and(3.2.5), with the latter proved by studying the orthogonal expansion of Vκ f on the unitball). The proof of (3.2.5) in Theorem 3.2.3 was given in [16]. The Funk–Hecke for-mula (3.2.4) was established in [71]. The results on the Cesaro summability for G = Zd

2were established in [13, 14, 37], which reduces to the classical results on the spherical har-monics [3, 5, 48] when κ = 0. The convolution and the translation operators were definedin [72], and used to study weighted best approximation on the sphere. The maximal func-tion Mκ f was defined in [73], the results in Section 3.5 were established in [12]. In thecase of G = Zd

2 and hκ(x) = ∏di=1 |xi|κi , we can consider the weighted Hardy–Littlewood

maximal function defined by

Mκ f (x) := sup0<θ≤π

∫c(x,θ) | f (y)|h2

κ(y)dσ(y)∫c(x,θ) h2

κ(y)dσ(y), (3.6.1)

for f ∈ L1(Sd−1;h2κ) where c(x,θ) := {y ∈ Sd−1 : arccos〈x,y〉 ≤ θ}, and prove, using

the explicit formula for Vκ , that Mκ f dominates the maximal function Mκ f defined inSection 3.5,

Mκ f (x)≤ c ∑ε∈Zd

2

Mκ f (xε), x ∈ Sd−1.

The lack of an explicit formula for the intertwining operator Vκ has been an obstaclefor deriving deeper results relying on the essence of reflection groups. At the moment,little information is known on the intertwining operator for reflection groups other thanZd

2.

Chapter 4

Littlewood–Paley Theory and the

Multiplier Theorem

The main result of this chapter is a Marcinkiewitcz multiplier theorem for h-harmonicexpansions. Its proof uses general Littlewood–Paley theory for a symmetric diffusionsemi-group. Several Littlewood–Paley type g-functions are introduced and studied viathe Cesaro means for h-harmonic expansions. These g-functions provide new equivalentnorms for the space Lp(h2

κ ;Sd−1), and play crucial roles in the proof of the multipliertheorem.

In Section 4.1, several vector-valued inequalities for self-adjoint operators onL2(h2

κ ;Sd−1) are established, which will be used in the proof of the main result. A briefdescription of the general Littlewood–Paley–Stein theory is given in Section 4.2, wherea Littlewood–Paley g-function defined via the Poisson semi-group for h-harmonics is in-troduced and studied as well. The weighted Littlewood–Paley theory on the sphere isdeveloped in Section 4.3, where two new g-functions defined via the Cesaro means playessential roles. With the help of the Littlewood–Paley theory, a Marcinkiewitcz type mul-tiplier theorem for h-harmonic expansions is proved in Section 4.4, which is then appliedto obtain a refined Littlewood–Paley inequality in Section 4.5.

4.1 Vector-valued inequalities for self-adjoint operators

Here we establish several vector-valued inequalities, which will play important roles inlater sections.

Theorem 4.1.1. Let {Tk}∞k=0 be a sequence of self-adjoint linear operators on the space

L2(h2κ ;Sd−1). Assume that there exists a positive operator N , not necessarily linear,

which is bounded on Lp(h2κ ;Sd−1) for all 1 < p < ∞ and pointwisely controls the maximal

operator of {Tk}:sup

k∈Z+

|Tk f (x)| ≤ cN f (x), ∀x ∈ Sd−1. (4.1.1)




87

35

4

36 Chapter 4. Littlewood–Paley Theory and the Multiplier Theorem

Then for any sequence {n j} of nonnegative integers and any { f j} ⊂ L2(h2κ ;Sd−1),∥∥∥( ∞

∑j=0

∣∣Tn j( f j)∣∣2

)1/2∥∥∥κ,p

≤ c′∥∥∥( ∞

∑j=0

∣∣ f j∣∣2

)1/2∥∥∥κ,p

. (4.1.2)

Proof. The proof of (4.1.2) follows the approach of [49, p.104-5], which uses a general-ization of the Riesz convexity theorem for sequences of functions. Let Lp(�q) denote thespace of all sequences { fk} of functions for which the norm

‖( fk)‖Lp(�q) :=

(∫Sd−1

( ∞

∑j=0

| f j(x)|q)p/q

h2κ(x)dσ(x)

)1/p

is finite. If T is a bounded operator on Lp0(�q0) and on Lp1(�q1), with 1 ≤ p0, p1,q0,q1 ≤∞, then the Riesz convexity theorem states that T is also bounded on Lpt (�qt ), where

1pt

=1− t

p0+

tp1

,1qt

=1− tq0

+t

q1, 0 ≤ t ≤ 1.

We apply this theorem to the operator T mapping the sequence { f j} to the sequence{Tn j( f j)}. By (4.1.1), T is bounded on Lp(�p). It is also bounded on Lp(�∞) since∥∥∥sup

j≥0

∣∣Tn j( f j)∣∣∥∥∥

κ,p≤

∥∥∥N(

supj≥0

| f j|)∥∥∥

κ,p≤ c

∥∥∥supj≥0

| f j|∥∥∥

κ,p.

Hence, the Riesz convexity theorem shows that T is bounded on Lp(�q) if 1 < p ≤ q ≤ ∞.In particular, T is bounded on Lp(�2) if 1 < p ≤ 2. The case 2 < p < ∞ follows by thestandard duality argument, since the dual space of Lp(�2) is Lp′(�2), where 1/p+1/p′ =1, under the pairing

〈( f j),(g j)〉 :=∫Sd−1 ∑

jf j(x)g j(x)h2

κ(x)dσ(x)

and T is self-adjoint under this paring since each Tj is self-adjoint in L2(h2κ ;Sd−1). �

The above proof of Theorem 4.1.1 actually yields the following Fefferman–Steininequality for the maximal function Mκ f associated to a reflection group.

Corollary 4.1.2. Let 1 < p ≤ 2 and let { f j} be a sequence of functions. Then∥∥∥∥(∑

j|Mκ f j|2

)1/2∥∥∥∥κ,p

≤ c∥∥∥∥(

∑j| f j|2

)1/2∥∥∥∥κ,p

. (4.1.3)

As a consequence of Theorem 4.1.1, we also have the following inequality for theCesaro means Sδ

n (hκ).

4.2. The Littlewood–Paley–Stein function 37

Corollary 4.1.3. For δ > λκ , 1 < p < ∞ and any sequence {n j} of nonnegative integers,∥∥∥∥( ∞

∑j=0

∣∣Sδn j(h2

κ ; f j)∣∣2

)1/2∥∥∥∥κ,p

≤ c∥∥∥∥( ∞

∑j=0

∣∣ f j∣∣2

)1/2∥∥∥∥κ,p

.

Proof. The Cesaro means Sδn (hκ) are clearly self-adjoint on L2(h2

κ ;Sd−1). By Theorem4.1.1,

supn∈Z+

|Sδn (hκ ; f )(x)| ≤CMκ f (x)+CMκ f (−x), x ∈ Sd−1, δ > λκ .

Since Mκ is a positive and bounded operator on Lp(h2κ ,S

d−1) for all 1 < p < ∞, thedesired conclusion follows directly from Theorem 4.1.1. �Definition 4.1.4. Let η ∈ C∞(R) be such that η(x) = 1 for |x| ≤ 1 and η(x) = 0 for|x| ≥ 2. For N = 1,2, . . ., we define the operator Lκ

N by

LκN( f ) =

∞

∑j=0

η( j

N

)projκj f .

The operator LκN is well behaved and has a highly localized kernel in the unweighted

case. Similar to the proof of Corollary 4.1.3, we can also deduce the following:

Corollary 4.1.5. For 1 < p < ∞ and any sequence {n j} of nonnegative integers,∥∥∥∥( ∞

∑j=0

∣∣Lκn j( f j)

∣∣2)1/2∥∥∥∥

κ,p≤ c

∥∥∥∥( ∞

∑j=0

∣∣ f j∣∣2

)1/2∥∥∥∥κ,p

.

4.2 The Littlewood–Paley–Stein function

Let (X ,μ) be a σ -finite measure space with a positive measure μ . Given 0 < p < ∞, wedenote by Lp(dμ) the usual Lebesgue space of functions on X with finite quasi-norm

‖ f‖Lp(dμ) :=(∫

X| f (x)|p dμ(x)

) 1p.

Definition 4.2.1. For a given symmetric diffusion semi-group {Tt}t≥0 on (X ,μ), theLittlewood–Paley–Stein function of f : X → C is defined by

g0( f )(x) :=(∫ ∞

0t∣∣∣ ∂∂ t

T t f (x)∣∣∣2

dt) 1

2, x ∈ X . (4.2.1)

A general Littlewood–Paley theory for a symmetric diffusion semi-group was de-veloped by E. M. Stein in his 1970 monograph [49]. In particular, the following resultwas established in [49, Theorem 10, p. 111]:

Theorem 4.2.2. If 1 < p < ∞, then for any f ∈ Lp(dμ),

‖g0( f )‖Lp(dμ) ≤ cp‖ f‖Lp(dμ).

If, in addition, f ∈ L1(dμ)∩Lp(dμ) and∫

X f dμ = 0, then the following inverse inequal-ity is also true:

‖ f‖Lp(dμ) ≤ cp‖g0( f )‖Lp(dμ).

According to Theorem 4.2.2, if 1 < p < ∞, f ∈ L1(dμ)∩Lp(dμ) and∫

X f dμ = 0,then

‖ f‖Lp(dμ) ∼ ‖g0( f )‖Lp(dμ).

Typical classical examples of the Littlewood–Paley–Stein function are given withthe Gauss semi-group etΔ and the Poisson semi-group e−t(−Δ)1/2

on the unit circle or Rd ,where Δ is the corresponding Laplace operator. The semi-group e−t(−Δ)1/2

is the one orig-inally considered by Littlewood and Paley, and it has the important additional propertyof being “subordinated” to another symmetric diffusion semi-group, namely etΔ, in thesense that

etΔ f =1√π

∫ ∞

0

e−u√

uet2Δ/4u f du.

More generally, if Tt = e−tA is a symmetric diffusion semi-group with generator −A, andPt = e−tA1/2

(in which case the above integral representation holds with −A in place ofΔ), we say that Pt is the subordinated semi-group of Tt .

The above general Littlewood–Paley theory is applicable to the h-harmonic expan-sions with a symmetric diffusion semi-group defined via Poisson integrals.

Definition 4.2.3. For a function f on the sphere Sd−1, define

g( f ) :=(∫ 1

0(1− r)

∣∣∣ ∂∂ r

Pκr f

∣∣∣2dr

) 12, (4.2.2)

where Pκr f denotes the Poisson integral of f given in (3.4.3).

As an application of Theorem 4.2.2, we have the following result.

Theorem 4.2.4. If 1 < p < ∞ and f ∈ Lp(h2κ ;Sd−1), then

‖g( f )‖κ,p ≤ cκ,p‖ f‖κ,p.

If, in addition,∫Sd−1 f (x)h2

κ(x)dσ(x) = 0, then the following inverse inequality holds:

‖ f‖κ,p ≤ cp‖g( f )‖κ,p. (4.2.3)

Proof. Define Tt := Pκe−t for t > 0. Lemma 3.4.5 then implies that Tt , for t > 0, is a

symmetric diffusion semi-group and hence, using Theorem 4.2.2,

c−1p ‖ f‖κ,p ≤ ‖g0( f )‖κ,p ≤ cp‖ f‖κ,p,

4.3. The Littlewood–Paley theory on the sphere 39

where the additional condition∫Sd−1 f (x)h2

κ(x)dσ(x)= 0 is required in the first inequality.Here, the g0-function is defined in terms of the Poisson semi-group Pκ

e−t and, performingthe change of variable r = e−t , we get

g0( f ) :=(∫ ∞

0

∣∣∣ ∂∂ t

(Pκe−t f )

∣∣∣2t dt

) 12=

(∫ 1

0

∣∣∣ ∂∂ r

Pκr f

∣∣∣2r| logr|dr

) 12.

To complete the proof, it suffices to show that

‖g0( f )‖κ,p ∼ ‖g( f )‖κ,p. (4.2.4)

Since | logr| ∼ 1− r as 1/2 ≤ r ≤ 1, and limr→0+ r logr = 0, we have that

g0( f )≤ cg( f )+ c(∫ 1/2

0

∣∣∣ ∂∂ r

Pκr f

∣∣∣2dr

) 12,

g( f )≤ cg0( f )+ c(∫ 1/2

0

∣∣∣ ∂∂ r

Pκr f

∣∣∣2dr

) 12.

However, by Lemma 3.4.5,

sup0<r≤1/2

∣∣∣ ∂∂ r

Pκr f

∣∣∣ ≤ ∞

∑j=1

j2− j+1|projκj f |,

which, using (3.3.4) and the positivity of Vκ , is bounded above by

c‖ f‖κ,1

∞

∑n=1

n22−n+1 maxt∈[−1,1]

|Cλκn (t)| ≤ c′‖ f‖κ,p < ∞.

Combining these inequalities we get the desired equation (4.2.4). �

4.3 The Littlewood–Paley theory on the sphere

In this section, two new Littlewood–Paley type g-functions are defined and studied via theCesaro (C, δ )-means for h-harmonic expansions. These g-functions play a central role inthe Littlewood–Paley theory on the sphere, and the main motivation for introducing themis to provide new equivalent norms for the spaces Lp(h2

κ ;Sd−1).

Definition 4.3.1. Given δ ≥ 0, the Littlewood–Paley function gδ ( f ) of f ∈ L(h2κ ;Sd−1)

is defined by

gδ ( f ) =( ∞

∑n=1

|Sδ+1n (hκ ; f )−Sδ

n (hκ ; f )|2n−1) 1

2. (4.3.1)

To define our next Littlewood–Paley function g∗δ ( f ), let {vk}∞k=1 be an arbitrarily

given sequence of positive numbers which satisfies the condition

supn∈N

n−1n

∑k=1

vk = M < ∞,

where M > 0 is a constant. We then fix the sequence {vk}∞k=1 and define g∗δ ( f ) as follows.

Definition 4.3.2. Given δ ≥ 0, the Littlewood–Paley function g∗δ ( f ) of f ∈ L1(h2κ ;Sd−1)

is defined by

g∗δ ( f ) :=( ∞

∑n=1

∣∣Sδ+1n (h2

κ ; f )−Sδn (h

2κ ; f )

∣∣2n−1vn

) 12.

The main result of this section is the following theorem.

Theorem 4.3.3. If f ∈ Lp(h2κ ;Sd−1) and

∫Sd−1 f (x)h2

κ(x)dσ(x) = 0, then

‖ f‖κ,p ≤ cp‖gδ ( f )‖κ,p, δ ≥ 0, 1 < p < ∞.

Conversely, if the vector-valued inequality,∥∥∥∥( ∞

∑j=1

|Sδn j(hκ ; f j)|2

) 12∥∥∥∥

κ,p≤ cp

∥∥∥∥( ∞

∑j=1

| f j|2) 1

2∥∥∥∥

κ,p, ∀{n j} ⊂ N, (4.3.2)

holds for all sequences { f j} ⊂ L(h2κ ;Sd−1), then the following inverse inequality holds:

‖g∗δ ( f )‖κ,p ≤ cpM‖ f‖κ,p, ∀ f ∈ Lp(h2κ ;Sd−1),

where the constant cp is independent of the sequence {vn}∞n=1.

Note that if, in particular, we choose v j = 1 for all j ∈N, then g∗δ ( f ) = gδ ( f ). Thus,for the Littlewood–Paley function gδ ( f ), we have the following immediate corollary.

Corollary 4.3.4. If 1 λκ , then

c−1p ‖ f‖κ,p ≤ ‖gδ ( f )‖κ,p ≤ cp‖ f‖κ,p,

where the additional condition∫Sd−1 f (x)h2

κ(x)dσ(x) = 0 is required in the first inequal-ity.

The proof of Theorem 4.3.3 is given in the following two subsections.

4.3.1 A crucial lemma

The proof of Theorem 4.3.3 requires a crucial lemma, which is proven in this subsection.We denote by |I| the length of a given interval I ⊂ R.

Lemma 4.3.5. Assume that 1 < p < ∞, δ ≥ 0 and∥∥∥∥( ∞

∑j=1

|Sδn j(hκ ; f j)|2

) 12∥∥∥∥

κ,p≤ cp

∥∥∥∥( ∞

∑j=1

| f j|2) 1

2∥∥∥∥

κ,p, ∀{n j} ⊂ N (4.3.3)

for any sequence of functions { f j} ⊂ L(h2κ ;Sd−1). If r j ∈ (0,1) and Ij is a subinterval of

[r j,1) for j = 1,2, . . ., then∥∥∥∥( ∞

∑j=1

|Sδn j(hκ ;Pκ

r jf j)|2

) 12∥∥∥∥

κ,p≤ cp

∥∥∥∥( ∞

∑j=1

1|I j|

∫I j

|Pκr f j|2 dr

) 12∥∥∥∥

κ,p, (4.3.4)

for all {n j}∞k=1 ⊂ N and { f j}∞

k=1 ⊂ L(h2κ ;Sd−1) with the constant cp being independent

of {r j}, {I j}, {n j} and { f j}.

Proof. For simplicity, we shall write Sδn f for the Cesaro (C,δ )-means Sδ

n (hκ ; f ) in theproof below. We first claim that for each {n j} ⊂ N and { f j} ⊂ L(h2

κ ;Sd−1),∥∥∥∥( ∞

∑j=1

|Sδn j

Pκr j

f j|2) 1

2∥∥∥∥

κ,p≤ cp

∥∥∥∥( ∞

∑j=1

| f j|2) 1

2∥∥∥∥

κ,p. (4.3.5)

To see this, we use Lemma 4.3.7 below to obtain

|Sδn j

Pκr j

f j|2 ≤ cn j

∑�=0

|bδ�,n j

||Sδ� f j|2, j = 1,2, . . . .

Summing over j and invoking (4.3.3), we deduce∥∥∥∥( ∞

∑j=1

|Sδn j

Pκr j

f j|2) 1

2∥∥∥∥

κ,p≤ cp

∥∥∥∥( ∞

∑j=1

n j

∑�=0

|bδ�,n j

|| f j|2) 1

2∥∥∥∥

κ,p

≤ c∥∥∥∥( ∞

∑j=1

| f j|2) 1

2∥∥∥∥

κ,p,

which proves (4.3.5).Next, we show that the desired inequality (4.3.4) follows from (4.3.5). To see this,

for each j ≥ 0 and n ≥ 1, we let {r j,i}2n

i=0 ⊂ I j be such that r j,i − r j,i−1 = 2−n|I j| forall 1 ≤ i ≤ 2n. Then, for each n ∈ N, R j,n := 2−n ∑2n

i=1 |Pκr j,i

f j|2 is a Riemann sum of theintegral 1

|I j |∫

I j|Pκ

r f j|2 dr. Thus, by Fatou’s theorem, it follows that∥∥∥∥( ∞

∑j=1

1|I j|

∫I j

|Pκr f j|2 dr

) 12∥∥∥∥

κ,p= lim

n→∞

∥∥∥∥(2−n

∞

∑j=1

2n

∑i=1

|Pκr j,i

f j|2) 1

2∥∥∥∥

κ,p.

On the other hand, since for each fixed n ∈ N, r j < r j,i for all 1 ≤ i ≤ n and j ∈ N,using (4.3.5) we have∥∥∥∥( ∞

∑j=1

|Sδn j

Pκr j

f j|2) 1

2∥∥∥∥

κ,p=

∥∥∥∥(2−n

2n

∑i=1

∞

∑j=1

|Sδn j

Pκr j/r j,i

(Pκr j,i

f j)|2) 1

2∥∥∥∥

κ,p

≤ cp

∥∥∥∥(2−n

2n

∑i=1

∞

∑j=1

|Sδn j(Pκ

r j,if j)|2

) 12∥∥∥∥

κ,p.

Thus, letting n → ∞, we obtain (4.3.4), and this completes the proof of the lemma. �The proof of the main theorem needs two more elementary lemmas. We introduce

the following notation: let sδj , j = 0,1, . . ., denote the Cesaro (C, δ )-means of a given

sequence {a j}∞j=0 of complex numbers. For a given r ∈ (0,1), define

sδn,r :=

n

∑j=0

Aδn− j

Aδn

a jr j.

Lemma 4.3.6. If δ ≥ 0, n ∈ N, and 1−n−1 ≤ r < 1, then

sδn = r−nsδ

n,r +n−1

∑j=0

aδj,nsδ

j,r,

where aδj,n are constants independent of the sequence {a j} and satisfying

max0≤ j≤n−1

|aδj,n| ≤ cδ (1− r).

Lemma 4.3.7. If δ ≥ 0 and 0 < r < 1, then

sδn,r =

n

∑j=0

bδj,nsδ

j , n = 1,2, . . . ,

where bδj,n are constants independent of {a j} and satisfying

n

∑j=0

|bδj,n| ≤ cδ .

The proofs of these two lemmas can be found in [3] and [16, Section 3.2], respec-tively.

4.3.2 Proof of Theorem 4.3.3

For simplicity, we shall write Sδn f for the Cesaro (C,δ )-means Sδ

n (hκ ; f ) in the proofbelow. We first prove the inequality ‖ f‖κ,p ≤ cp‖gδ ( f )‖κ,p. By Theorem 4.2.4, it sufficesto show that

g( f )(x)≤ cgδ ( f )(x), ∀x ∈ Sd−1. (4.3.6)

To see this, we note first that∣∣∣ ∂∂ r

Pκr f

∣∣∣ = (1− r)δ+1(1− r)−δ−1∣∣∣∣ ∞

∑j=0

jr j−1 projκj f∣∣∣∣

= (1− r)δ+1∣∣∣∣ ∞

∑n=1

( n

∑j=0

jAδn− j projκj f

)rn−1

∣∣∣∣.Since a quick computation shows that

Sδ+1n f −Sδ

n f =−(n+δ +1)−1(Aδn )

−1n

∑j=0

jAδn− j projκj f ,

it follows that ∣∣∣ ∂∂ r

Pκr f

∣∣∣ ≤ c(1− r)δ+1∞

∑n=1

nAδn∣∣Sδ+1

n f −Sδn f

∣∣rn−1,


which, by the Cauchy–Schwartz inequality, implies∣∣∣ ∂∂ r

Pκr f

∣∣∣2 ≤ c(1− r)2δ+2( ∞

∑n=1

nAδn∣∣Sδ+1

n f −Sδn f

∣∣2rn−1)( ∞

∑n=1

nAδn rn−1

)= c(1+δ )(1− r)δ

∞

∑n=1

nAδn |Sδ+1

n f −Sδn f |2rn−1.

Consequently,

|g( f )|2 =∫ 1

0

∣∣∣ ∂∂ r

Pκr f

∣∣∣2(1− r)dr ≤ c

∞

∑n=1

nAδn |Sδ+1

n f −Sδn f |2

∫ 1

0(1− r)1+δ rn−1 dr

≤ c∞

∑n=1

n−1|Sδ+1n f −Sδ

n f |2 = |gδ ( f )|2,

where the third step uses the fact that∫ 1

0 (1− r)δ+1rn−1 dr = Γ(δ+2)Γ(n)Γ(n+δ+2) ∼ n−δ−2. This

proves the desired inequality (4.3.6).We now turn to the proof of the second assertion in Theorem 4.3.3. Without loss of

generality, we may assume that n≤∑nj=1 v j ≤ 2n, since the desired conclusion for general

{v j} can be deduced by applying the result in this special case to the two sequences v j = 1and v j = M−1v j +1, respectively. For convenience, we define, for n = 1,2, . . .,

En f =−(n+1+δ )−1n

∑j=0

j projκj f .

It is easily seen that, for 0 ≤ j ≤ n,

Sδj (En f ) =

j+δ +1n+δ +1

(Sδ+1

j f −Sδj f

). (4.3.7)

Using Lemma 4.3.6 we obtain that, for any r ∈ [1−n−1,1),

Sδ+1n f −Sδ

n f = Sδn (En f ) = r−nPκ

r (Sδn (En f ))+

n−1

∑j=1

aδj,nPκ

r (Sδj (En f ))

= r−n(

Sδ+1n (Pκ

r f )−Sδn (P

κr f )

)+

n−1

∑j=1

j+δ +1n+δ +1

aδj,n

[Sδ+1

j (Pκr f )−Sδ

j (Pκr f )

], (4.3.8)

where |aδj,n| ≤ c(1− r) ≤ cn−1, and the last step uses (4.3.7). Now let μ1 = 1, and μn =

1+∑n−1i=1 vi for n > 1. Clearly, rn := 1− 1

μn∈ [1− n−1,1− (2n− 1)−1]. Thus, applying

(4.3.8) with r = rn, and setting fn = Pκrn f , we deduce that

|Sδ+1n f −Sδ

n f | ≤ c|Sδ+1n fn −Sδ

n fn|+ cn−2n−1

∑j=1

j|Sδ+1j ( fn)−Sδ

j ( fn)|,


which, using the Cauchy–Schwartz inequality again, yields

|Sδ+1n f −Sδ

n f |2 ≤ c|Sδ+1n fn −Sδ

n fn|2 + cn−3n−1

∑j=1

j2|Sδ+1j fn −Sδ

j fn|2. (4.3.9)

Therefore, by (4.3.9) and Corollary 4.2.4, to complete the proof of the second as-sertion in Theorem 4.3.3 it suffices to show the inequalities∥∥∥∥( ∞

∑n=1

n−1|Sδ+1n fn −Sδ

n fn|2vn

) 12∥∥∥∥

κ,p≤ cp‖g( f )‖κ,p (4.3.10)

and ∥∥∥∥( ∞

∑n=1

vn

n4

n−1

∑j=1

j2|Sδ+1j fn −Sδ

j fn|2) 1

2∥∥∥∥

κ,p≤ cp‖g( f )‖κ,p. (4.3.11)

To this end, let η ∈C∞(R) and Lκn be as in Definition 4.1.4. Define

Dn f =−2n

∑j=0

jη( j

n

)projκj f , n = 1,2, . . . ,

and observe that, for 1 ≤ j ≤ n ≤ N,

Sδ+1j fn −Sδ

j fn = ( j+δ +1)−1Pκrn(S

δj (DN f )). (4.3.12)

Thus, using Lemma 4.3.5 and (4.3.12) with j = n, we obtain∥∥∥∥( N

∑n=1

n−1|Sδ+1n fn −Sδ

n fn|2vn

) 12∥∥∥∥

κ,p≤ c

∥∥∥∥( N

∑n=1

vn

n3 |Pκrn(S

δn (DN f ))|2

) 12∥∥∥∥

κ,p

≤ c∥∥∥∥( N

∑n=1

vn

n31

rn+1 − rn

∫ rn+1

rn

|Pκr (DN f )|2 dr

) 12∥∥∥∥

κ,p.

Since

|Pκr (DN f )|= r

∣∣∣LκN

( ∂∂ r

Pκr f

)∣∣∣,applying Corollary 4.1.5 to the Riemann sums of the integrals

∫ rn+1rn

, we obtain∥∥∥∥( N

∑n=1

vn

n31

rn+1 − rn

∫ rn+1

rn

|Pκr (DN f )|2 dr

) 12∥∥∥∥

κ,p

≤ cp

∥∥∥∥( N

∑n=1

vn

n31

rn+1 − rn

∫ rn+1

rn

∣∣∣ ∂∂ r

Pκr f

∣∣∣2dr

) 12∥∥∥∥

κ,p.

Since rn+1 − rn =vn

μnμn+1∼ vn

n2 and 1− r ∼ 1n for all r ∈ [rn,rn+1], it follows that∥∥∥∥( N

∑n=1

vn

n31

rn+1 − rn

∫ rn+1

rn

∣∣∣ ∂∂ r

Pκr f

∣∣∣2dr

) 12∥∥∥∥

κ,p≤ cp

∥∥∥∥( ∞

∑n=1

1n

∫ rn+1

rn

∣∣∣ ∂∂ r

Pκr f

∣∣∣2dr

) 12∥∥∥∥

κ,p

≤ cp‖g( f )‖κ,p.

4.4. The Marcinkiewicz type multiplier theorem 45

Putting the above together, and letting N → ∞, we obtain (4.3.10).The proof of (4.3.11) is similar. In fact, using Lemma 4.3.5 and (4.3.12), we have∥∥∥∥( N

∑n=1

vn

n4

n−1

∑j=1

j2|Sδ+1� fn −Sδ

� fn|2) 1

2∥∥∥∥

κ,p≤ c

∥∥∥∥( N

∑n=1

vn

n4

n−1

∑j=1

|Pκrn(S

δj DN f )|2

) 12∥∥∥∥

κ,p

≤ cp

∥∥∥∥( ∞

∑n=1

vn

n4

n−1

∑j=1

1rn+1 − rn

∫ rn+1

rn

| ∂∂ r

Pκr f |2 dr

) 12∥∥∥∥

κ,p

≤ cp

∥∥∥∥( ∞

∑n=1

vn

n31

rn+1 − rn

∫ rn+1

rn

| ∂∂ r

Pκr f |2 dr

) 12∥∥∥∥

κ,p≤ cp‖g( f )‖κ,p.

Letting N → ∞ yields (4.3.11). �

4.4 The Marcinkiewicz type multiplier theorem

In this section, we prove the Marcinkiewicz type multiplier theorem for spherical h-harmonic expansions. The conditions on the multiplier are stated in terms of the forwarddifference, defined below.

Definition 4.4.1. Given a sequence {a j}∞j=0 of complex numbers, define

�a j = a j −a j+1, �n+1a j =�na j −�na j+1.

Theorem 4.4.2. Let {μ j}∞j=0 be a sequence of complex numbers satisfying

(A0) supj|μ j| ≤ M < ∞,

(An0) supj≥1

2 j(n0−1)2 j+1

∑l=2 j

|�n0 μl | ≤ M < ∞,

where n0 is the smallest integer bigger than or equal to λκ + 1. Then {μ j} defines anLp(h2

κ ;Sd−1) multiplier for all 1 < p < ∞; namely,∥∥∥∥∥ ∞

∑j=0

μ j projκj f

∥∥∥∥∥κ,p

≤ cpM‖ f‖κ,p, 1 < p < ∞,

where the constant cp is independent from {μ j} and f .

Remark 4.4.3. If {μ j} is a sequence of complex numbers satisfying the condition (Ak) forsome positive integer k, then, as can be easily verified from the definition, {μ j} satisfiesthe condition (Ai) for all 1 ≤ i ≤ k, with a possibly different absolute constant M.

The proof of Theorem 4.4.2 relies on the following lemma, whose proof can befound in [3] and [16, Lemma 3.3.3].

Lemma 4.4.4. Let {μ j}∞j=0 be a bounded sequence of complex numbers satisfying the

condition (Aδ+1) for some nonnegative integer δ . Assume that {a j}∞j=0 is another se-

quence of complex numbers. Let sδn and σδ

n denote the Cesaro (C,δ )-means of the se-quences {a j}∞

j=0 and {a jμ j}∞j=0, respectively. Then

σδn = μnsδ

n +n−1

∑�=0

Cδ�,nsδ

� ,

where the constants Cδ�,N are independent of {a j}∞

j=0 and satisfy

∣∣∣Cδ�,n

∣∣∣ ≤ cδ+1

∑k=1

(�+1)k−1∣∣∣�kμ�

∣∣∣ , �= 0,1, . . . ,n−1. (4.4.1)

Proof of Theorem 4.4.2. Without loss of generality, we may assume μ0 = 0 and M = 1.Let δ be the smallest integer bigger than λκ . Then (4.3.2) follows from Theorem 4.1.3.Now let F = ∑∞

j=1 μ j projκj f . By Theorem 4.3.3 and Corollary 4.3.4, it suffices to show

gδ (F)≤ c( ∞

∑n=1

|Sδ+1n f −Sδ

n f |2vnn−1) 1

2=: g∗δ ( f ), (4.4.2)

for every sequence of positive numbers {vn} satisfying the condition

supN∈N

N−1N

∑j=1

v j ≤ cM < ∞.

For the proof of (4.4.2), we use Lemma 4.4.4 to obtain

Sδ+1n F −Sδ

n F =−1

n+δ +1(Aδ

n )−1

n

∑j=0

Aδn− jμ j j projκj f

=1

n+δ +1

(μnσδ

n +n−1

∑�=0

Cδ�,nσδ

�

),

where

σδ� =−(Aδ

� )−1

�

∑j=0

Aδ�− j j projκj f = (�+δ +1)

(Sδ+1� f −Sδ

� f).

It then follows by (4.4.1) that

|Sδ+1n F −Sδ

n F | ≤ |μn||Sδ+1n f −Sδ

n f |+Cn−1δ+1

∑j=1

n−1

∑�=1

� j|� jμ�||Sδ+1� f −Sδ

� f |.

On the other hand, using Remark 4.4.3, we deduce from condition (Aδ+1) that

δ+1

∑j=1

n−1

∑�=1

� j|� jμ�| ≤ cn. (4.4.3)

4.5. A Littlewood–Paley inequality 47

Thus, using (4.4.3) and the Cauchy–Schwartz inequality,

|gδ (F)|2 ≤ c|gδ ( f )|2 + cδ+1

∑j=1

∞

∑n=1

n−2(n−1

∑�=1

� j|� jμ�||Sδ+1� f −Sδ

� f |2)

≤ c|gδ ( f )|2 + c∞

∑�=1

|Sδ+1� f −Sδ

� f |2δ+1

∑j=1

� j−1|� jμ�|

≤ c∞

∑n=1

|Sδ+1n f −Sδ

n f |2vnn−1,

where vn = 1+δ+1

∑j=1

|� jμn|n j. However, using (4.4.3) once again,

n−1n

∑�=1

v� = 1+δ+1

∑j=1

n−1n

∑�=1

� j|� jμ�| ≤ c.

This proves the desired equation (4.4.2) and completes the proof of Theorem 4.4.2. �

4.5 A Littlewood–Paley inequality

In this section, as an application of the Marcinkiewicz multiplier theorem, we prove auseful Littlewood–Paley type inequality. Let us start with the following definition.

Definition 4.5.1. Given a compactly supported continuous function θ : [0,∞) → R, wedefine a sequence of operators Δκ

θ , j by Δκθ ,0( f ) = projκ0 ( f ), and

Δκθ , j( f ) :=

∞

∑n=0

θ( n

2 j

)projκn ( f ), j = 1,2, . . . .

Theorem 4.5.2. Let m be the smallest positive integer greater than λκ + 1. If θ is acompactly supported function in Cm[0,∞) with suppθ ⊂ (a,b) for some 0 < a < b < ∞,then, for all f ∈ Lp(h2

κ ;Sd−1) with 1 0, (4.5.2)

and∫Sd−1 f (x)h2

κ(x)dσ(x) = 0, then∥∥∥∥( ∞

∑j=0

∣∣Δκθ , j f

∣∣2)1/2∥∥∥∥

κ,p∼ ‖ f‖κ,p, 1 < p < ∞. (4.5.3)

Proof. Firstly, let us prove (4.5.1). Let {ξ j}∞j=0 be a sequence of independent random

variables taking the values ±1 and having zero mean. Then, by the Khinchine inequality,for any sequence {a j} of complex numbers,

(E

∣∣∣∣∣ ∞

∑j=0

a jξ j

∣∣∣∣∣p)1/p

∼(

∞

∑j=0

|a j|2) 1

2

, 0 < p < ∞, (4.5.4)

where E denotes the expectation of random variables. Now consider the (random) linearoperator

T f =∞

∑j=0

ξ jΔκθ , j f . (4.5.5)

Directly from the definition of Δκθ , j f , T f can be rewritten in the form

T f =∞

∑k=1

A(k)projκk f , A(u) :=∞

∑j=0

θ( u

2 j

)ξ j.

Since θ ∈Cm[0,∞) is supported in a finite interval (a,b)⊂ (0,∞), it follows by a straight-forward computation that∣∣∣∣( d

du

)r

A(u)∣∣∣∣ ≤ cru−r, u ≥ 1, r = 0,1, . . . ,m,

which, in particular, implies that

|�rA(k)| ≤ c′rk−r, r = 0,1, . . .m, k ≥ 1,

where the constants cr and c′r are independent of the random variables ξ j. We now applythe Marcinkiewicz multiplier theorem (Theorem 4.4.2) with μk = A(k) to deduce that

‖T f‖κ,p ≤ cp‖ f‖κ,p, 1 < p < ∞, (4.5.6)

where cp is a constant depending only on p, d and κ . Combining (4.5.4) and (4.5.5) with(4.5.6), we conclude that∥∥∥∥∥

(∞

∑j=0

∣∣Δκθ , j f

∣∣2

) 12∥∥∥∥∥

κ,p

∼ (E‖T f‖p

κ,p)1/p ≤ cp‖ f‖κ,p,

which proves the desired inequality (4.5.1).Secondly, we prove the inverse inequality∥∥∥∥∥

(∞

∑j=1

(Δκ

θ , j f)2

)1/2∥∥∥∥∥κ,p

≥ c′p‖ f‖κ,p, (4.5.7)

4.5. A Littlewood–Paley inequality 49

for f ∈ Lp(h2κ ;Sd−1) with 1 0. (4.5.8)

This assumption implies that, for every spherical polynomial g,

∞

∑j=0

(Δκθ , j ◦Δκ

θ , j)g = g−projκ0 g. (4.5.9)

Now, for f ∈ Lp(h2κ ;Sd−1) with

∫Sd−1 f (x)h2

κ(x)dσ(x) = 0 and ε > 0, there is a g ∈Lq(h2

κ ;Sd−1) with ‖g‖κ,1 = 1, where 1p +

1q = 1, such that

‖ f‖κ,p − ε/2 ≤ 1ωd

∫Sd−1

f (x)g(x)h2κ(x)dσ(x).

Let gn be a spherical polynomial such that ‖g−gn‖κ,p < ε/2. Then it follows readily that‖ f‖κ,p − ε ≤ 1

ωd|∫Sd−1 f gnh2

κ dσ(x)|. Using (4.5.9), we have

1ωd

∣∣∣∣∫Sd−1

f gnh2κ dσ(x)

∣∣∣∣ = 1ωd

∣∣∣∣∣∫Sd−1

∞

∑j=0

Δκθ , j f (x)Δκ

θ , jgn(x)h2κ(x)dσ(x)

∣∣∣∣∣≤ c

∥∥∥∥∥(

∞

∑j=0

|Δκθ , j f |2

) 12∥∥∥∥∥

κ,p

∥∥∥∥∥(

∞

∑j=0

|Δκθ , jgn|2

) 12∥∥∥∥∥

κ,q

≤ c‖gn‖κ,q

∥∥∥∥∥(

∞

∑j=0

|Δκθ , j f (x)|2

) 12∥∥∥∥∥

κ,p

≤ c

∥∥∥∥∥(

∞

∑j=0

|Δκθ , j f (x)|2

) 12∥∥∥∥∥

κ,p

.

This proves (4.5.7) under the additional condition (4.5.8).Finally, we show that, for (4.5.7) to hold, condition (4.5.8) can be relaxed to (4.5.2).

To this end, we define

θ(x) :=θ(x)(

∑∞j=0 |θ(2− jx)|2

) 12.

It is obvious that θ ∈Cm[0,∞), supp θ ⊂ (a,b)⊂ (0,∞), and

∞

∑j=0

θ(2− jx) = 1, ∀x > 0.

Thus, using the already proven case of the inequality (4.5.7), we have

‖ f‖κ,p ≤ cp

∥∥∥∥∥(

∞

∑j=1

(Δκ

θ , j f)2

)1/2∥∥∥∥∥κ,p

. (4.5.10)

Next, let φ ∈C∞[0,∞) be such that φ(x) = 1 for x ∈ [a,b], and suppφ ⊂ (a1,b1) for some0 < a1 < a < b < b1 < ∞. Define

ψ(x) =φ(x)(

∑∞j=0 |θ(2− jx)|2

) 12.

Then θ(x) = θ(x)ψ(x), and hence Δκθ , j

= Δκψ, j ◦Δκ

θ , j. Thus, we may rewrite (4.5.10) inthe form

‖ f‖κ,p ≤ cp

∥∥∥∥∥(

∞

∑j=0

|Δκψ, jg j|2

) 12∥∥∥∥∥

κ,p

, (4.5.11)

with g j :=Δκθ , j f . On the other hand, since ψ ∈Cm[0,∞) has compact support, and m−1>

λκ , using Theorem 3.5.4 and summation by parts finitely many times, it follows that

supj∈N

|Δκψ, jg(x)| ≤ cκ Sλκ+1

∗ (g)(x)≤CMκ g(x)+CMκ g(−x).

Thus, by Theorem 4.1.1,∥∥∥∥∥(

∞

∑j=0

|Δκψ, jg j|2

) 12∥∥∥∥∥

κ,p

≤ cp

∥∥∥∥∥(

∞

∑j=0

|g j|2) 1

2∥∥∥∥∥

κ,p

= cp

∥∥∥∥∥(

∞

∑j=0

|Δκθ , j f |2

) 12∥∥∥∥∥

κ,p

,

which shows the desired inverse inequality. �


A general Littlewood–Paley theory for a symmetric diffusion semi-group was developedby E. M. Stein in his 1970 monograph [49]. A vector-valued extension of this generaltheory was developed in [33]. The main references for Section 4.2 are [3, 33, 49].

The multiplier theorem (Theorem 4.4.2) and its analogue on the unit ball and thesimplex were proved in [12]. The Littlewood–Paley theory in Section 4.3 and the proof ofTheorem 4.4.2 follow the argument in Bonami and Clerc [3], who established the theoryin the unweighted setting.

Applications of the refined Littlewood–Paley inequality, Theorem 4.5.2, can befound in [7, 9].

Chapter 5

Sharp Jackson and Sharp

Marchaud Inequalities

The goal of this chapter is to prove two inequalities, the sharp Jackson inequality andthe sharp Marchaud inequality, for the h-harmonic expansions on the sphere Sd−1, whichare useful in the embedding theory of function spaces. The multiplier theorem and theLittlewood–Paley inequality established in the prior chapter play crucial roles in theirproofs.

As a motivation, these inequalities for trigonometric polynomial approximation onthe circle are stated in Section 5.1. Section 5.2 contains several useful properties andresults on weighted moduli of smoothness, including the direct Jackson inequality andits inverse. A weighted K-functional that is equivalent with the weighted modulus ofsmoothness is defined in Section 5.3, using fractional powers of the h-Laplace–Beltramioperator. Weighted sharp Marchaud and sharp Jackson inequalities are proved in Sections5.4 and 5.5, respectively. Finally, the optimality of the parameters in the sharp Marchaudinequality and the sharp Jackson inequality is proven in Section 5.6.

5.1 Introduction

For trigonometric polynomial approximation of functions on the unit circle S1 (identifiedwith [−π,π]), M. Timan [56] proved that, for 1 < p < ∞,

n−r{ n

∑k=1

ksr−1Ek( f )sp

}1/s ≤C(r, p)ωr( f ,n−1)p, s = max(p,2), (5.1.1)

where r ∈N, Ek( f )p is the best approximation of f ∈ Lp(S1) by trigonometric polynomi-als of degree at most k,

Ek( f )p = min{‖ f −Tn‖Lp(S1) : Tn ∈ span

k<n{sinkt,coskt}

},




87

51

5

52 Chapter 5. Sharp Jackson and Sharp Marchaud Inequalities

and ωr( f , t)p denotes the r-th order modulus of smoothness of f ∈ Lp(S1),

ωr( f , t)p = sup|h|≤t

∥∥∥ r

∑j=0

(−1)r− j(

rj

)f (·+ jh)

∥∥∥Lp(S1)

.

We call an inequality of the type (5.1.1) a sharp Jackson inequality, since it improves theclassical Jackson inequality, En( f )p ≤Cωr( f ,1/n)p, for 1 < p < ∞.

An estimate of ωr( f , t)p in the direction opposite to (5.1.1) was proved by M. Timan[55] as well: for 1 < p < ∞ and r ∈ N,

ωr( f ,1/n)p ≤ c(r, p)n−r{ n

∑k=1

krq−1Ek( f )qp

}1/q, q = min(p,2). (5.1.2)

This estimate is, in fact, equivalent to the following inequality on moduli of smoothness:for 1 < p < ∞ and r ∈ N,

ωr( f , t)p ≤ c(r, p)tr{∫ 1/2

t

ωr+1( f ,u)qp

uqr+1 du}1/q

, q = min(p,2). (5.1.3)

The inequality (5.1.3) is stronger than the usual Marchaud inequality

ωr( f , t)p ≤ c(r, p)tr∫ 1/2

t

ωr+1( f ,u)p

ur+1 du,

and accordingly is sometimes called the sharp Marchaud inequality.Similar to the sharp Marchaud inequality (5.1.3), one has in the other direction the

following inequality on moduli of smoothness, which is equivalent to the sharp Jacksoninequality (5.1.1): for 1 < p < ∞,

tr{∫ 1/2

t

ωr+1( f ,u)sp

usr+1 du}1/s ≤Cωr( f , t)p, s = max(p,2). (5.1.4)

Of particular interest is the case p = 2, for which (5.1.1) combined with (5.1.2)yields

ωr( f ,1/n)2 ∼ n−r{ n

∑k=1

k2r−1Ek( f )22

}1/2.

5.2 Moduli of smoothness and best approximation

Let G be a finite reflection group and let hκ be the weight function defined in (2.1.2),which is invariant under G. Then hκ is a homogeneous function of degree ∑v∈R+

κv. Recallthat

λκ = ∑v∈R+

κv +d −2

2.

5.2. Moduli of smoothness and best approximation 53

Associated with the weight h2κ , a generalized translation operator T κ

θ is defined in (3.5.1)for all θ ∈ R, which we write as

projκn (Tκ

θ f ) =Cλκ

n (cosθ)Cλκ

n (1)projκn f , n = 0,1, . . . .

The operators T κθ are uniformly bounded on Lp(h2

κ ;Sd−1), as shown in (3.5.3),

‖T κθ f‖κ,p ≤ ‖ f‖κ,p, 1 ≤ p ≤ ∞.

When κ = 0, T 0θ is the usual translation operator on Sd−1, aka spherical mean operator.

For r > 0 and 0 < θ < π , we define the r-th order difference operator

(I −T κθ )r/2 :=

∞

∑n=0

(−1)n(

r/2n

)(T κ

θ )n,

in a distributional sense, by

projκn[(I −T κ

θ )r/2 f]=

(1− Cλκ

n (cosθ)Cλκ

n (1)

)r/2

projκn f , n = 0,1,2, . . . .

Definition 5.2.1. Let r > 0 and 0 < θ < π . For f ∈ Lp(h2κ ;Sd−1) and 1 ≤ p < ∞, or

f ∈C(Sd−1) and p = ∞, the weighted r-th order modulus of smoothness is defined as

ωr( f , t)κ,p := sup0<θ≤t

‖(I −T κθ )r/2 f‖κ,p, 0 < t < π. (5.2.1)

This definition makes sense, since the next proposition shows that ωr( f , t)κ,p satis-fies the basic properties of the usual moduli:

Proposition 5.2.2. Let f ∈ Lp(h2κ) if 1 ≤ p < ∞ and f ∈C(Sd−1) if p = ∞. Then

1. ωr( f , t)κ,p ≤ 2r+2‖ f‖κ,p;

2. ωr( f , t)κ,p → 0 if t → 0+;

3. ωr( f , t)κ,p is monotone nondecreasing on (0,π);

4. ωr( f +g, t)κ,p ≤ ωr( f , t)κ,p +ωr(g, t)κ,p;

5. for 0 < s < r,ωr( f , t)κ,p ≤ 2(r−s)+2ωs( f , t)κ,p.

The proof of Proposition 5.2.2 can be found in [16, Chapter 10] and [72].

Definition 5.2.3. For f ∈ Lp(h2κ) and 1 ≤ p < ∞, or f ∈C(Sd−1) and p = ∞, the weighted

best approximation of f by spherical polynomials of degree at most n is defined as

En( f )κ,p := infP∈Πn−1(Sd−1)

‖ f −P‖κ,p.

We will need the near best approximation operator defined via a cut-off function forh-spherical harmonic expansions.

Definition 5.2.4. Let η be a C∞ function on [0,∞) such that η(t) = 1 for 0 ≤ t ≤ 1 andη(t) = 0 for t ≥ 2. For f ∈ L(h2

κ), we define

Lκn f :=

2n

∑j=0

η( j

n

)projκj f , n = 1,2, . . . . (5.2.2)

The following proposition collects several useful results on the operator Lκn .

Proposition 5.2.5. Let f ∈ Lp(h2κ) if 1 ≤ p < ∞ and f ∈C(Sd−1) if p = ∞. Then

(1) Lκn f ∈ Π2n−1(S

d−1) and Lκn f = f for f ∈ Πn(S

d−1);

(2) for n ∈ N, ‖Lκn f‖κ,p ≤ c‖ f‖κ,p;

(3) for n ∈ N, ‖ f −Lκn f‖κ,p ≤ (1+ c)En( f )κ,p.

Theorem 5.2.6. Let f ∈ Lp(h2κ ;Sd−1) if 1 ≤ p < ∞, and f ∈C(Sd−1) if p = ∞. Then for

any r > 0 and n ∈ N,En( f )κ,p ≤ cωr( f ,n−1)κ,p, (5.2.3)

and

ωr( f ,n−1)κ,p ≤ cn−rn

∑k=0

(k+1)r−1Ek( f )κ,p. (5.2.4)

The proofs of Proposition 5.2.5 and Theorem 5.2.6 can be found in [16, Chapter10].

5.3 Weighted Sobolev spaces and K-functionals

Recall that the space H dn (h2

κ) of h-spherical harmonics on Sd−1 of degree n is an eigen-vector space of the h-Laplace–Beltrami operator Δh,0; namely,

H dn (h2

κ) ={

f ∈C2(Sd−1) : Δh,0 f =−n(n+2λκ) f}, n = 0,1, . . . .

Accordingly, we can define fractional powers of Δh,0 and the weighted Sobolev spaces asfollows.

Definition 5.3.1. For r > 0 and 1 ≤ p ≤ ∞, a function f ∈ Lp(h2κ ;Sd−1) is said to belong

to the weighted Sobolev space W rp (h

2κ) if there exists a function g ∈ Lp(h2

κ ;Sd−1), whichwe denote by (−Δh,0)

r/2 f , such that

projκn[(−Δh,0)

r/2 f]= (n(n+2λκ))

r/2 projκn f , n = 0,1, . . . , (5.3.1)

5.3. Weighted Sobolev spaces and K-functionals 55

where we assume f ,g ∈C(Sd−1) when p = ∞. The norm in the weighted Sobolev spaceW r

p (h2κ) is defined by

‖ f‖W rp (h2

κ ):= ‖ f‖κ,p +‖(−Δh,0)

r/2 f‖κ,p.

The fractional spherical h-Laplacian (−Δh,0)r/2 is then a linear operator on the space

W rp (h

2κ) defined by (5.3.1).

Let η ∈C∞[0,∞) be as in Definition 5.2.4, and let θ(x) = η(2x)−η(4x). Let �κθ , j,

j = 0,1, . . ., be the operators defined in Definition 4.5.1. Obviously, �κθ , j( f ) := Lκ

2 j−1 f −Lκ

2 j−2 f and, for f ∈ Lp(h2κ ;Sd−1) if 1 ≤ p < ∞ or f ∈C(Sd−1) if p = ∞,

f =∞

∑j=0

�κθ , j f , (5.3.2)

where the series converges in the norm of Lp(h2κ).

A Littlewood–Paley type inequality holds on the weighted Sobolev spaces W rp (h

2κ).

Theorem 5.3.2. For 1 < p < ∞, γ ≥ 0, and f ∈W γp (h2

κ),∥∥∥{ ∞

∑j=1

22 jγ(�κθ , j f

)2}1/2∥∥∥

κ,p∼ ‖(−Δh,0)

γ/2 f‖κ,p, (5.3.3)

and ∥∥∥{ ∞

∑j=0

22 jγ(�κθ , j f

)2}1/2∥∥∥

κ,p∼ ‖ f‖W

γp (h2

κ ), (5.3.4)

where the constants of equivalence depend only on p, d and γ.

Theorem 5.3.2 is a consequence of the Marcinkiewicz multiplier theorem, Theorem4.4.2, and its proof runs along the same line as that of Theorem 4.5.2.

For f ∈ Lp(h2κ), we define its K-functional in terms of the h-spherical Laplacian as

follows.

Definition 5.3.3. Given r > 0, the r-th K-functional of f ∈ Lp(h2κ) is

Kr( f ; t)κ,p := infg∈W r

p (h2κ )

{‖ f −g‖κ,p + tr

∥∥∥(−Δh,0)r/2 g

∥∥∥κ,p

}. (5.3.5)

We have the following realization theorem of the K-functional, whose proof can befound in [16, Chapter 10] and [21].

Theorem 5.3.4. Let f ∈ Lp(h2κ) if 1 ≤ p < ∞ and f ∈C(Sd−1) if p = ∞. If t ∈ (0,1) and

n is a positive integer such that n ∼ t−1, then

Kr( f , t)κ,p ∼ ‖ f −Lκn f‖κ,p +n−r

∥∥∥(−Δh,0)r/2Lκ

n f∥∥∥

κ,p. (5.3.6)

It turns out that the weighted modulus of smoothness ωr( f , t)κ,p and the weightedK-functional Kr( f ; t)κ,p are equivalent.

Theorem 5.3.5. Let f ∈ Lp(h2κ) if 1 ≤ p < ∞ and let f ∈C(Sd−1) if p = ∞. If t ∈ (0, π

2 )and r > 0, then

ωr( f , t)κ,p ∼∥∥∥(I −T κ

t )r/2 f∥∥∥

κ,p∼ Kr( f , t)κ,p.

The proof of Theorem 5.3.5 can be found in [16, Chapter 10]. Theorem 5.3.5 com-bined with Theorem 5.2.6 yields the following direct Jackson inequality and the inverseinequality.

Theorem 5.3.6. Let f ∈ Lp(h2κ ;Sd−1) if 1 ≤ p < ∞, and f ∈C(Sd−1) if p = ∞. Then for

any r > 0 and n ∈ N,En( f )κ,p ≤ cKr( f ;n−1)κ,p,

and

Kr( f ,n−1)κ,p ≤ cn−rn

∑k=0

(k+1)r−1Ek( f )κ,p.

5.4 The sharp Marchaud inequality

In this section, we will prove the following sharp Marchaud inequality.

Theorem 5.4.1. For α > 0, 1 < p < ∞ and q = min(p,2),

ωα( f , t)κ,p ≤ ctα(∫ 1

t

ωα+1( f ,u)qκ,p

uqα+1 du) 1

q

. (5.4.1)

Using Theorem 5.2.6, it can be easily seen that Theorem 5.4.1 is, in fact, equivalentto the following sharp inverse inequality.

Corollary 5.4.2. For 1 0,

ωα( f ,1/n)κ,p ≤ c(α, p)n−α( n

∑j=1

jαq−1E j( f )qκ,p

)1/q. (5.4.2)

The proof of Theorem 5.4.1 relies only on the Cesaro summability of the orthogonalexpansions, and on Theorem 4.2.4 (the Littlewood–Paley–Stein theorem).

Proof of Theorem 5.4.1. By Theorem 5.3.5 and (5.3.6), it suffices to show that

2−mα‖(−Δh,0)α/2(Lκ

2m( f ))‖κ,p ≤ c2−mα

(m

∑j=0

2 jαqKα+1( f ,2− j)qκ,p

) 1q

, (5.4.3)

where q = min{p,2}. For convenience, we set F = (−Δh,0)α/2(Lκ

2m( f )), and use Sδn ( f )

to denote the Cesaro (C,δ )-means Sδn (h

2κ ; f ) defined in (3.3.7).

5.4. The sharp Marchaud inequality 57

For the proof of (5.4.3) we claim that for, δ ≥ 0,

g(F)≤ c

(∞

∑j=0

2− j(1+q)2 j+1−1

∑i=2 j

|Sδi ((−Δh,0)

1/2F)|q) 1

q

, (5.4.4)

where g( f ) denotes the the Littlewood–Paley–Stein g-function defined in (4.2.2).For the moment, we take the claim (5.4.4) for granted and proceed with the proof.

Using (5.4.4) and Theorem 4.2.4, we obtain

‖F‖κ,p ≤C‖g(F)‖κ,p ≤ c

(∞

∑j=0

2− j(1+q)2 j+1−1

∑i=2 j

‖Sδi ((−Δh,0)

1/2F)‖qκ,p

) 1q

, (5.4.5)

where the last step uses the Minkowski inequality for p > 2. We break the first sum on theright side of (5.4.5) into two parts: ∑m−4

j=0 and ∑∞j=m−3. Observe that if 2 j ≤ i ≤ 2 j+1 −1

and 0 ≤ j ≤ m−4, then

Sδi ((−Δh,0)

1/2F) = Lκ2 j+2(Sδ

i ((−Δh,0)1/2F)) = Sδ

i (Lκ2 j+2((−Δh,0)

1/2F))

= Sδi ((−Δh,0)

(α+1)/2(Lκ2 j+2( f ))). (5.4.6)

Thus, if δ > λκ ,

(m−4

∑j=0

2− j(1+q)2 j+1−1

∑i=2 j

‖Sδi ((−Δh,0)

1/2F)‖qκ,p

) 1q

≤ c

(m−4

∑j=0

2− jq∥∥∥(−Δh,0)

(α+1)/2(Lκ2 j+2( f ))

∥∥∥q

κ,p

) 1q

≤ c

(m−4

∑j=0

2 jαqKα+1( f ,2− j)qκ,p

) 1q

, (5.4.7)

where the second step uses (5.4.6) and the Cesaro (C,δ )-summability of h-harmonic ex-pansions for δ > λκ , and the last step uses (5.3.6). However, on the other hand, for δ > λκ ,

( ∞

∑j=m−3

2− j(1+q)2 j+1−1

∑i=2 j

‖Sδi ((−Δh,0)

1/2F)‖qκ,p

) 1q ≤ c

(∞

∑j=m−3

2− jq

) 1q

‖(−Δh,0)1/2F‖κ,p

≤ c2−m‖(−Δh,0)1/2F‖κ,p = c2−m‖(−Δh,0)

(α+1)/2(Lκ2m( f ))‖κ,p

≤ c2mα Kα+1( f ,2−m)κ,p, (5.4.8)

where the last step uses the realization (5.3.6). Therefore, combining (5.4.5), (5.4.7), and(5.4.8), we deduce the desired estimate (5.4.3).

It remains to prove claim (5.4.4). Note that

∂∂ r

Pκr (F) =

∞

∑i=1

iri−1 projκi (F) = (1− r)δ+1∞

∑i=1

(i

∑j=1

jAδi− j projκj (F)

)ri−1,

where we used the identity ∑∞k=0 Aδ

k rk = (1− r)−1−δ in the last step. Thus,∣∣∣ ∂∂ r

Pκr (F)

∣∣∣ ≤ (1− r)δ+1∞

∑i=1

Aδi |Sδ

i ((−Δh,0)1/2F)|ri−1

≤ c(1− r)1+δ∞

∑j=0

2 jδ r2 j−12 j+1−1

∑i=2 j

|Sδi ((−Δh,0)

1/2F)|,

which, using the Cauchy–Schwartz inequality, implies∣∣∣ ∂∂ r

Pκr (F)

∣∣∣2 ≤ c(1− r)2+2δ( ∞

∑j=0

2 jδ r2 j−1( 2 j+1−1

∑i=2 j

|Sδi ((−Δh,0)

1/2F)|)2)( ∞

∑�=0

2�δ r2�−1)

= c∞

∑j=0

[2 jδ ( 2 j+1−1

∑i=2 j

|Sδi ((−Δh,0)

1/2F)|)2∞

∑�=0

2�δ r2�+2 j−2(1− r)2+2δ].

(5.4.9)

On the other hand, a straightforward calculation shows that∞

∑�=0

2�δ∫ 1

0r2�+2 j−2(1− r)2+2δ r| logr|dr ≤ c

∞

∑�=0

2�δ∫ 1

0r2�+2 j−2(1− r)3+2δ dr

= c∞

∑�=0

2�δΓ(2�+2 j −1)Γ(4+2δ )

Γ(2�+2 j +3+2δ )≤ c

∞

∑�=0

2�δ (2�+2 j)−4−2δ

≤ c2− j(δ+4).

Thus, using (5.4.9) and (4.2.2), we conclude that

g(F)≤ c( ∞

∑j=0

2−4 j( 2 j+1−1

∑i=2 j

|Sδi ((−Δh,0)

1/2F)|)2) 1

2, (5.4.10)

which implies claim (5.4.4) for p > 2.Finally, for 1 < p ≤ 2, we use (5.4.10) and Holder’s inequality to obtain

|g(F)|p ≤ c∞

∑j=0

2−2 jp

(2 j+1−1

∑i=2 j

|Sδi ((−Δh,0)

1/2F)|)p

≤ c∞

∑j=0

2− j(p+1)2 j+1−1

∑i=2 j

|Sδi ((−Δh,0)

1/2F)|p,

which proves (5.4.4) for 1 < p ≤ 2 as well. �

5.5. The sharp Jackson inequality 59

5.5 The sharp Jackson inequality

The main goal in this section is to prove the following sharp Jackson inequality.

Theorem 5.5.1. For f ∈ Lp(h2κ ;Sd−1), 1 0, and s = max(p,2),

tr{

∑1≤ j≤1/t

jsr−1E j( f )sκ,p

}1/s ≤ cωr( f , t)κ,p. (5.5.1)

Using Hardy’s inequality and Theorem 5.2.6, it is easily seen that Theorem 5.5.1 is,in fact, equivalent to the following corollary:

Corollary 5.5.2. For f ∈ Lp(h2κ ;Sd−1), 1 < p < ∞, and s = max(p,2),

tr{∫ 1/2

t

ωr+1( f ,u)sκ,p

urs+1 du}1/s ≤ cωr( f , t)κ,p.

Proof of Theorem 5.5.1. Obviously, it suffices to show that

2−nr( n

∑j=1

2 jrsE2 j( f )sκ,p

)1/s ≤ cKr( f ,2−n)κ,p. (5.5.2)

Setting gn = Lκ2n−1 f , we have

E2n( f )κ,p ≤ ‖ f −gn‖κ,p ≤ cKr(

f ,2−nr)κ,p.

As Em( f −gn)κ,p ≤ ‖ f −gn‖κ,p for all m ∈ N, we obtain

E2 j( f )κ,p ≤ E2 j( f −gn)κ,p +E2 j(gn)κ,p ≤ ‖ f −gn‖κ,p +E2 j(gn)κ,p .

We can now write

2−nr( n

∑j=1

2 jrsE2 j( f )sκ,p

)1/s

≤ 2−nr( n

∑j=1

2 jrsE2 j( f −gn)sκ,p

)1/s+2−nr

( n

∑j=1

2 jrsE2 j(gn)sκ,p

)1/s

≤ 2r

(2rs −1)1/s ‖ f −gn‖κ,p +2−nr( n

∑j=1

2 jrsE2 j(gn)sκ,p

)1/s.

Therefore, for the proof of (5.5.2), it remains to show that

2−nr( n

∑j=1

2 jrsE2 j(gn)sκ,p

)1/s ≤ cKr(

f ,2−n)κ,p,

which, using (5.3.6), is a direct consequence of the inequality

n

∑j=1

2 jrsE2 j(gn)sκ,p ≤ c‖(−Δh,0)

r/2gn‖sκ,p . (5.5.3)

For the proof of (5.5.3), we write for j ≤ n,

E2 j+1(gn)κ,p ≤ ‖gn −L2 j gn‖κ,p = ‖L2ngn −L2 j gn‖κ,p =∥∥∥ n+1

∑�= j+2

�κθ ,�gn

∥∥∥κ,p

.

Note that Lκ2i(Lκ

2n f −Lκ2 j f ) = 0 for i < j ≤ n, and �κ

θ ,i(Lκ2ngn −Lκ

2 j gn) = 0 for i > n+1.Thus, applying Theorem 5.3.2 to the function Lκ

2ngn −Lκ2 j gn, we deduce that

‖Lκ2ngn −Lκ

2 j gn‖κ,p ∼∥∥∥( n+1

∑�= j+1

(�κθ ,�gn)

2)1/2∥∥∥

κ,p.

Thus, proving (5.5.3) reduces to showing that

n+1

∑j=0

2 jrs∥∥∥( n+1

∑�= j+1

(�κθ ,�gn)

2)1/2∥∥∥s

κ,p≤ c‖(−Δh,0)

r/2gn‖sκ,p. (5.5.4)

We prove (5.5.4) separately for 1 < p ≤ 2, in which case s = 2, and for 2 < p < ∞,in which case s = p. For 1 < p ≤ 2 we use ‖ f‖q+‖g‖q ≤ ‖| f |+ |g|‖q for the quasi-norm‖ · ‖q when q ≤ 1, and obtain

n+1

∑j=1

2 jr2∥∥∥ n+1

∑�= j+1

(�κθ ,�gn)

2∥∥∥

p/2≤

∥∥∥ n+1

∑j=1

2 jr2n+1

∑�= j+1

(�κθ ,�gn)

2∥∥∥

p/2

≤ c∥∥∥( n+1

∑�=2

(�κθ ,�gn)

22�r2)1/2∥∥∥2

κ,p≤ c‖(−Δh,0)

r/2gn‖sκ,p,

where the last step uses Theorem 5.3.2. This proves (5.5.4) for the case 1 < p ≤ 2.Finally, we prove (5.5.4) for the case of 2 < p < ∞. Setting E := {( j, �) : j, � ∈

Z, 0 ≤ j ≤ �−1 ≤ n}, we have

∫Sd−1

n+1

∑j=0

2 jrp( n+1

∑�= j+1

(�κθ ,�gn)

2)p/2

=∫Sd−1

(n+1

∑j=0

2 jrp∣∣∣ n+1

∑�=1

χE( j, �)|�κθ ,�gn|2

∣∣∣p/2) 2p

p2

≤∫Sd−1

(n+1

∑�=1

|�κθ ,�gn|2

(n+1

∑j=0

2 jrpχE( j, �)) 2

p) p

2 ≤ c∥∥∥( n+1

∑�=2

|�κθ ,�(gn)|22�r2

)1/2∥∥∥p

κ,p,

where the second step uses the weighted Minkowskii inequality. This together with (5.3.3)proves (5.5.4) for p > 2. �

5.6. Optimality of the power in the Marchaud inequality 61

5.6 Optimality of the power in the Marchaud inequality

In this section, we show the optimality of the powers s and q in the sharp Jackson inequal-ity (5.5.1) and the sharp Marchaud inequality (5.4.1). More precisely, we have

Theorem 5.6.1. For 1 0 : sup

fsupn∈N

n−r(

∑nk=1 krs−1Ek( f )s

κ,p

)1/s

ωr( f ,n−1)κ,p< ∞

}, (5.6.1)

min{p,2}= max

{q > 0 : sup

fsup

n

ωr( f ,n−1)κ,p

n−r(

∑nk=1 krq−1Ek( f )q

κ,p

) 1q< ∞

}, (5.6.2)

where the supremums sup f are taken over all functions f ∈ Lp(h2κ) that are not constant.

By slight modifications of the examples in [9], we can deduce the optimality (5.6.1)for all 1 < p < ∞ and the optimality (5.6.2) for 1 < p ≤ 2. The main goal in this sectionis to show that (5.6.2) holds for 2 ≤ p < ∞ as well. By Theorem 5.3.5, it is sufficient toconstruct a sequence of functions fn such that

Kr( fn,2−n)κ,p ∼ c2−nr( n

∑k=1

22krE2k( fn)2κ,p

)1/2, 2 ≤ p < ∞ (5.6.3)

and

limn→∞

Kr( fn,2−n)κ,p

2−nr(

∑nk=1 2qkrE2k( fn)

qκ,p

)1/q = ∞, ∀q > 2. (5.6.4)

The construction of the sequence of functions fn with the above properties relies onthe following crucial proposition, whose proof can be found in [11].

Proposition 5.6.2. Let X be a linear subspace of ΠdN with dimX ≥ ε dimΠd

N for someε ∈ (0,1). Then there exists a function f ∈ X such that ‖ f‖κ,p ∼ 1 for all 0 < p ≤ ∞ withthe constants of equivalence depending only on ε , d, κ and p, when p is small.

Proofs of (5.6.3) and (5.6.4). For each j ∈ N, let

Xj :=⊕

2 j−1<k≤2 j

H dk .

Since dimH dk (h2

κ)∼ kd−2, it follows that

dimXj ∼ 2 j(d−1) ∼ dimΠd2 j .

Thus, using Proposition 5.6.2, there exists a spherical polynomial Pj ∈ ⊕2 j−1<k≤2 j

H dk (h2

κ)

such that ‖Pj‖∞ ∼ ‖Pj‖2 ∼ 1 for each j ∈ N. Let fn = ∑nj=1 2− jrPj. Using (5.3.5), we

obtain

Kr( fn,2−n)κ,p ≥ c2−nr‖(−Δh,0)r2 fn‖κ,p ≥ c2−nr‖

n

∑j=0

2− jr(−Δh,0)r2 Pj‖2

= c2−nr( n

∑j=0

2−2 jr‖(−Δh,0)r2 Pj‖2

2

) 12 ∼ 2−nr

( n

∑j=0

‖Pj‖22

) 12

∼ 2−nr√n.

On the other hand,

E2 j( fn)κ,p ≤ ‖n

∑i= j+1

2−irPi‖κ,p ≤n

∑i= j+1

2−ir‖Pi‖κ,p ≤ c2− jr.

Thus, for any 2 ≤ q < ∞,

2−nr( n

∑j=1

2 jqrE2 j( fn)qκ,p

)1/q ≤ c2−nr( n

∑j=1

2 jqr2− jqr)1/q ≤ c2−nrn1/q

≤ cn1q− 1

2 Kr( fn,2−n)κ,p,

which implies (5.6.4) for q > 2, and the lower estimate of (5.6.3) for q = 2. Finally, theupper estimate of (5.6.3) follows directly from (5.4.2). �


Inequalities (5.1.1) and (5.1.2) were proved by M. Timan [55] and Zygmund [75]. Theywere generalized in several articles (see [7, 8, 20, 22, 61]) and described in the texts [19,p. 210], [57, p.338 (12)], and [58, (4.88), p. 191]. Other useful inequalities on moduli ofsmoothness and their applications in embedding theory can be found in [24, 47, 54].

The weighted moduli of smoothness (5.2.1) and K-functionals (5.3.5) were definedand studied in [72], where the direct and inverse theorem, namely Theorem 5.2.6, andthe equivalence ωr( f , t)κ,p ∼ Kr( f , t)κ,p, as well as several other useful properties ofωr( f , t)κ,p were established; see also [74]. For polynomial approximation on the un-weighted sphere, we refer the reader to the book [64].

Most of the results in Section 5.3 for the K-functionals were proved by Ditzian [21]in a more general setting, where only the Cesaro summability was assumed.

The proof of Theorem 5.5.1, the sharp Jackson inequality, follows along the samelines as [9], where the theorem was shown in a more general setting. A very elegantalternative proof of the sharp Jackson inequality was recently discovered in [22] where,instead of the Littlewood–Paley inequality, only semi-groups and convexity properties ofLp-spaces are used. The method in [22] works also for more general Banach spaces.

5.7. Notes and further results 63

The proof of the sharp Marchaud inequality, namely Theorem 5.4.1, follows alongthe same lines as [7] and [8]. An alternative approach to the sharp Marchaud inequalitieswithout using the Littlewood–Paley inequalities can be found in [20].

The proof of Proposition 5.6.2 is from [11], whereas its idea can be traced back to[67].

Chapter 6

Dunkl Transform

The Dunkl transform is a generalization of the Fourier transform and is an isometry inL2(Rd ,h2

κ) with hκ being a reflection invariant weight function. In this chapter we studythe Dunkl transform from the point of view of harmonic analysis. In Section 6.1 we showthat the Dunkl transform is an isometry in L2 space with respect to the measure h2

κ(x)dxon Rd and it preserves Schwartz class of functions. The inversion formula when bothf and its Dunkl transform are in L1 is proved in Section 6.2, for which we consideran approximation operator defined by a generalized convolution with the delation of theGaussian kernel. The convolution structure is defined in terms of a generalized translationoperator, which is defined in the Dunkl transform side; this translation operator is studiedin Section 6.3 and its boundedness is established in some restricted classes of functions.The boundedness of the generalized convolution operator is studied in Section 6.4 andused to study the summability of the inverse Dunkl transform. Finally, in Section 6.5,we consider analogues of the Hardy–Littlewood maximal functions in the weighted Lp

spaces and prove that they are strong type (p, p) and weak type (1,1), which lead toalmost everywhere convergence of the summability methods.

6.1 Dunkl transform: L2 theory

Recall that Vκ is the intertwining operator associated with a reflection group G and amultiplicity function κ . We define

E(x,y) :=V (x)κ e〈x,y〉, x,y ∈ Rd .

Recall that En(x,y) = V (x)κ (〈x,y〉n/n!). Since the function y �→ fy(x) := e〈x,y〉 is in A(Bd)

and ‖ fy‖A = e‖y‖, the sum E(x,y) = ∑∞n=0 En(x,y) converges uniformly and absolutely on

compact sets. In particular, by Proposition 2.3.8, E is symmetric, E(x,y) = E(y,x).The function E(x, iy) plays the role of ei〈x,y〉 in the usual case of the Fourier trans-

form. Since Vκ is a positive operator, |E(x, iy)| ≤ 1.




87

65

6

66 Chapter 6. Dunkl Transform

Definition 6.1.1. Let hκ be defined as in (2.1.2). For f ∈ L1(Rd ,h2κ), the Dunkl transform

is defined by

Fκ f (y) := bh

∫Rd

f (x)E(x,−iy)h2κ(x)dx, y ∈ Rd ,

where bh is the constant

bh =

(∫Rd

h2κ(x)e

−‖x‖2/2dx)−1

=ch

(2π)d/2 ,

and ch is as in (2.1.6).

If κ = 0, then Vκ = id and the Dunkl transform coincides with the usual Fouriertransform.

Theorem 6.1.2. For f ∈ L1(Rd ,h2κ), the Dunkl transform Fκ f is a bounded continuous

function, and |Fκ f (x)| ≤ ‖ f‖κ,1, x ∈ Rd, where ‖ · ‖κ,p denotes the norm of Lp(Rd ;h2κ).

Proof. Since |E(x, iy)| ≤ 1, the inequality |Fκ f (x)| ≤ ‖ f‖κ,1 follows immediately. Thecontinuity follows from the continuity of E(x,−iy) and the dominated convergence theo-rem. �

Proposition 6.1.3. Let ν(z) = z21 + · · ·+ z2

d, zi ∈ C. For y,z ∈ Cd,

bh

∫Rd

E(x,y)E(x,z)h2κ(x)e

−‖x‖2/2dx = e(ν(y)+ν(z))/2E(y,z). (6.1.1)

Proof. First we prove that, for p being a polynomial on Rd and y ∈ Cd ,

bh

∫Rd

(e−Δh/2 p(x)

)E(x,y)h2

κ(x)e−‖x‖2/2dx = eν(y)/2 p(y). (6.1.2)

Let m be an integer larger than the degree of p, fix y ∈ Cd , and let qm(x) = ∑mj=0 E j(x,y).

Decomposing p into homogeneous components shows that 〈qm, p〉D = p(y). By (3.1.16),then,

p(y) = 〈qm, p〉D = ch

∫Rd

(e−Δh/2 p(x)

)(e−Δh/2qm(x)

)h2

κ(x)e−‖x‖2/2dx.

Since Δ(x)h En(x,y) = ν(y)En−2(x,y), it follows that

e−Δh/2qm(x) =m

∑j=0

∑l≤ j/2

1l!

(−ν(y)2

)lE j−2l(x,y)

= ∑l≤m/2

1l!

(−ν(y)2

)l m−2l

∑j=0

E j(x,y).

6.1. Dunkl transform: L2 theory 67

The double sum converges to e−ν(y)/2E(x,y) as m → ∞. Since Vκ is positive, |E j(x,y)| ≤e‖x‖·‖y‖, the terms in the double sum are dominated by

∞

∑l=0

‖y‖2l

l!2l

∞

∑s=0

‖x‖s‖y‖s

s!= exp

(‖y‖2

2+‖x‖ · ‖y‖

),

which is integrable with respect to e−‖x‖2/2dx. Hence, by the dominated convergencetheorem,

p(y) = eν(y)/2bh

∫Rd(e−Δh/2 p(x))E(x,y)hκ(x)2e−‖x‖2/2dx.

Multiplying both sides by e−ν(y) completes the proof of (6.1.2). Setting p(x) = pm(x) =∑m

j=0 E j(x,z) in (6.1.2), we see that (6.1.1) follows from pm(x)→E(x,z) and e−Δh/2 pm(x)→e−ν(z)/2E(x,z), as m → ∞, upon using the dominated convergence and Fubini theorems.

�The analog of the heat kernel for the Dunkl transform is defined by

qκt (x) := (2t)−(λκ+1)e−t‖x‖2/4, x ∈ Rd , t > 0, (6.1.3)

where λκ = γκ +d−2

2 as before.

Corollary 6.1.4. For t > 0,Fκ qκ

t (x) = e−t‖x‖2. (6.1.4)

As an analog of the Fourier transform, we show that the Dunkl transform is anisometry of L2(Rd ,h2

κ). First we give an orthogonal basis for L2(h2κ). Let Y ∈ H d

n (h2κ).

Defineφm(Y ;x) = Ln+λκ

m (‖x‖2)Y (x)e−‖x‖2/2, x ∈ Rd . (6.1.5)

Proposition 6.1.5. For k, l,m,n ∈ N0, the integral

ch

(2π)d/2

∫Rd

φm(Yn;x)φk(Yl ;x)h2κ(x)dx

= δmkδnl(λκ +1)n+m

2λκ+1mωd

ωκd

∫Sd−1

[Yn(x)]2h2κ(x)dσ .

Proof. Using spherical polar coordinates, the first integral equals

ch

∫ ∞

0Ln+λκ

m (r2)Ll+λκk (r2)e−r2

rn+l+2γκ+d−1dr21−d/2

Γ(d/2)

∫Sd−1

YnYlh2κ dσ .

The second integral is zero if n = l. Assume n = l, make the change of variable r2 = t.The first integral equals (1/2)δmkΓ(n+λk + 1+m)/m!, which gives the constant in theformula by (2.1.8). �

In particular, if {Yk,n} denotes an orthogonal basis of H dn (h2

κ), then {φm(Yk,n;x)}forms an orthogonal basis of L2(Rd ,h2

κ).


Theorem 6.1.6. For m,n = 0,1,2, . . ., Y ∈ H dn (h2

κ) and y ∈ Rd,

(Fκ φm(Y ))(y) = (−i)n+2mφm(Y ;y).

Proof. By (6.1.2) and (3.1.17),

bh

∫Rd

Ln+λκm (‖x‖2/2)p(x)E(x,y)h2

κ(x)e−‖x‖2/2 dx

= (−1)m(m!2m)−1eν(y)/2ν(y)m p(y).

We change the argument in the Laguerre polynomial by using the identity

Lαm(t) =

m

∑j=0

2 j (α +1)m

(α +1) j

(−1)m− j

(m− j)!Lα

j (t/2), t ∈ R,

which can be derived from the generating function of the Laguerre polynomials. Usingthis expansion together with the above integral, we obtain

bh

∫Rd

Ln+λkm (‖x‖2)p(x)e−‖x‖2/2E(x,y)h2

κ(x)dx

= eν(y)/2 p(y)(−1)m (n+λκ +1)m

m!

m

∑j=0

(−m) j

(n+λκ +1) j

(−ν(y)) j

j!.

We now replace y by −i y for y ∈ Rd , so that ν(y) becomes −‖y‖2 and p(y) becomes(−1)m(−i)n p(y), and the sum yields a Laguerre polynomial. Consequently, the integralequals (−1)m(−i)n p(y)Ln+λk

m (‖y‖2)e−‖y‖2/2. �Since the functions {φm(Y ) : Y ∈ H d

n (h2k),m ≥ 0} constitutes a basis of L2(Rd ,h2

κ)and the eigenvalues are powers of i =

√−1, this proves the isometry properties, whichare the analog of the Plancherel theorem stated below.

Theorem 6.1.7. The Dunkl transform extends to an isometry of L2(Rd ,h2κ) onto itself.

The square of the transform is the central involution; that is, if f ∈ L2(Rd ,h2κ), Fκ f = g,

then Fκ g(x) = f (−x) for almost all x ∈ Rd.

As a consequence, the inverse of the Dunkl transform is given by

f (x) = bh

∫Rd

Fκ f (y)E(ix,y)h2κ(y)dy, (6.1.6)

which holds in L2(Rd ,h2κ). In particular, it holds for the Schwartz class of functions.

As another analog to the Fourier transform, the Dunkl transform of a radial functionis also radial, and it can be expressed in terms of the Bessel function Jα(t), defined by

Jα(t) :=(t/2)α

√π Γ(α +1/2)

∫ 1

−1eits(1− s2)α−1/2ds (6.1.7)

=( t

2

)α ∞

∑n=0

(−1)n

n!Γ(n+α +1)

( t2

)2n.


Theorem 6.1.8. Suppose f is a radial function in L1(Rd ,h2κ); f (x) = f0(‖x‖) for almost

all x ∈ Rd. The Dunkl transform Fκ f is also radial and has the form Fκ f (x) = F0(‖x‖)for all x ∈ Rd with

F0(‖x‖) = F0(r) =1

ωdrλκ

∫ ∞

0f0(s)Jλκ (rs)sλκ+1ds.

Proof. Using polar coordinates and (3.2.6), we obtain

Fκ f (y) = bh

∫ ∞

0f0(s)sd−1+2γκ

∫Sd−1

E(sx′,y)h2κ(x

′)dσ(x′)ds

= bhcλκ ωκd

∫ ∞

0f0(s)s2λκ+1

∫ 1

−1eis‖y‖t(1− t2)λκ− 1

2 dt ds,

from which the stated result follows by the definition of Jα(t) and, using (2.1.8), puttingconstants together. �

As a consequence, we see that the Dunkl transform of f0(‖x‖) is a Hankel transformin ‖x‖. In general, the Hankel transform Hα is defined on the positive reals R+. Forα >−1/2,

Hα f (s) :=1

Γ(α +1)

∫ ∞

0f (r)

Jα(rs)(rs)α r2α+1dr. (6.1.8)

The inverse Hankel transform is given by

f (r) =1

Γ(α +1)

∫ ∞

0Hα f (s)

Jα(rs)(rs)α s2α+1ds, (6.1.9)

which holds under mild conditions on f ; for example, it holds if f is piecewise continuousand of bounded variation in every finite subinterval of (0,∞), and

√r f ∈ L1(R+) ([65, p.

456]).Example. Consider hκ(x) = |x|κ , κ ≥ 0, on the real line. Then the group is Z2, and

E(x,−iy) = Γ(κ +1/2)(|xy|/2)−κ+1/2 [Jκ−1/2(|xy|)− isign(xy)Jκ+1/2(|xy|)] ,

so that the Dunkl transform is related to the Hankel transform.Indeed, in this case, the intertwining operator Vκ is given in (2.3.2), hence,

E(x,−iy) = cκ

∫ 1

−1e−isxy(1+ s)(1− s2)κ−1ds,

so that, integrating by parts,

E(x,−iy) = cκ

∫ 1

−1e−isxy(1− s2)κ−1ds− i x

2κcκ

∫ 1

−1e−isxy(1− s2)κ ds

= Γ(κ +1/2)(|xy|/2

)−κ+1/2[Jκ−1/2(|x|)− isign(x)Jκ+1/2(|x|)].

The following proposition is useful for dealing with the Dunkl transform of func-tions that involve h-harmonics.

Proposition 6.1.9. Let f ∈ H dn (h2

κ), n = 0,1,2, . . .. If y ∈ Rd, ρ > 0, then the function

g(y) =ωd

ωκd

∫Sd−1

f (x)E(x,−i yρ)h2κ(x)dσ(x)

satisfies Δhg =−ρ2g and

g(y) = (−i)nΓ(λκ +1)(ρ‖y‖

2

)−λκf( y‖y‖

)Jn+λκ (ρ‖y‖).

Proof. First, let y ∈ C. Since f is h-harmonic, e−Δh/2 f = f . In the formula from (6.1.2),

bh

∫Rd

f (x)E(x,y)h2κ(x)e

−‖x‖2/2dx = eν(y)/2 f (y),

the part that is homogeneous of degree n+2m in y, m = 0,1,2, . . ., yields the equation

bh

∫Rd

f (x)En+2m(x,y)h2κ(x)e

−‖x‖2/2dx =ν(y)m

2mm!f (y).

Then, using the integral formula (2.1.7) and the fact that∫Sd−1

f (x)E j(x,y)h2κ(x)dσ(x) = 0

if j < n or j ≡ n mod 2, we conclude that

ωd

ωκd

∫Sd−1

f (x)E(x,y)h2κ(x)dσ(x)

=∞

∑m=0

12n+m(d/2+ γκ)n+m

ch

∫Rd

f (x)En+2m(x,y)h2κ(x)e

−‖x‖2/2dx.

Replace y by −iρy for ρ > 0, y ∈ Rd . Let A = n+λκ . This leads to the expression for gin terms of JA. To find Δhg we can interchange the integral and Δ(y)

h , because the resulting

integral of a series ∑∞n=0 Δ(y)

h En(x,−i y) converges absolutely. Indeed

Δ(y)h E(x,−iρy) = Δ(y)

h E(−iρx,y)

=N

∑j=1

(−iρx j)2E(−iρx,y) =−ρ2‖x‖2E(x,−iρy).

But ‖x‖2 = 1 on Sd−1, and so Δhg =−ρ2g. �

Let us denote by {Yj,n : 1 ≤ j ≤ dim H dn (h2

κ)} an orthonormal basis of H dn (h2

κ).We can prove a Paley-Wiener theorem for the Dunkl transform. Let S denote the spaceof Schwartz class of functions on Rd .


Theorem 6.1.10. Let f ∈ S and R be a positive number. Then f is supported in {x ∈Rd : ‖x‖ ≤ R} if and only if for every j and n, the function

Fj,n(ρ) = ρ−n∫Sd−1

Fκ f (ρx)Yj,n(x)h2κ(x)dσ(x)

extends to an entire function of ρ ∈ C satisfying the estimate

|Fj,n(ρ)| ≤ c j,neR‖ Imρ‖.

Proof. By the definition of Fκ f and Proposition 6.1.9,∫Sd−1

Fκ f (ρx)Yj,n(x)h2κ(x)dσ(x)

= c∫Rd

(∫Sd−1

E(y,−iρx)Yj,n(x)h2κ(x)dσ(x)

)f (y)h2

κ(y)dy

= c∫Rd

f (y)Yj,n(y′)Jλk+n(ρ‖y‖)(ρ‖y‖)λk

h2κ(y)dy

= cρn∫ ∞

0f j,n(r)

Jλk+n(rρ)(rρ)λk+n r2λκ+2n+1dr,

where c is a constant and

f j,n(r) =∫Sd−1

f (ry′)Yj,n(y′)h2κ(y

′)dσ(y′).

Thus, Fj,n is the Hankel transform of order λκ + n of the function f j,n(r). The theoremthen follows from the Paley–Wiener theorem for the Hankel transform (see, for example,[36]). �Corollary 6.1.11. A function f ∈ S is supported in {x ∈ Rd : ‖x‖ ≤ R} if and only ifFκ f extends to an entire function of ζ ∈ Cd which satisfies

|Fκ f (ζ )| ≤ ceR‖ Imζ‖.

Proof. The direct part follows from the fact that E(x,−iζ ) is entire and |E(x,−iζ )| ≤ce‖x‖·‖ Imζ‖. For the converse we look at∫

Sd−1Fκ f (ρy′)Yj,n(y′)h2

κ(y′)dσ(y′), ρ ∈ C.

This is certainly entire and, from the proof of the previous theorem, has a zero of order nat the origin. Hence,

ρ−n∫Sd−1

Fκ f (ρy′)Yj,n(y′)h2κ(y

′)dσ(y′)

is an entire function of exponential type R, from which the converse follows from thetheorem. �


6.2 Dunkl transform: L1 theory

Let S denote the space of Schwartz functions on Rd . The inversion formula of the Dunkltransform holds for f ∈ S .

Theorem 6.2.1. Let f ∈ S . Then for y ∈ Rd, FκD j f (y) = i y jFκ f (y) for j = 1, . . . ,d.Furthermore, if g j(x) = x j f (x), then Fκ g j(y) = iD jFκ f (y), y ∈Rd. The operator −iD jis densely defined on L2(Rd ,h2

κ) and is self-adjoint.

Proof. From the definition of D j, it is not difficult to prove that if f ,g ∈ S , then∫Rd(D j f )gh2

κ dx =−∫Rd

f (D jg)h2κ dx, j = 1, . . . ,d.

For fixed y∈Rd , put g(x)=E(x,−i y) in the above identity. Then D jg(x)=−i y jE(x,−i y)and FκD j f (y) = (−1)(−i y j)Fκ f (y). The multiplication operator defined by Mj f (y) =y j f (y), j = 1, . . . ,d, is densely defined and self-adjoint on L2(Rd ,h2

κ). Furthermore, −iD jis the inverse image of Mj under the Dunkl transform. �

In particular, this shows that if f ∈ S , then Fκ f ∈ S . The assumption f ∈ S canof course be substantially relaxed.

Theorem 6.2.2. If f ∈ L1(Rd ;h2κ), then f ∈C0(R

d).

Proof. The space S is dense in L1(Rd ;h2κ). For each f ∈ L1(Rd ;h2

κ), there are functionsfn ∈ S such that ‖ f − fn‖κ,1 → 0. Since Fκ fn ∈ S ⊂ C0(R

d) and Fκ fn convergesuniformly to Fκ f by ‖Fκ f‖∞ ≤ ‖ f‖κ,1, it follows that Fκ f ∈C0(R

d). �We want to show that if both f and Fκ f are in L1(Rd ;h2

κ), then the inversiontheorem holds. For this purpose, we use a generalized convolution operator. Recall thatthe usual convolution f ∗ g is defined in terms of the translation τy f = f (· − y) of Rd ,which satisfies, taking the Fourier transform, Fκ τy f (x) = e−i〈x,y〉Fκ f (x). The translationoperator works for the Lebesgue measure since it leaves L1(Rd) invariant. It is not obviouswhat operation is translation invariant for L1(Rd ,h2

κ). We define it in the Dunkl transformside.

Definition 6.2.3. Let y ∈ Rd be given. The generalized translation operator f �→ τy f isdefined on L2(Rd ;h2

κ) by the relation

Fκ τy f (x) := E(y,−ix)Fκ f (x), x ∈ Rd . (6.2.1)

The definition makes sense as the Dunkl transform is an isometry of L2(Rd ;h2κ)

onto itself and the function E(y,−ix) is bounded. However, none of the other propertiesof the usual translation operator is obvious; for example, boundedness in Lp, translationinvariance, or positivity. Some of these properties will be studied below and in the nextsection.

We start with an example. For f ∈ S we can write

τy f (x) = bh

∫Rd

E(ix,ξ )E(−iy,ξ )Fκ f (ξ )h2κ(ξ )dξ . (6.2.2)


Proposition 6.2.4. For t > 0 and x ∈ Rd,

τye−t‖x‖2= e−t(‖x‖2+‖y‖2)E(

√2tx,

√2ty). (6.2.3)

Proof. This follows immediately from (6.1.1), (6.1.4) and (6.2.2). �

Proposition 6.2.5. Assume that f ,g ∈ S . Then

(1)∫Rd

τy f (ξ )g(ξ )h2κ(ξ )dξ =

∫Rd

f (ξ )τ−yg(ξ )h2κ(ξ )dξ .

(2) τy f (x) = τ−x f (−y).

Proof. The property (2) follows from the definition, since E(λx,ξ ) = E(x,λξ ) for anyλ ∈ C. If f ,g ∈ S , then both integrals in (1) are well defined. From the definition,∫Rd

τy f (ξ )g(ξ )h2κ(ξ )dξ = bh

∫Rd

(∫Rd

E(ix,ξ )E(−iy,ξ )Fκ f (ξ )h2κ(ξ )dξ

)g(x)h2

κ(x)dx

=∫Rd

Fκ f (ξ )Fκ g(−ξ )E(−iy,ξ )h2κ(ξ )dξ

by the inversion theorem applied to g. We also have∫Rd

f (ξ )τ−yg(ξ )h2κ(ξ )dξ = bh

∫Rd

(∫Rd

E(ix,ξ )E(iy,ξ )Fκ g(ξ )h2κ(ξ )dξ

)f (x)h2

κ(x)dx

=∫Rd

Fκ f (−ξ )Fκ g(ξ )E(iy,ξ )h2κ(ξ )dξ

=∫Rd

Fκ f (ξ )Fκ g(−ξ )E(−iy,ξ )h2κ(ξ )dξ

by the inversion theorem applied to f . This proves (1). �

Proposition 6.2.6. Let f ∈S be supported in {x ∈Rd : ‖x‖ ≤ R}. Then τy f is supportedin {x ∈ Rd : ‖x‖ ≤ R+‖y‖}.

Proof. Let g(x) = τy f (x). Then, by Corollary 6.1.11, Fκ g(ξ ) = E(y,−iξ )Fκ f (ξ ) ex-tends to Cd as an entire function of type R+ ‖y‖. Hence, the stated result follows fromCorollary 6.1.11. �

Theorem 6.2.7. If f ∈C∞0 (R

d) is supported in ‖x‖ ≤ R, then

‖τy f − f‖p ≤ c f ‖y‖(R+‖y‖) 2λκ+2p

for 1 ≤ p ≤ ∞. Consequently, limy→0 ‖τy f − f‖κ,p = 0.

Proof. From the definition we have

τy f (x)− f (x) = bh

∫Rd

(E(y,−iξ )−1)E(x, iξ )Fκ f (ξ )h2κ(ξ )dξ .

Using the mean value theorem and estimates on the derivatives of E(x, iξ ), we obtain theestimate

‖τy f − f‖∞ ≤ c‖y‖∫Rd

‖ξ‖|Fκ f (ξ )|h2κ(ξ )dξ .

As f is supported in ‖x‖ ≤ R and τy f is supported in ‖x‖ ≤ (R+‖y‖), we can restrict theintegration domain above to ‖x‖ ≤ (R+‖y‖) and conclude, accordingly, that

‖τy f − f‖p ≤ c f ‖y‖(R+‖y‖) 2λκ+2p ,

which goes to zero as y goes to zero. �The generalized translation operator is used to define a convolution structure:

Definition 6.2.8. For f ,g ∈ L2(Rd ;h2κ) we define

f ∗κ g(x) := bh

∫Rd

f (y)τxg(y)h2κ(y)dy,

where g(y) := g(−y).

Since τxg ∈ L2(Rd ;h2κ) the above convolution is well defined. By (6.2.1), we can

also write the definition as

f ∗κ g(x) = bh

∫Rd

Fκ f (ξ )Fκ g(ξ )E(ix,ξ )h2κ(ξ )dξ . (6.2.4)

Recall that qκt (x) = (2t)−λk+1e−t‖x‖2

denotes the heat kernel. Changing variableshows that bh

∫Rd qκ

t (x)h2κ(x)dx = 1.

Lemma 6.2.9. For f ∈ L1(Rd ;h2κ),

limt→0+

‖ f ∗κ qκt − f‖κ,1 = 0.

Proof. By (1) in Proposition 6.2.5 with f = qκt and g = 1, bh

∫Rd τxqκ

t (y)h2κ(y)dy = 1.

Since τuqκt ≥ 0 by (6.2.3), it follows then that

‖ f ∗qκt ‖κ,1 ≤ ‖ f‖κ,1.

For a given ε > 0 we choose g ∈ S such that ‖g− f‖κ,1 < ε/3. The triangle inequalitythen leads to

‖ f ∗κ qκt − f‖κ,1 ≤ 2

3ε +‖g∗κ qκ

t −g‖κ,1.

Since g ∈ S , it follows that

g∗κ qκt (x) =

∫Rd

g(y)τx(qκt )(y)h

2κ(y)dy =

∫Rd

τ−xg(y)qκt (−y)h2

κ(y)dy.

We also know that τ−xg(y) = τ−yg(x). Therefore,

g∗κ qκt (x) =

∫Rd

τyg(x)qκt (y)h

2κ(y)dy.


In view of this,

g∗κ qκt (x)−g(x) =

∫Rd

(τyg(x)−g(x))qκt (y)h

2κ(y)dy,

which implies then

‖g∗κ qκt −g‖κ,1 ≤

∫Rd

‖τyg−g‖κ,1|qκt (y)|h2

κ(y)dy.

If g is supported in ‖x‖ ≤ R, then the estimate in Theorem 6.2.7 gives

‖g∗κ qκt −g‖κ,1 ≤ c

∫Rd

‖y‖(R+‖y‖)2λκ+2 |qκt (y)|h2

κ(y)dy

≤ ct∫Rd

‖y‖(R+‖ty‖)2λk+1 e−‖y‖2h2

κ(y)dy,

which can be made smaller than ε/3 by choosing ε small. This completes the proof ofthe lemma. �Theorem 6.2.10. If both f and Fκ f ∈ L1(Rd ,h2

κ), then for almost all x ∈ Rd,

f (x) = bh

∫Rd

Fκ f (y)E(ix,y)h2κ(y)dy.

Proof. Since Fκ qκt (x) = e−‖x‖2

, for f ∈ S we have

f ∗qκt (x) = bh

∫Rd

Fκ f (y)e−t‖y‖2E(ix,y)h2

κ(y)dy.

This extends to f ∈ L1(h2κ ;Rd) since the convolution operator extends to L1(h2

κ ;Rd) asa bounded operator by ‖ f ∗κ qκ

t ‖κ,1 ≤ ‖ f‖κ,1. Letting t → 0+, applying Lemma 6.2.9 tothe left-hand side and the dominant convergence theorem to the right-hand side, we seethat the inversion formula follows almost everywhere. �

For the convenience of applications, we summarize some of the most useful prop-erties of the Dunkl transform established in this and the preceding section as follows.

Theorem 6.2.11. (i) If f ∈ L1(Rd ;h2κ), then Fκ f ∈C(Rd) and lim

‖ξ‖→∞Fκ f (ξ ) = 0.

(ii) The Dunkl transform Fκ is an isomorphism of the Schwartz class S (Rd) ontoitself, and F 2

κ f (x) = f (−x).

(iii) The Dunkl transform Fκ on S (Rd) extends uniquely to an isometric isomorphismon L2(Rd ;h2

κ), i.e., ‖ f‖κ,2 = ‖Fκ f‖κ,2.

(iv) If f and Fκ f are both in L1(Rd ;h2κ), then the following inverse formula holds:

f (x) = cκ

∫Rd

Fκ f (y)Eκ(ix,y)h2κ(y)dy, x ∈ Rd .


(v) If f ,g ∈ L2(Rd ;h2κ), then∫

RdFκ f (x)g(x)h2

κ(x)dx =∫Rd

f (x)Fκ g(x)h2κ(x)dx.

(vi) Given ε > 0, let fε(x) = ε−2−2γκ f (ε−1x). Then Fκ fε(ξ ) = Fκ f (εξ ).

(vii) If f (x) = f0(‖x‖) is radial, then Fκ f (ξ ) = Hλκ f0(‖ξ‖) is again a radial function,where Hα denotes the Hankel transform defined by

Hα g(s) =1

Γ(α +1)

∫ ∞

0g(r)

Jα(rs)(rs)α r2α+1 dr,

and Jα denotes the Bessel function of the first kind.

6.3 Generalized translation operator

Here we study properties of the generalized translation operator. It is convenient to definea class of functions on which (6.2.1) holds pointwisely:

Aκ(Rd) := { f ∈ L1(Rd ;h2

κ) : Fκ f ∈ L1(Rd ;h2κ)}.

This is a subspace of the intersection of L1(Rd ;h2κ) and L∞ and, hence, a subspace of

L2(Rd ;h2κ). The assumption on f in Proposition 6.2.5 can be relaxed as follows:

Proposition 6.3.1. Assume that f ∈ Aκ(Rd) and g ∈ L1(Rd ;h2

κ) is bounded. Then

(1)∫Rd

τy f (ξ )g(ξ )h2κ(ξ )dξ =

∫Rd

f (ξ )τ−yg(ξ )h2κ(ξ )dξ .

(2) τy f (x) = τ−x f (−y).

Proof. The proof of (2) is the same as before. If both f and g are in Aκ(Rd), then the

proof of Proposition 6.2.5 works. Suppose now f ∈ Aκ(Rd), g ∈ L1(Rd ;h2

κ)∩L∞. Sinceg ∈ L2(Rd ;h2

κ), τyg is defined as an L2 function. As f is in L2(Rd ;h2κ) and bounded, both

integrals are finite. The relation∫Rd

f (ξ )Fκ g(ξ )h2κ(ξ )dξ =

∫Rd

Fκ f (ξ )g(ξ )h2κ(ξ )dξ ,

which is true for Schwartz class functions, remains true for f ,g ∈ L2(Rd ;h2κ) as well.

Using this we get∫Rd

τy f (x)g(x)h2κ(x)dx =

∫Rd

τy f (−x)g(−x)h2κ(x)dx

=∫Rd

E(y,−iξ )Fκ f (ξ )Fκ g(−ξ )h2κ(ξ )dξ .

By the same argument, the integral on the right-hand side is also given by the same ex-pression. Hence (1) is proved. �

6.3. Generalized translation operator 77

6.3.1 Translation operator on radial functions

A function is called radial, if it depends only on ‖x‖. For such a function, there is anexplicit expression of the generalized translation operator.

Theorem 6.3.2. Let f (x) = f0(‖x‖). Assume f ∈ Aκ(Rd). Then for almost every x ∈ Rd,

τy f (x) =Vκ

[f0

(√‖x‖2 +‖y‖2 −2‖x‖ ‖y‖〈x′, ·〉

)]( y‖y‖

)where x′ =

x‖x‖ .

Proof. Since f is radial, its Dunkl transform is also a radial function, which we denoteby Fκ f0(r). Using (6.2.1), the property (iv) of Theorem 6.2.11, and the spherical-polarcoordinates ξ = rξ ′, we get

τy f (x) = bh

∫ ∞

0r2λκ+1

[∫Sd−1

E(x,−irξ ′)E(y, irξ ′)h2κ(ξ

′)dσ(ξ ′)]Fκ f0(r)dr.

We compute the inner integral first. For each y′ ∈ Sd−1, the reproducing kernel Pn(h2κ ;y′, ·)

is an element of H dn (h2

κ). Hence, Proposition 6.1.9 shows that

ωd

ωκd

∫Sd−1

E(x,−irξ ′)Pn(h2κ ;y′,ξ ′)h2

κ(ξ′)dσ(ξ ′)

= (−i)n2λκ (r‖x‖)−λκ Jn+λκ (r‖x‖)Pn(h2κ ;y′,x′),

where x′ = x/‖x‖, which implies that in L2(Sd−1,h2κ),

E(x,−irξ ′) = cκ,d

∞

∑n=0

(−i)n(r‖x‖)−λκ Jn+λκ (r‖x‖)Pn(h2κ ;ξ ′,x′).

Replacing ξ ′ by −ξ ′ gives the expansion of E(x, irξ ′). Hence, using the reproducingproperty of Pn(h2

κ ; ·, ·), we get

ωd

ωκd

∫Sd−1

E(x,−irξ ′)E(y, irξ ′)h2κ(ξ

′)dσ(ξ ′)

= c′κ,d∞

∑n=0

Jn+λκ (r‖x‖)(r‖x‖)λκ

Jn+λκ (r‖y‖)(r‖y‖)λκ

Pn(h2κ ;y′,x′)

= c′κ,dVκ

[∞

∑n=0

Jn+λκ (r‖x‖)(r‖x‖)λκ

Jn+λκ (r‖y‖)(r‖y‖)λκ

n+λκ

λκCλκ

n (〈x′, ·〉)](y′).

By the addition formula for Bessel functions ([2, p. 215]), the last expression is equal to

cVκ

[Jλκ (r

√|x‖2 +‖y‖2 −2‖x‖ ‖y‖〈x′, ·〉)

rλκ (‖x‖2 +‖y‖2 −2‖x‖ ‖y‖〈x′, ·〉)λκ/2

](y′),


where c is a constant. Consequently, we conclude that

τy f (x) = cVκ

[∫ ∞

0r2λκ+1 Jλκ (rz(x,y, ·))

(rz(x,y, ·))λκFκ f0(r)dr

](y′)

= cVκ[Hλκ Fκ f0(z(x,y, ·))

](y′),

where z(x,y, ·) =√‖x‖2 +‖y‖2 −2‖x‖ ‖y‖〈x′, ·〉 and c is a constant independent of f .

By Theorem 6.1.8, Fκ f (x) = cHλκ f0(‖x‖), thus it follows from the inversion formula ofthe Hankel transform (6.1.9) that τy f (x) = cVκ [ f0(z(x,y, ·))](y′). The constant c can bedetermined by setting f (x)≡ 1. This completes the proof. �

The condition in Theorem 6.3.2 can be relaxed somewhat, see Lemma 7.2.4 below.An immediate consequence of the explicit expression of τy is the following:

Theorem 6.3.3. Let f ∈Aκ(Rd) be radial and nonnegative. Then τy f ≥ 0,τy f ∈L1(Rd ;h2

κ)and ∫

Rdτy f (x)h2

κ(x)dx =∫Rd

f (x)h2κ(x)dx. (6.3.1)

Proof. As f is radial, the explicit formula in Proposition 6.3.2 shows that τy f ≥ 0 sinceVκ is a positive operator. Taking g(x) = e−t‖x‖2

and making use of (6.2.3) we obtain from(1) of Proposition 6.2.5 that∫

Rdτy f (x)e−t‖x‖2

h2κ(x)dx =

∫Rd

f (x)e−t(‖x‖2+‖y‖2)E(√

2tx,√

2ty)h2κ(x)dx.

As |E(x,y)| ≤ e‖x‖‖y‖, we can take limit as t → 0 to get

limt→0

∫Rd

τy f (x)e−t‖x‖2h2

κ(x)dx =∫Rd

f (x)h2κ(x)dx.

Since τy f ≥ 0, the monotone convergence theorem applied to the integral on the leftcompletes the proof. �

We can relax the conditions on f as follows.

Theorem 6.3.4. Let f ∈ L1(Rd ;h2κ)∩L∞ be radial and nonnegative. Then τy f ≥ 0,τy f ∈

L1(Rd ;h2κ) and (6.3.1) holds.

Proof. Since f is radial and nonnegative, the convolution f ∗κ qκt is also radial and non-

negative. Since f is both in L1(Rd ;h2κ) and L2(Rd ;h2

κ), f ∗κ qκt ∈ L1(Rd ,h2

κ) becauseqκ

t ∈ Aκ(Rd) and, by the Plancherel theorem and Holder’s inequality, ‖Fκ f ∗κ qκ

t ‖κ,1 =‖Fκ f ·Fκ qκ

t ‖κ,1 ≤ ‖ f‖κ,2‖qκt ‖κ,2. Hence, f ∗κ qκ

t ∈ Aκ(Rd). Thus, by Theorem 6.3.3,

τy( f ∗κ qκt )(x) ≥ 0. Since f ∈ L2(Rd ;h2

κ), it is easy to see that ‖ f ∗κ qκt − f‖κ,2 → 0.

Since τy is bounded on L2(Rd ;h2κ), we have τy( f ∗κ qt)→ τy f in L2(Rd ;h2

κ) as t → 0. Bypassing to a subsequence if necessary, we can assume that the convergence is also almosteverywhere. This gives us

limt→0

τy( f ∗κ qt)(x) = τyg(x)≥ 0


for almost every x. Since τy f is nonnegative, we let t → 0 and can apply the monotoneconvergence theorem to∫

Rdτy f (x)e−t‖x‖2

h2κ(x)dx =

∫Rd

f (x)e−t(‖x‖2+‖y‖2)E(√

2tx,√

2ty)h2κ(x)dx

and get (6.3.1). �Let Lp

rad(Rd ;h2

κ) denote the space of all radial functions in Lp(Rd ;h2κ).

Theorem 6.3.5. The generalized translation operator τy can be extended to all radialfunctions in Lp(Rd ;h2

κ), 1 ≤ p ≤ 2, and τy : Lprad(R

d ,h2κ) → Lp(Rd ;h2

κ) is a boundedoperator.

Proof. For a radial function f ∈ L1(Rd ;h2κ)∩L∞, the inequality −| f | ≤ f ≤ | f | together

with the nonnegativity of τy on radial functions in L1(Rd ;h2κ)∩L∞ shows that |τy f (x)| ≤

τy| f |(x). Hence ∫Rd

|τy f (x)|h2κ(x)dx ≤

∫Rd

| f |(x)h2κ(x)dx ≤ ‖ f‖κ,1.

We also have ‖τy f‖κ,2 ≤ ‖ f‖κ,2. By interpolation between L1 and L2, then ‖τy f‖κ,p ≤‖ f‖κ,p for all 1 ≤ p ≤ 2 and all f ∈ Lp

rad(Rd ;h2

κ). This proves the theorem. �

Theorem 6.3.6. For every f ∈ L1rad(R

d ;h2κ),∫

Rdτy f (x)h2

κ(x)dx =∫Rd

f (x)h2κ(x)dx.

Proof. Choose radial functions fn ∈Aκ(Rd) such that fn → f and τy fn → τy f in L1(Rd ;h2

κ).Since ∫

Rdτy fn(x)g(x)h2

κ(x)dx =∫Rd

fn(x)τ−yg(x)h2κ(x)dx

for every g ∈ Aκ(Rd) we get, taking limit as n tends to infinity,∫Rd

τy f (x)g(x)h2κ(x)dx =

∫Rd

f (x)τ−yg(x)h2κ(x)dx.

Now take g(x) = e−t‖x‖2and take the limit as t goes to 0. Since τy f ∈ L1(Rd ;h2

κ), thedominated convergence theorem shows that∫

Rdτy f (x)h2

κ(x)dx =∫Rd

f (x)h2κ(x)dx

for f ∈ L1(Rd ;h2κ). �

For non-radial functions, say in the Schwartz class, it is known that τy is not posi-tive when the group G is either Zd

2 or the symmetric group, and this should be the casefor all other reflection groups. It remains an open problem if τy f can be defined for allf ∈ L1(Rd ;h2

κ) when the group G is not Zd2. The case G = Zd

2 is discussed in the nextsubsection.


6.3.2 Translation operator for G = Zd2

Recall that the weight function hκ , invariant under the group Zd2, takes the form

hκ(x) =d

∏i=1

|xi|κi , κi ≥ 0.

The explicit formula (2.3.2) for the intertwining operator Vκ for Zd2 allows us to derive an

explicit formula for τy. Let us first consider the case d = 1.

Theorem 6.3.7. For G = Z2 and hκ(x) = |x|κ on R,

τy f (x) =12

∫ 1

−1f(√

x2 + y2 −2xyt)(

1+x− y√

x2 + y2 −2xyt

)Φκ(t)dt

+12

∫ 1

−1f(−

√x2 + y2 −2xyt

)(1− x− y√

x2 + y2 −2xyt

)Φκ(t)dt, (6.3.2)

where Φκ(t) = bκ(1+ t)(1− t2)κ−1.

Proof. In this case, f radial means that f is an even function. Using the explicit formulaof Vκ in (2.3.2), the formula in Theorem 6.3.2 shows that if f is even, then

τy f (x) =∫ 1

−1f(√

x2 + y2 −2xyt)

Φκ(t)dt.

Making use of the fact that the derivative of an even function is odd, we derive a formulafor τy f ′ using the fact that Dτy = τyD . In this simple case the Dunkl operator D is givenby

D f (x) = f ′(x)+κf (x)− f (−x)

x.

On the one hand, since a radial function is invariant under the difference part, we have

Dτy f (x) = τyD f (x) = τy f ′(x).

On the other hand, for f even, a simple computation shows that

τy f (x)− τy f (−x)x

=1x

∫ 1

−1

(∫ t

−t

dds

f(√

x2 + y2 −2xys)

ds)

Φκ(t)dt

=−y∫ 1

−1

(∫ t

−t

f ′(√

x2 + y2 −2xys)

√x2 + y2 −2xys

ds)

Φκ(t)dt

=−2ybκ

∫ 1

0

(∫ t

−t

f ′(√

x2 + y2 −2xys)

√x2 + y2 −2xys

ds)

t(1− t2)κ−1dt

=−ybκ

∫ 1

−1

f ′(√

x2 + y2 −2xys)

√x2 + y2 −2xys

(∫ 1

|s|t(1− t2)κ−1dt

)ds


=− yκ

∫ 1

−1

f ′(√

x2 + y2 −2xys)

√x2 + y2 −2xys

(1− s)Φκ(s)ds.

Together with the formula for τy f ′(x), this leads to

Dτy f (x) =∫ 1

−1f ′

(√x2 + y2 −2xys

) x− y√x2 + y2 −2xys

Φκ(s)ds.

Consequently, replacing f ′ by any odd function fo, we conclude that

τy fo(x) =∫ 1

−1fo

(√x2 + y2 −2xys

) x− y√x2 + y2 −2xys

Φκ(s)ds.

Any function f can be written as f = fe + fo, where fe(x) = ( f (x) + f (−x))/2 is theeven part and fo(x) = ( f (x)− f (−x))/2 is the odd part, from which the stated formulafollows. �

The explicit formula readily extends to the case of G = Zd2 and the product weight

function.

Theorem 6.3.8. For G = Zd2 and hκ(x) = ∏d

i=1 |xi|κi on Rd,

τy f (x) = τy1 · · ·τyd f (x), y = (y1, . . . ,yd) ∈ Rd .

Proof. For G = Zd2, the explicit formula of Vκ in (2.3.2) shows that

E(ix,y) = E(ix1,y1) · · ·E(ixd ,yd)

for x,y ∈Rd , from which the theorem follows upon taking the Dunkl transform of τy. �Note that the explicit formula also shows that τy f is not a positive operator. For

example, we have

τy(x1 − x2)2 = [(x1 − y1)− (x2 − y2)]

2 +4κ1

2κ1x1y1 +

4κ2

2κ2x2y2.

Choosing x1 =−y1 = 1 and x2 =−y2 = 1 shows that τy(x1 − x2)2 is not positive.

Using the formula for τy, we can establish the boundedness of the translation oper-ator in the case of Zd

2.

Theorem 6.3.9. For each y ∈ Rd, the generalized translation operator τy is a boundedoperator on Lp(Rd ,h2

κ). More precisely, ‖τy f‖κ,p ≤ 3‖ f‖κ,p, 1 ≤ p ≤ ∞.

Proof. The product nature of τy and hκ means that we only have to consider the cased = 1. We have∣∣∣∣∣

∫ 1

−1f(√

x2 + y2 −2xyt)(

1+x− y√

x2 + y2 −2xyt

)Φκ(t)dt

∣∣∣∣∣≤ ‖ f‖∞ + cκ

∫ 1

−1

∣∣∣ f(√

x2 + y2 −2xyt)∣∣∣ |x− y|√

x2 + y2 −2xyt(1+ t)(1− t2)κ−1dt.

Since (x− y)(1+ t) = (x− yt)− (y− xt), it follows that

|x− y|√x2 + y2 −2xyt

(1+ t)≤ 2.

Consequently, the above integral is bounded by 3‖ f‖∞. Hence, by the explicit formula ofτy f , we get ‖τy f‖∞ ≤ 3‖ f‖∞.

Next we consider the case p = 1. For f ∈ L1(Rd ,h2κ) the mapping g �→ Lg defined

by Lg = ch∫Rd τy f (x)g(x)h2

κ(x)dx is a linear functional L on C0(Rd). Using the property

(1) of Proposition 6.3.1 we get∣∣∣∣∫Rd

g(x)τy f (x)h2κ(x)dx

∣∣∣∣ = ∣∣∣∣∫Rd

τ−yg(x) f (x)h2κ(x)dx

∣∣∣∣≤ ‖τyg‖∞‖ f‖κ,1 ≤ 3‖g‖∞‖ f‖κ,1,

where we have used the fact that ‖τyg‖∞ ≤ 3‖g‖∞. Hence, L is a bounded linear functionalon C0(R

d). By the Riesz representation theorem, τy f (x)dx is the unique regular measureproviding the integral representation of L. Consequently,

‖τy f‖κ,1 = sup‖g‖∞=1

∣∣∣∣∫Rd

g(x) f (x)h2κ(x)dx

∣∣∣∣ ≤ 3‖ f‖κ,1.

Finally, interpolation shows that the same holds for 1 < p < ∞. �

6.4 Generalized convolution and summability

The convolution f ∗κ g in Definition 6.2.8 is defined for f ,g ∈ L2(Rd ;h2κ). By (6.2.4), it

satisfies the relations

Fκ( f ∗k g) = Fκ f ·Fκ g and f ∗κ g = g∗κ f . (6.4.1)

6.4.1 Convolution with radial functions

If g ∈ L1(Rd ;h2κ), then Fκ g is bounded so that, by the Plancherel theorem,

‖ f ∗κ g‖κ,2 ≤ ‖Fκ g‖∞‖ f‖k,2 ≤ ‖g‖κ,1‖ f‖k,2.

Since we do not know if the generalized translation operator is bounded in Lp(Rd ;h2κ), the

usual proof of Young’s inequality does not apply. For convolution with radial functionswe can state the following theorem.

Theorem 6.4.1. Let g be a bounded radial function in L1(Rd ;h2κ). Then the map f �→

f ∗κ g extends to all Lp(Rd ;h2κ), 1 ≤ p ≤ ∞, as a bounded operator. In particular,

‖ f ∗κ g‖κ,p ≤ ‖g‖κ,1‖ f‖κ,p. (6.4.2)

6.4. Generalized convolution and summability 83

Proof. For g ∈ L1(Rd ;h2κ), bounded and radial, we have |τyg| ≤ τy|g|, which shows that∫Rd

|τyg(x)|h2κ(x)dx ≤

∫Rd

|g(x)|h2κ(x)dx.

Therefore, ∫Rd

| f ∗κ g(x)|h2κ(x)dx ≤ ‖ f‖κ,1‖g‖κ,1.

We also have ‖ f ∗κ g‖∞ ≤ ‖ f‖∞‖g‖κ,1. By the Riesz–Thorin interpolation theorem, weobtain ‖ f ∗κ g‖κ,p ≤ ‖g‖κ,1‖ f‖κ,p. �

For φ ∈ L1(Rd ;h2κ) and ε > 0, we define the dilation φε by

φε(x) = ε−(2γκ+d)φ(x/ε) = ε−(2λκ+2)φ(x/ε).

A change of variables shows that∫Rd

φε(x)h2κ(x)dx =

∫Rd

φ(x)h2κ(x)dx, for all ε > 0.

Theorem 6.4.2. Let φ ∈ L1(Rd ;h2κ) be a bounded radial function and assume that it

satisfies ch∫Rd φ(x)h2

κ(x)dx = 1. Then for f ∈ Lp(Rd ;h2κ), 1 ≤ p < ∞, and f ∈ C0(R

d),p = ∞,

limε→0

‖ f ∗κ φε − f‖κ,p = 0.

Proof. For a given η > 0 we choose g ∈ C∞0 such that ‖g− f‖κ,p < η/3. The triangle

inequality and (6.4.2) lead to

‖ f ∗κ φε − f‖κ,p ≤ 23

η +‖g∗κ φε −g‖κ,p.

Since φ is radial, we can choose a radial function ψ ∈C∞0 such that

‖φ −ψ‖κ,1 ≤ (12‖g‖κ,p)−1η .

If we let a = ch∫Rd ψ(y)h2

κ(y)dy, then, by the triangle inequality and (6.4.2),

‖g∗κ φε −g‖κ,p ≤ ‖g‖κ,p‖φ −ψ‖κ,1 +‖g∗κ ψε −ag‖κ,p + |a−1|‖g‖κ,p

≤ η/6+‖g∗κ ψε −ag‖κ,p,

since ‖g‖κ,p‖φ −ψ‖κ,1 ≤ η12 and

|a−1|=∣∣∣∣ch

∫Rd

(φε(x)−ψε(x))h2κ(x)dx

∣∣∣∣ ≤ (12‖g‖κ,p)−1η .

Thus,

‖ f ∗κ φε − f‖κ,p ≤ 56

η +‖g∗κ ψε −ag‖κ,p.


Hence it suffices to show that ‖g∗κ ψε −ag‖κ,p ≤ η/6. Now g ∈ Aκ(Rd), hence

g∗κ φε(x) =∫Rd

g(y)τxφε(y)h2κ(y)dy =

∫Rd

τ−xg(y)φε(−y)h2κ(y)dy.

We also know that τ−xg(y) = τ−yg(x), as g ∈C∞0 . Therefore,

g∗κ φε(x) =∫Rd

τyg(x)φε(y)h2κ(y)dy.

In view of this

g∗κ ψε(x)−ag(x) =∫Rd

(τyg(x)−g(x))ψε(y)h2κ(y)dy,

which gives, by Minkowski’s integral inequality,

‖g∗κ ψε −ag‖κ,p ≤∫Rd

‖τyg−g‖κ,pψε(y)|h2κ(y)dy.

Now, using Theorem 6.2.7, the rest of the proof follows as in the proof of Lemma 6.2.9.�

6.4.2 Summability of the inverse Dunkl transform

With Theorem 6.4.2 established, we can now extend our proof of the inversion formula inTheorem 6.2.10 to a more general summability method of the inverse Dunkl transform.

Let Φ ∈ L1(Rd ;h2κ) be continuous at 0 and assume Φ(0) = 1. For f ∈ S and ε > 0

define

Tε f (x) = ch

∫Rd

Fκ f (y)E(ix,y)Φ(−εy)h2κ(y)dy.

It is clear that Tε extends to the whole of L2 as a bounded operator: this follows fromPlancherel’s theorem. Let us study the convergence of Tε f as ε → 0. Note that T0 f = fby the inversion formula for the Dunkl transform.

If Tε f can be extended to all f ∈ Lp(Rd ;h2κ) and if Tε f → f in Lp(Rd ;h2

κ), we saythat the inverse Dunkl transform is Φ-summable.

Proposition 6.4.3. Suppose both Φ and φ = Fκ Φ belong to L1(Rd ;h2κ). If Φ is radial,

thenTε f (x) = ( f ∗κ φε)(x)

for all f ∈ L2(Rd ;h2κ) and ε > 0.

Proof. Under the hypothesis on Φ, both Tε and the operator taking f into ( f ∗κ φε) extendto L2(Rd ;h2

κ) as bounded operators. So it is enough to verify Tε f (x) = ( f ∗κ φε)(x) for

6.4. Generalized convolution and summability 85

all f in the Schwartz class. By the definition of the Dunkl transform,

Tε f (x) = ch

∫Rd

Fκ τ−x f (y)Φ(−εy)h2κ(y)dy

= ch

∫Rd

τ−x f (ξ )ch

∫Rd

Φ(−εy)E(y,−iξ )h2κ(y)dyh2

κ(ξ )dξ

= chε−(d+2γκ )∫Rd

τ−x f (ξ )Fκ Φ(−ε−1ξ )h2κ(ξ )dξ

= ( f ∗κ φε)(x),

where we have changed variable ξ �→ −ξ and used the fact that τ−x f (−ξ ) = τξ f (x). �

Theorem 6.4.4. Let Φ(x) ∈ L1(Rd ;h2κ) be radial and assume that Fκ Φ ∈ L1(Rd ;h2

κ) isbounded and Φ(0) = 1. For f ∈ Lp(Rd ;h2

κ), Tε f converges to f in Lp(Rd ;h2κ) as ε → 0,

for 1 ≤ p < ∞.

Proof. Since f ∗κ φε agrees with Tε f on L2(Rd ;h2κ) by the previous theorem, and f ∗κ φε

is bounded in Lp(Rd ;h2κ), Tε f can be extended to Lp(Rd ;h2

κ). The convergence of Tε f tof now follows from Theorem 6.4.2. �

By choosing specific radial functions Φ, we consider several examples of summa-bility methods.

Heat kernel transform. We consider f ∗κ qκt , where qκ

t is the heat kernel defined in(6.1.3). Recall the Dunkl Laplacian Δh defined in (2.2.3).

Theorem 6.4.5. Suppose f ∈ Lp(Rd ;h2κ), 1 ≤ p < ∞ or f ∈C0(R

d), p = ∞.

1. The heat transform

Ht f (x) := ( f ∗κ qκt )(x) = ch

∫Rd

f (y)τyqκt (x)h

2κ(y)dy, t > 0,

converges to f in Lp(Rd ;h2κ) as t → 0.

2. Define H0 f (x) = f (x). Then the function Ht f (x) solves the initial value problem

Δhu(x, t) = ∂tu(x, t), u(x,0) = f (x), (x, t) ∈ Rd × [0,∞).

Proof. By (6.1.4), Fκ qκt = e−t‖x‖2

, so that (1) follows by setting Φ(x) = e−‖x‖2and ε =√

t. Using the spherical-polar form of Δh in (3.1.5), it is easy to see that qκt satisfies the

heat equation Δhu(x, t) = ∂tu(x, t), so that (2) holds. �

Poisson integral. The Poisson kernel is defined by

P(x,ε) := cd,κε

(ε2 +‖x‖2)λk+32, cd,κ = 2λκ+1 Γ(λκ +

32 )√

π. (6.4.3)

Theorem 6.4.6. Suppose f ∈ Lp(Rd ;h2κ), 1 ≤ p < ∞, or f ∈ C0(R

d), p = ∞. Then thePoisson integral f ∗κ Pε converges to f in Lp(Rd ;h2

κ).

Proof. Set Φ(x) = e−‖x‖. Then P(x,1) =Fκ Φ(x) and P(x,ε) = ε−λκ−2P(x/ε,1). Indeed,this can be proved as for the ordinary Fourier transform using the integral relation betweene−t and e−t2

in (3.4.8) and the fact that the Dunkl transform of e−‖x‖2/4 is e−‖x‖2, as shown

by (6.1.4). Since Φ(0) = 1, it readily follows that ch∫Rd P(x,ε)h2

κ(x)dx = 1. Thus, theconvergence is established by Theorem 6.4.4. �Bochner–Riesz means. Here we consider

Φ(x) =

{(1−‖x‖2)δ , ‖x‖ ≤ 1,0, otherwise,

where δ > 0. As in the case of the ordinary Fourier transform, we take ε = 1/R whereR > 0. Then the Bochner–Riesz means of order δ are defined by

SδR f (x) := ch

∫‖y‖≤R

(1− ‖y‖

R

)δFκ f (y)E(ix,y)h2

κ(y)dy. (6.4.4)

Theorem 6.4.7. If f ∈ Lp(Rd ;h2κ), 1 ≤ p ≤ ∞ and δ > d−1

2 + γκ , then

‖SδR f − f‖κ,p → 0, as R → ∞.

Proof. The proof follows as in the case of ordinary Fourier transform [52, p. 171]. FromTheorem 6.1.8 and the properties of the Bessel function, we have

Fκ Φ(x) = 2λκ‖x‖−λκ−δ−1Jλκ+δ+1(‖x‖).

It is known that Jα(r) = O(r−1/2), so that Fκ Φ ∈ L1(Rd ,h2κ) under the condition δ >

λκ +1/2. �We note that λκ = (d −2)/2+ γκ , where γκ is the sum of all (nonnegative) param-

eters in the weight function. If all parameters are zero, then hκ(x)≡ 1 and we are back tothe classical Fourier transform, for which the index (d −1)/2 is the critical index for theBochner–Riesz means.

6.4.3 Convolution operator for Zd2

In the case of hκ(x) associated with the group Zd2, the translation operator τy is bounded on

Lp(Rd ,h2κ). Hence, the standard proof can be used to establish the following inequality:

Theorem 6.4.8. Let G = Zd2 . Let p,q,r ≥ 1 and p−1 = q−1 + r−1 −1. Let f ∈ Lq(Rd ,h2

κ)and g ∈ Lr(Rd ,h2

κ). Then

‖ f ∗κ g‖κ,p ≤ c‖ f‖κ,q‖g‖κ,r.

6.5. Maximal function 87

The boundedness of τy allows us to remove the assumption that φ is radial in Theo-rem 6.4.2 when G = Zd

2.

Theorem 6.4.9. Let φ ∈ L1(Rd ,h2κ) with

∫Rd φ(x)h2

κ(x)dx = 1. Then for f ∈ Lp(Rd ,h2κ)

if 1 ≤ p < ∞, or f ∈C0(Rd) if p = ∞,

limε→0

‖ f ∗κ φε − f‖κ,p = 0, 1 ≤ p ≤ ∞.

Proof. For f ∈ Lp(Rd ,h2κ) we write f = f1+ f2, where f1 is in C∞

0 with compact support,and ‖ f2‖κ,p ≤ δ . Then the second term of the inequality

‖τy f (x)− f (x)‖κ,p ≤ ‖τy f1(x)− f1(x)‖κ,p +‖τy f2(x)− f2(x)‖κ,p

is bounded by (1+ c)δ , as τy is a bounded operator, and the first term goes to zero asε → 0 by Theorem 6.2.7. This proves that ‖τy f (x)− f (x)‖κ,p → 0 as y → 0. We havethen

ch

∫Rd

| f ∗κ gε(x)− f (x)|ph2κ(x)dx

= ch

∫Rd

∣∣∣∣ch

∫Rd(τy f (x)− f (x))gε(y)h2

κ(y)dy∣∣∣∣p

h2κ(x)dx

≤ ch

∫Rd

‖τy f − f‖pκ,p|gε(x)|h2

κ(x)dx

= ch

∫Rd

‖τεy f − f‖pκ,p|g(x)|h2

κ(x)dx,

which goes to zero as ε → 0. �

All results on the summability for the inverse Dunkl transform in the previous sec-tion are proved for a generic reflection group, hence they all hold for the case of Zd

2.

6.5 Maximal function

We study the Hardy–Littlewood maximal function in weighted spaces.

6.5.1 Boundedness of maximal function

Let Br = B(0,r) denote the ball of radius r centered at the origin, and let χBr denote itscharacteristic function.

Definition 6.5.1. For f ∈ L1(Rd ;h2κ), we define the maximal function Mκ f by

Mκ f (x) := supr>0

∣∣∫Rd f (y)τxχBr(y)h

2κ(y)dy

∣∣∫Br

h2κ(y)dy

.

The function χBr is radial. If ϕ ∈ C∞0 (R

d) is a radial function such that χBr(x) ≤ϕ(x), then, by Theorem 6.3.3, τyχBr(x) ≤ τyϕ(x). As τyϕ is bounded, τyχBr is boundedand compactly supported so that it belongs to L1(Rd ;h2

κ). Hence, Mκ f is well defined forf ∈ L1(Rd ;h2

κ). Using spherical polar coordinates, we also have∫Br

h2κ(y)dy =

∫ r

0sd−1+2γκ ds

∫Sd−1

h2κ(ξ )dσ(ξ ) =

ωκd

2λκ +2r2λκ+2.

By definition, we can also write Mκ f as

Mκ f (x) = supr>0

1dκ r2λκ+2 | f ∗κ χBr(x)|, dκ :=

ωκd

2λκ +2.

Since τyχBr ≥ 0, we have Mκ f (x)≤ Mκ | f |(x).Theorem 6.5.2. The maximal function is bounded on Lp(Rd ;h2

κ) for 1< p≤∞; moreoverit is of weak type (1,1), that is, for f ∈ L1(Rd ;h2

κ) and α > 0,∫E(a)

h2κ(x)dx ≤ c

α‖ f‖κ,1,

where E(α) = {x : Mκ f (x)> α} and c is a constant independent of α and f .

Proof. Without loss of generality we can assume that f ≥ 0. Let Pε(x) := P(x,ε) be thePoisson kernel defined in (6.4.3), and let σ := 2λκ + 3. For j ≥ 0, define Br, j := {x :2− j−1r ≤ ‖x‖ ≤ 2− jr}. Then

χBr, j(y) = (2− jr)σ (2− jr)−σ χBr, j(y)

≤ c(2− jr)σ−1 2− jr((2− jr)2 +‖y‖2)σ/2 χBr, j(y)

≤ c(2− jr)σ−1P2− jr(y),

where c is a constant independent of r and j. Since both χBr and Pε are bounded, integrableradial functions, it follows from Theorem 6.3.3 that

τxχBr, j(y)≤ c(2− jr)σ−1τxP2− jr(y).

This shows that for any positive integer m∫Rd

f (y)m

∑j=0

τxχBr, j(y)h2κ(y)dy ≤ c

∞

∑j=0

(2− jr)σ−1∫Rd

f (y)τxP2− jr(y)h2κ(y)dy

≤ crd+2γκ supt>0

f ∗κ Pt(x).

As ∑mj=0 χBr, j(y) converges to χBr(y) in L1(Rd ;h2

κ), the boundedness of τx on L1rad(R

d ;h2κ)

shows that ∑mj=0 τxχBr, j(y) converges to τxχBr(y) in L1(Rd ;h2

κ). By passing to a subse-quence if necessary, we can assume that ∑m

j=0 τxχBr, j(y) converges to τxχBr(y) for almost

every y. Thus all the functions involved are uniformly bounded by τxχBr(y). This showsthat ∑m

j=0 τxχBr, j(y) converges to τxχBr(y) in Lp′(Rd ;h2κ) and hence

limm→∞

∫Rd

f (y)m

∑j=0

τxχBr, j(y)h2κ(y)dy =

∫Rd

f (y)τxχBr(y)h2κ(y)dy.

Thus we have provedf ∗κ χBr(x)≤ crd+2γκ sup

t>0f ∗κ Pt(x),

which gives the inequality Mκ f (x) ≤ cP∗ f (x), where P∗ f (x) = supt>0 f ∗κ Pt(x) is themaximal function associated to the Poisson semi-group.

Therefore, it is enough to prove the boundedness of P∗ f . By looking at the Dunkltransforms of the Poisson kernel and the heat kernel we conclude, as in the proof ofLemma 3.4.7, that

f ∗κ Pt(x) =t√2π

∫ ∞

0( f ∗κ qs)(x)e−t2/2ss−3/2ds,

which allows us to conclude that

P∗ f (x)≤ csupt>0

1t

∫ t

0Qs(| f |)(x)ds,

where Qs f (x) = f ∗κ qs(x) is the heat semi-group. Hence, using the Hopf–Dunford–Schwartz ergodic theorem (Theorem 3.4.3), we conclude the boundedness of P∗ f onLp(Rd ;h2

κ) for 1 < p ≤ ∞, and the weak type (1,1). �

The maximal function can be used to study almost everywhere convergence of theconvolution f ∗κ ϕε when φ satisfies moderate conditions.

Theorem 6.5.3. Let φ ∈ Aκ(Rd) be a real valued radial function which satisfies |φ(x)| ≤

c(1+‖x‖)−2λκ−3. Thensupε>0

| f ∗κ φε(x)| ≤ cMκ f (x).

Consequently, f ∗κ φε(x)→ f (x) for almost every x as ε goes to 0, for all f in Lp(Rd ;h2κ),

1 ≤ p < ∞.

Proof. We can assume that both f and φ are nonnegative. Writing

φε(y) =∞

∑j=−∞

φε(y)χε2 j≤‖y‖≤ε2 j+1(y),

we have ∣∣∣∣τx

[ m

∑j=−m

φε χε2 j≤‖y‖≤ε2 j+1

](y)

∣∣∣∣ ≤ cm

∑j=−m

(1+2 j)−2λκ−3ε−2λκ−2.


This shows that∫Rd

f (y)τx

[φε(y)

m

∑j=−m

χε2 j≤‖y‖≤ε2 j+1

](y)h2

κ(y)dy

≤ cε−2λκ−2m

∑j=−m

(1+2 j)−2λκ−3(ε2 j)2λκ+2Mκ f (x)≤ cMκ f (x).

Since |φ(y)| ≤ c(1+‖y‖)−2λκ−3 ≤ cP1(y) it follows that |τxφ(y)| ≤ cτxP1(y) is bounded.Arguing as in the previous theorem, we can show that the left-hand side of the aboveinequality converges to f ∗κ φε(x). Thus we obtain

supε>0

| f ∗κ φε(x)| ≤ cMκ f (x),

from which the proof of the almost everywhere convergence follows via the standardargument. �

6.5.2 Convolution versus maximal function for Zd2

In the case of Zd2, the conditions of the last theorem can be relaxed. For this, we need the

spherical mean operator defined on Aκ(Rd) by

Sr f (x) :=1

ωκd

∫Sd−1

τry f (x)h2κ(y)dσ(y).

If f ∈ Aκ(Rd) and g(x) = g0(‖x‖) is an integrable radial function, then, using spherical-

polar coordinates, the generalized convolution f ∗κ g can be expressed in terms of thespherical mean:

( f ∗κ g)(x) = ch

∫Rd

τy f (x)g(y)h2κ(y)dy

= ch

∫ ∞

0r2λκ+1g0(r)

(∫Sd−1

τry′ f (x)h2κ(y

′)dy′)

dr

= chωκd

∫ ∞

0Sr f (x)g0(r)r2λκ+1dr.

The spherical mean operator is bounded.

Theorem 6.5.4. Let G = Zd2 . For f ∈ Lp(Rd ,h2

κ),

‖Sr f‖κ,p ≤ c‖ f‖κ,p, 1 ≤ p ≤ ∞.

Furthermore, ‖Sr f − f‖κ,p → 0 as r → 0+.

Proof. Using Holder’s inequality,

|Sr f (x)|p ≤ 1ωκ

d

∫Sd−1

|τry f (x)|ph2κ(y)dσ(y).

Hence, a simple computation shows that

ch

∫Rd

|Sr f (x)|ph2κ(x)dx ≤ ch

∫Rd

1ωκ

d

(∫Sd−1

|τry f (x)|ph2κ(y)dω(y)

)h2

κ(x)dx

=1

ωκd

∫Sd−1

‖τry f‖pκ,ph2

κ(y)dσ(y)

≤ c‖ f‖κ,p.

Furthermore,

‖Sr f − f‖pκ,p ≤

1ωκ

d

∫Sd−1

‖τry f − f‖pκ,ph2

κ(y)dσ(y),

which goes to zero as r → 0, since ‖τry f − f‖κ,p → 0. �Theorem 6.5.5. Set G = Zd

2 . Let φ(x) = φ0(‖x‖) ∈ L1(Rd ;h2κ) be a radial function. As-

sume that φ0 is differentiable, limr→∞ φ0(r) = 0, and∫ ∞

0 r2λκ+2|φ0(r)|dr < ∞. Then

|( f ∗κ φ)(x)| ≤ cMκ f (x).

In particular, if φ ∈ L1(Rd ;h2κ) and ch

∫Rd φ(x)h2

κ(x)dx = 1, then

1. for 1 ≤ p ≤ ∞, f ∗κ φε converges to f as ε → 0 in Lp(Rd ;h2κ);

2. for f ∈ L1(Rd ,h2κ), ( f ∗κ φε)(x) converges to f (x) as ε → 0 for almost all x ∈ Rd.

Proof. By definition of the spherical mean St f , we can also write

Mκ f (x) = supr>0

∣∣∫ r0 t2λκ+1St f (x)dt

∣∣∫ r0 t2λκ+1dt

.

Since |Mκ f (x)| ≤ cMκ | f |(x), we can assume f (x)≥ 0. The assumption on φ0 shows that

limr→∞

φ0(r)∫ r

0St f (x)t2λκ+1dt = lim

r→∞φ0(r)

∫Rd

τy f (x)h2κ(y)dy

= limr→∞

φ0(r)∫Rd

f (y)h2κ(y)dy = 0.

Hence, using the spherical-polar coordinates and integrating by parts, we get

( f ∗κ φ)(x) =∫ ∞

0φ0(r)r2λκ+1Sr f (x)dr

=−∫ ∞

0

(∫ r

0St f (x)t2λκ+1dt

)φ ′(r)dr,

which implies that

|( f ∗κ φ)(x)| ≤ cMκ f (x)∫ ∞

0r2λκ+2|φ ′

0(r)|dr.

Boundedness of the last integral proves the maximal inequality. �

We can further enhance Theorem 6.5.5 by removing the assumption that φ is radial.For this purpose, we make the following simple observation about the maximal function.If f is nonnegative, then we can drop the absolute value sign in the definition of themaximal function, even though τy f may not be nonnegative.

Lemma 6.5.6. If f ∈ L1(Rd ,h2κ) is a nonnegative function, then

Mκ f (x) = supr>0

∫Br

τy f (x)h2κ(y)dy∫

Brh2

κ(y)dy.

In particular, if f and g are two nonnegative functions, then

Mκ f +Mκ g = Mκ( f +g).

Proof. Since τyχBr(x) is nonnegative, we have that

( f ∗κ χBr)(x) =∫Rd

f (y)τyχBr(x)h2κ(y)dy

is nonnegative if f is nonnegative. Hence, we can drop the absolute value symbol in thedefinition of Mκ f . �

Theorem 6.5.7. Set G = Zd2 . Let φ ∈ L1(Rd ,h2

κ) and let ψ(x) = ψ0(‖x‖) ∈ L1(Rd ,h2κ)

be a nonnegative radial function such that |φ(x)| ≤ ψ(x). Assume that ψ0 is differen-tiable, limr→∞ ψ0(r) = 0, and

∫ ∞0 r2λκ+2|ψ0(r)|dr < ∞. Then supε>0 | f ∗κ φε(x)| is of

weak type (1,1). In particular, if φ ∈ L1(Rd ,h2κ) and ch

∫Rd φ(x)h2

κ(x)dx = 1, then, forf ∈ L1(Rd ,h2

κ), ( f ∗κ φε)(x) converges to f (x) as ε → 0 for almost all x ∈ Rd.

Proof. Since Mκ f (x) ≤ Mκ | f |(x), we can assume that f (x) ≥ 0. The proof uses the ex-plicit formula for τy f . Let us first consider the case d = 1. Since ψ is an even function,τyψ is given by

τy f (x) =∫ 1

−1f(√

x2 + y2 −2xyt)

Φκ(t)dt,

according to (6.3.2). Since (x− y)(1+ t) = (x− yt)− (y− xt), we have

|x− y|√x2 + y2 −2xyt

(1+ t)≤ 2.

Consequently, by the explicit formula for τy f , see (6.3.2), the inequality |φ(x)| ≤ ψ(x)implies

|τyφ(x)| ≤ τyψ(x)+2τy,1ψ(x),

where τy,1ψ is defined by

τy,1ψ(x) = bκ

∫ 1

−1f(√

x2 + y2 −2xyt)(1− t2)κ−1dt.


Note that τy,1ψ differs from τyψ by the factor 1+ t in the weight function. The change ofvariables t �→ −t and y �→ −y in the integrals shows that∫

Rf (y)τy,1ψ(x)h2

κ(y)dy =∫R

F(y)τyψ(x)h2κ(y)dy,

where F(y) = ( f (y)+ f (−y))/2. It follows that

|( f ∗κ φ)(x)|=∣∣∣∣ch

∫R

f (y)τyφ(x)h2κ(y)dy

∣∣∣∣ ≤ ( f ∗κ ψ)(x)+2(F ∗κ ψ)(x).

The same consideration can be extended to the case of Zd2 for d > 1. Let {e1, . . . ,ed} be the

standard Euclidean basis. For δ j =±1 define xδ j = x− (1+δ j)x je j (that is, multiplyingthe j-th component of x by δ j gives xδ j). For 1 ≤ j ≤ d we define

Fj1,..., jk = 2−k ∑(δ j1 ,...,δ jk )∈Zk

2

f (xδ j1 · · ·δ jk).

In particular,

Fj(x) = (F(x)+F(xδ j))/2, Fj1, j2(x) = (F(x)+F(xδ j1)+F(xδ j2)+F(xδ j1δ j2))/4,

and the last sum is over Zd2, F1,...,d(x) = 2−d ∑σ∈Zd

2f (xσ). Following the proof in the case

d = 1 it is not hard to see that

|( f ∗κ φ)(x)| ≤ ( f ∗κ ψ)(x)+2d

∑j=1

(Fj ∗κ ψ)(x)+4 ∑j1 = j2

(Fj1, j2 ∗κ ψ)(x)

+ · · ·+2d(F1,...,d ∗κ ψ)(x).

For G = Zd2, the explicit formula of τy shows that Mκ f (x) is even in each of its variables.

Hence, applying the result of the previous theorem to each of the above terms, we get

|( f ∗κ φ)(x)| ≤ Mκ f (x)+2d

∑j=1

Mκ Fj(x)+4 ∑j1 = j2

Mκ Fj1, j2(x)

+ · · ·+2dMκ F1,...,d(x).

Since all Fj are clearly nonnegative, by Lemma 6.5.6, the last expression can bewritten as Mκ H, where H is the sum of all functions involved. Consequently, since‖Fj1,..., jd‖κ,1 ≤ ‖ f‖κ,1,∫

{x:( f∗κ φ)(x)≥a}h2

κ(y)dy ≤ c‖H‖κ,1

a≤ cd

‖ f‖κ,1

a.

Hence, f ∗κ φ is of weak type (1,1), from which the almost everywhere convergencefollows as usual. �

The Dunkl transform was introduced in [27], where the L2 isometry was established aswe have seen in Section 6.1. The Dunkl transform was studied in [34], where the boundfor E(x,y) was established for general parameters κ with Reκ ≥ 0 and without the pos-itivity of Vκ when κ is real and nonnegative, and where the main results of the L1 the-ory (Theorem 6.2.10) were also established. Our development in Section 6.2, based onthe convolution operator, follows closely the approach of the classical harmonic analysis(see, for example, [52]). The generalized translation operator was studied in [39, 63] forsmooth functions, and the starting point in [63] was the expression

τy f (x) =V (x)κ ⊗V (y)

κ[(V−1

κ f )(x+ y)],

where V−1κ denotes the inverse of Vκ , which satisfies V−1

κ f (x) = e−〈y,D〉 f (x)|y=0. Thisexpression by itself, however, does not provide much useful information on τy f . Theformula for τy for radial functions was proven in [44] under more restrictive conditions,our proof here is taken from an earlier version of [59] and the rest of our development inSection 6.3 follows from the latter paper. Our Section 6.4, convolution and summabilityalso follows the treatment in [59]. Some of the summability method, such as the heattransform and Poisson transforms were studied earlier in [42, 45]. The maximal functionswere defined and studied in [59]. Further results on maximal functions were obtained by[1, 17, 18]. Paley–Wiener theorem (Theorem 6.1.10) was proved in [59]. A much moregeneral study of Paley–Wiener theorems for Dunkl transform was given in [35], see also[63].

Many results for the classical Fourier transform can be extended to the Dunkl trans-forms. For example, in the distributional sense

Fκ

( P(x)‖x‖2λκ+2+n−α

)= dα

n,κP(x)

‖x‖n+α , dαn,κ = i−n 2λκ+1−α Γ( n+α

2 )

Γ(λκ +1+ n−α2 )

,

where P ∈ H dn (h2

κ) and 0 < Re{α} < 2λκ + 2, which allows one to define analoguesof the Riesz potentials and Bessel potentials for the Dunkl transforms and to study theirboundedness in Lp spaces [60]. For the latter purpose, however, we need the boundednessof the translation operator, which however is known, as shown in Section 6.3, to hold,only for G = Zd

2 or radial functions. Thus, the boundedness can be established at thispoint only for G = Zd

2; see [32, 59].The simplest non-trivial case of the Dunkl transform is the one on the real line

associated with the weight function |x|κ that is invariant under Z2. If f is even, then itagrees with the Hankel transform. For general functions, the difference part comes in andneeds to be dealt with. Nevertheless, the structure is relatively simple and many tools inharmonic analysis are accessible. There are numerous papers on Dunkl transforms on thereal line. Interested readers should check MathSciNet or arXiv.

Chapter 7

Multiplier Theorems for the

Dunkl Transform

For a family of weight functions invariant under a finite reflection group, we prove atransference theorem between the Lp multiplier of h-harmonic expansions on Sd and thatof the Dunkl transform. This theorem is stated together with some related definitionsand notations in Section 7.1. The proof of this transference theorem is, however, ratherlong, so we split it into three parts, which are given in the Sections 7.2, 7.3, and 7.4,respectively. The transference theorem allows us to deduce several useful results for theDunkl transform on Rd from the corresponding results for the h-harmonic expansions onSd . This is done in the last two sections, 7.5 and 7.6. More precisely, in Section 7.5, thetransference theorem combined with Theorem 4.4.2, Theorem 4.5.2 is used to establisha Hormander type multiplier theorem and the Littlewood–Paley inequality for the Dunkltransform on Rd . In Section 7.5, we apply the transference theorem and Theorem 3.3.6 todeduce the convergence of the Bochner–Riesz means of order above the critical index inthe weighted Lp spaces for the group G = Zd

2.

7.1 Introduction

Let hκ denote the weight function on Rd defined by (2.1.2), invariant under a finite reflec-tion group G generated by a reduced root system R in Rd . Throughout this chapter, theroot system R is normalized so that 〈α,α〉= 2 for all α ∈ R, and κ denotes a nonnegativemultiplicative function on R. For each g∈G, we denote by g′ the reflection on Rd+1 givenby

x′g′ = (xg,xd+1) for x′ = (x,xd+1) with x ∈ Rd and xd+1 ∈ R.

Then G′ := {g′ : g ∈ G} is a finite reflection group on Rd+1 with a reduced root systemR′ := {(α,0) : α ∈ R}. Let κ ′ denote the nonnegative multiplicity function on R′ givenby κ ′(α,0) = κα for α ∈ R. We denote by Vκ ′ the intertwining operator on C(Rd+1)




87 7

95

96 Chapter 7. Multiplier Theorems for the Dunkl Transform

associated with the reflection group G′ and the multiplicity function κ ′. Define

hκ ′(x,xd+1) := hκ(x) = ∏α∈R+

|〈x,α〉|κα , x ∈ Rd , xd+1 ∈ R,

where R+ is an arbitrary but fixed positive subsystem of R. Recall that H d+1n (h2

κ ′) de-notes the space of h-harmonics of degree n on the sphere Sd , and projκ

′n : L2(Sd ;h2

κ ′)→H d+1

n (h2κ ′) denotes the orthogonal projection onto the space H d+1

n (h2κ ′). We will keep

the notations G′, κ ′, hκ ′ and Vκ ′ throughout this chapter. Also, recall that for a given1 ≤ p ≤ ∞, Lp(Rd ;h2

κ) denotes the weighted Lebesgue space on Rd endowed with thenorm

‖ f‖κ,p :=(∫

Rd| f (y)|ph2

κ(y)dy) 1

p

,

with the usual change when p = ∞.One of the main goals in this chapter is to show the following transference theo-

rem between the Lp multiplier of h-harmonic expansions on Sd and that of the Dunkltransform.

Theorem 7.1.1. Let m : [0,∞)→ R be a continuous and bounded function, and let Uε ,ε > 0, be a family of multiplier operators on L2(Sd ;h2

κ ′) given by

projκ′

n (Uε f ) = m(εn)projκ′

n f , n = 0,1, . . . . (7.1.1)

Assume thatsupε>0

‖Uε f‖Lp(Sd ;h2κ ′ )

≤ A‖ f‖Lp(Sd ;h2κ ′ ), ∀ f ∈C(Sd) (7.1.2)

for some 1 ≤ p ≤ ∞. Then the function m(‖ · ‖) defines an Lp(Rd ;h2κ) multiplier; that is,

‖Tm f‖Lp(Rd ;h2κ )≤ cd,κ A‖ f‖Lp(Rd ;h2

κ ), ∀ f ∈ S (Rd),

where Tm is an operator initially defined on L2(Rd ;h2κ) by

Fκ(Tm f )(ξ ) = m(‖ξ‖)Fκ f (ξ ), f ∈ L2(Rd ;h2κ), ξ ∈ Rd . (7.1.3)

The proof of Theorem 7.1.1 is rather long, so we break it into three parts, given inSections 7.2, 7.3 and 7.4, respectively. The first part contains several technical lemmasthat are crucial for the proof. The second part proves the conclusion under the additionalassumption that |m(t)| ≤ c1e−c2t for all t > 0 and some c1,c2 > 0. The third part showshow the additional decaying condition on m can be relaxed to yield the desired conclusion.

7.2 Proof of Theorem 7.1.1: part I

A series of technical lemmas used in the proof of Theorem 7.1.1 are proved below. Thefirst lemma reveals a connection between the Dunkl intertwining operators Vκ and Vκ ′ .

7.2. Proof of Theorem 7.1.1: part I 97

Lemma 7.2.1. If f ∈ Πd+1, then for any x ∈ Rd and xd+1 ∈ R,

Vκ ′ f (x,xd+1) =Vκ [ f (·,xd+1)](x) =∫Rd

f (ξ ,xd+1)dμκx (ξ ), (7.2.1)

where dμκx is the Borel measure in the integral representation of Vκ in Theorem 2.3.4.

Proof. Clearly, the second equality in (7.2.1) follows directly from Theorem 2.3.4. Toshow the first equality, we set Vκ,1 f (x,xd+1) = Vκ [ f (·,xd+1)](x) for f ∈ C(Rd+1) andx ∈ Rd . Since Vκ ′ is a linear operator uniquely determined by (2.3.1), it suffices to showthat the following conditions are satisfied:

Vκ,1(Pd+1n )⊂ Pd+1

n , Vκ,1(1) = 1, and Dκ ′,iVκ,1 =Vκ,1∂i, 1 ≤ i ≤ d +1,

where we used the notation Dκ,i rather than Di to denote the Dunkl operators introducedin Definition 2.2.1 to emphasize their dependence on the multiplicative function κ . In-deed, these conditions can be easily verified using the properties of Vκ in (2.3.1), and thefollowing identities, which follow directly from (2.2.1):

Dκ ′,ig(x,xd+1) = Dκ,i

[g(·,xd+1)

](x), 1 ≤ i ≤ d,

Dκ ′,d+1g(x,xd+1) = ∂d+1g(x,xd+1), for g ∈ Πd+1, x ∈ Rd , and xd+1 ∈ R.

This completes the proof of Lemma 7.2.1. �

To formulate the next lemma, we define the mapping ψ : Rd → Sd by

ψ(x) := (ξ sin‖x‖,cos‖x‖) for x = ‖x‖ξ ∈ Rd and ξ ∈ Sd−1.

Given N ≥ 1, we denote by NSd := {x ∈ Rd+1 : ‖x‖ = N} the sphere of radius N inRd+1, and define the mapping ψN : Rd → NSd by

ψN(x) := Nψ( x

N

)=

(Nξ sin

‖x‖N

,N cos‖x‖N

)(7.2.2)

with x = ‖x‖ξ ∈ Rd and ξ ∈ Sd−1.

Lemma 7.2.2. If f : NSd → R is supported in the set {x ∈ NSd : arccos(N−1xd+1)≤ 1},then∫

Sdf (Nx)h2

κ ′(x)dσ(x) = N−2λκ−2∫

B(0,N)f(ψN(x)

)h2

κ(x)(

sin(‖x‖/N)

‖x‖/N

)2λκ+1

dx,

where B(0,N) = {y ∈ Rd : ‖y‖ ≤ N}, and λκ = d−22 + |κ|.

Proof. First, using the polar coordinates transformation

(ξ ,θ) ∈ Sd−1 × [0,π]→ x := (ξ sinθ ,cosθ) ∈ Sd ,


and the fact that dσ(x) = sind−1 θ dθdσ(ξ ), we obtain∫Sd

f (Nx)h2κ ′(x)dσ(x)

=∫ π

0

[∫Sd−1

f (Nξ sinθ ,N cosθ)h2κ ′(ξ sinθ ,cosθ)dσ(ξ )

](sinθ)d−1 dθ

=∫ 1

0

[∫Sd−1

f (Nξ sinθ ,N cosθ)h2κ(θξ )dσ(ξ )

](sinθ

θ

)d−1+2γκ

θ d−1 dθ ,

where the last step uses the identity hκ ′(y,yd+1)= hκ(y), the fact that h2κ is a homogeneous

function of degree 2γκ , and the assumption that f is supported in the set {x ∈ NSd :arccos(N−1xd+1) ≤ 1}. Using the usual spherical coordinates transformation in Rd , thelast double integral equals∫

‖y‖≤1f(

Nysin‖y‖‖y‖ ,N cos‖y‖

)h2

κ(y)(

sin‖y‖‖y‖

)2λκ+1

dy

= N−d−2γκ∫‖x‖≤N

f(

Nx‖x‖ sin

‖x‖N

,N cos‖x‖N

)h2

κ(x)(

sin(‖x‖/N)

‖x‖/N

)2λκ+1

dx

= N−2λκ−2∫

B(0,N)f (ψNx)h2

κ(x)(

sin(‖x‖/N)

‖x‖/N

)2λκ+1

dx,

where the first step uses the homogeneity of the weight hκ and the change of variablesy = x/N. This proves the desired formula. �Remark 7.2.3. It is easily seen that the restriction ψN

∣∣B(0,N)

of the mapping ψN on

B(0,N) is a bijection from B(0,N) to {x ∈ NSd : arccos(N−1xd+1) ≤ 1}. Thus, givena function f : B(0,N) → R, there exists a unique function fN supported in {x ∈ NSd :arccos(N−1xd+1)≤ 1} such that

fN(ψNx) = f (x), ∀x ∈ B(0,N). (7.2.3)

On the other hand, using Lemma 7.2.2, we have∫Sd

fN(Nx)h2κ ′(x)dσ(x) = N−2λκ−2

∫B(0,N)

f (x)h2κ(x)

(sin(‖x‖/N)

‖x‖/N

)2λκ+1

dx. (7.2.4)

The formula (7.2.4) will play an important role in our proof of Theorem 7.1.1.The third lemma asserts that the conclusion of Theorem 6.3.2 holds under a slightly

weaker condition.

Lemma 7.2.4. If f (x) = f0(‖x‖) is a continuous radial function in L2(Rd ;h2κ), then for

almost every y ∈ Rd and almost every x ∈ Rd,

τy f (x) =Vκ

[f0

(√‖x‖2 +‖y‖2 −2‖y‖〈x, ·〉

)]( y‖y‖

). (7.2.5)

7.2. Proof of Theorem 7.1.1: part I 99

Proof. We first choose a sequence of even, C∞ functions g j on R satisfying

sup|t|≤2 j+1

|g j(t)− f0(t)| ≤ 2− j

(∫ 2 j

0s2λκ+1 ds

)− 12

.

Let ϕ j be an even, C∞ function on R such that χ[2− j ,2 j ](|t|) ≤ ϕ j(t) ≤ χ[2− j−1,2 j+1](|t|),and let f j(x)≡ f j,0(‖x‖) := g j(‖x‖)ϕ j(‖x‖) for x ∈Rd . Then it is easily seen that { f j} isa sequence of radial Schwartz functions on Rd satisfying

limj→∞

sup2− j≤|t|≤2 j

| f j,0(t)− f0(t)|= 0 (7.2.6)

andlimj→∞

‖ f j − f‖L2(Rd ;h2κ )= 0. (7.2.7)

Since each f j is a radial Schwartz function, by Theorem 6.3.2, we obtain

τy( f j)(x) =∫‖ξ‖≤1

f j,0

(√‖x‖2 +‖y‖2 −2‖y‖〈x,ξ 〉

)dμκ

y/‖y‖(ξ ). (7.2.8)

Next, we fix y ∈ Rd , and set

An ≡ An(y) := {x ∈ Rd : 2−n ≤∣∣∣‖x‖−‖y‖

∣∣∣ ≤ ‖x‖+‖y‖ ≤ 2n},

for n ∈ N and n ≥ n0(y) := [log‖y‖/ log2]+1. Since

(‖x‖−‖y‖)2 ≤ ‖x‖2 +‖y‖2 −2‖y‖|〈x,ξ 〉| ≤ (‖x‖+‖y‖)2

for all ‖ξ‖ ≤ 1, it follows by (7.2.6) that

limj→∞

f j,0

(√‖x‖2 +‖y‖2 −2‖y‖〈x,ξ 〉

)= f0

(√‖x‖2 +‖y‖2 −2‖y‖〈x,ξ 〉

)uniformly for x ∈ An(y) and ‖ξ‖ ≤ 1. This, together with (7.2.8) and Theorem 2.3.4implies that

limj→∞

τy( f j)(x) =∫‖ξ‖≤1

f0

(√‖x‖2 +‖y‖2 −2‖y‖〈x,ξ 〉

)dμκ

y/‖y‖(ξ )

=Vκ

[f0

(√‖x‖2 +‖y‖2 −2‖y‖〈x, ·〉

)]( y‖y‖

)for every x ∈ An(y)\{0} and n ≥ n0(y). On the other hand, however, by (7.2.7), we have

limj→∞

‖τy( f j)− τy f‖κ,2 = 0


for all y ∈ Rd . Thus,

τy( f )(x) =Vκ

[f0

(√‖x‖2 +‖y‖2 −2‖y‖〈x, ·〉

)]( y‖y‖

),

for almost every x ∈ An(y) and all n ≥ n0(y). Finally, observing that the set

Rd \( ∞⋃

n=n0(y)

An(y))= {x ∈ Rd : ‖x‖= ‖y‖}

has measure zero in Rd , we deduce the desired conclusion. �

Remark 7.2.5. By Theorem 2.3.4 and the support condition on the measure dμκx ,

Vκ F(rx) =∫Rd

F(rξ )dμκx (ξ ), for all F ∈C(Rd), x ∈ Rd , and r > 0. (7.2.9)

Thus, (7.2.5) can be rewritten more symmetrically as

τy f (x) =Vκ

[f0

(√‖x‖2 +‖y‖2 −2〈x, ·〉

)](y). (7.2.10)

Lemma 7.2.6. Let Φ ∈ L1(R, |x|2λκ+1) be an even, bounded function on R, and let TΦ bethe operator L2(Rd ;h2

κ)→ L2(Rd ;h2κ) defined by

Fκ(TΦ f )(ξ ) := Fκ f (ξ )Φ(‖ξ‖), f ∈ L2(Rd ;h2κ).

Then TΦ has an integral representation

TΦ f (x) =∫Rd

f (y)K(x,y)h2κ(y)dy,

valid for f ∈ S (Rd) and almost every x ∈ Rd, where

K(x,y) = c∫ ∞

0Φ(s)Vκ

[Jλκ

(s√‖x‖2 +‖y‖2 −2〈x, ·〉)(

s√‖x‖2 +‖y‖2 −2〈x, ·〉)λκ

](y)s2λκ+1 ds. (7.2.11)

Furthermore, K(x,y) = K(y,x) for almost every x ∈ Rd and almost every y ∈ Rd.

Proof. Let g(x) = Hλκ Φ(‖x‖), where x ∈Rd and Hα denotes the Hankel transform. SinceΦ is an even function in L1(R, |x|2λκ+1)∩L∞(R), it follows by the properties of the Hankeltransform that g is a continuous radial function in L2(Rd ;h2

κ) and Fκ g(ξ ) = Φ(‖ξ‖).Thus, using (6.4.1), we have

TΦ f (x) = f ∗κ g(x) =∫Rd

f (y)τyg(x)h2κ(y)dy

7.3. Proof of Theorem 7.1.1: part II 101

for f ∈ L2(Rd ;h2κ). Since g is a continuous radial function in L2(Rd ;h2

κ), by Lemma 7.2.4and Remark 7.2.5 it follows that

K(x,y) : = τyg(x) =Vκ

[Hλκ Φ

(√‖x‖2 +‖y‖2 +2〈x, ·〉

)](y)

= c∫ ∞

0Φ(s)Vκ

[Jλκ

(s√‖x‖2 +‖y‖2 −2〈x, ·〉)(

s√‖x‖2 +‖y‖2 −2〈x, ·〉)λκ

](y)s2λκ+1 ds,

where the last step uses (2.3.4), the inequality∣∣∣Φ(s)Jλκ (rs)(rs)λκ

∣∣∣ ≤ c|Φ(s)|

and Fubini’s theorem. This proves (7.2.11). The equality K(x,y) = K(y,x) follows fromthe fact that τxg(y) = τyg(x). �

Our final lemma is a well-known result for ultraspherical polynomials (see, for in-stance, [53, (8.1.1), p.192]):

Lemma 7.2.7. For z ∈ C and μ ≥ 0,

limk→∞

k1−2μCμk

(cos

zk

)=

Γ(μ + 12 )

Γ(2μ)

( z2

)−μ+ 12

Jμ− 12(z). (7.2.12)

This formula holds uniformly in every bounded region of the complex z-plane.

7.3 Proof of Theorem 7.1.1: part II

In this section, we shall prove Theorem 7.1.1 under the additional assumption that |m(t)| ≤c1e−c2t for all t > 0 and some c1,c2 > 0. By Lemma 7.2.6, the operator Tm has the integralrepresentation

Tm f (x) =∫Rd

f (y)K(x,y)h2κ(y)dy,

where K(x,y) is given by (7.2.11) with Φ = m. Thus, it is sufficient to prove that

I :=∣∣∣∫

Rd

∫Rd

f (y)g(x)K(x,y)h2κ(x)h

2κ(y)dxdy

∣∣∣ ≤ cA (7.3.1)

whenever f ∈ Lp(Rd ;h2κ) and g ∈ Lp′(Rd ;h2

κ) both have compact supports and satisfy‖ f‖Lp(Rd ;h2

κ )= ‖g‖Lp′ (Rd ;h2

κ )= 1. Here and in what follows, 1

p +1p′ = 1.

To this end, we choose a sufficiently large positive number N so that the supports off and g are both contained in the ball B(0,N). By Remark 7.2.3, there exist functions fNand gN , both supported in {x ∈ NSd : arccos(N−1xd+1)≤ 1} and satisfying

fN(ψN(x)

)= f (x), gN

(ψN(x)

)= g(x), x ∈ Rd , (7.3.2)


where ψN is defined by (7.2.2). It is easily seen from (7.2.4) that

‖ fN(N·)‖Lp(Sd ;h2κ ′ )

≤ cN− 2λκ+2p , ‖gN(N·)‖Lp′ (Sd ;h2

κ ′ )≤ cN− 2λκ+2

p′ .

Thus, using (3.2.1), (3.2.3), (7.1.1), and the assumption (7.1.2) with ε = 1N , we obtain

IN := N2λκ+2

×∣∣∣∫

Sd

[∫Sd

( ∞

∑n=0

m(N−1n)Pκ ′n (x,y)

)fN(Ny)gN(Nx)h2

κ ′(x)h2κ ′(y)dσ(x)

)dσ(y)

∣∣∣≤ cA, (7.3.3)

where Pκ ′n (x,y) = n+λκ+1/2

λκ+1/2 Vκ ′ [Cλκ+1/2n (〈x, ·〉)](y). Setting

HN(x,y) = N−2λκ−2∞

∑n=0

m(N−1n)Pκ ′n

(ψ(

xN),ψ(

yN)),

and invoking (7.3.2) and Lemma 7.2.2, we obtain

IN =∣∣∣∫

Rd

[∫Rd

HN(x,y) f (y)g(x)h2κ(x)h

2κ(y)

(sin(‖x‖/N)

‖x‖/N

)2λκ+1

(7.3.4)

×(

sin(‖y‖/N)

‖y‖/N

)2λκ+1

dx]dy

∣∣∣.On the other hand, setting

bN(ρ,x,y) = N−2λκ−2∞

∑n=0

m(nN)Pκ ′

n

(ψ(

xN),ψ(

yN))(∫ n+1

N

nN

t2λκ+1 dt

)−1

χ[ nN , n+1

N )(ρ),

we haveHN(x,y) =

∫ ∞

0bN(ρ,x,y)ρ2λκ+1 dρ.

Hence, by (7.3.4),

IN =∣∣∣∫

Rd

[∫Rd

(∫ ∞

0bN(ρ,x,y)ρ2λκ+1 dρ

)f (y)g(x)h2

κ(x)h2κ(y) (7.3.5)

×(

sin(‖x‖/N)

‖x‖/N

)2λκ+1 (sin(‖y‖/N)

‖y‖/N

)2λκ+1

dx]dy

∣∣∣.The key ingredient in our proof is to show that limN→∞ IN = cI, where c is a constant

depending only on d and κ . In fact, once this is proven, then the desired estimate (7.3.1)will follow immediately from (7.3.3).

To show limN→∞ IN = cI, we make the following two assertions:

7.3. Proof of Theorem 7.1.1: part II 103

Assertion 1. For any N > 0 and x,y ∈ Rd ,

|bN(ρ,x,y)| ≤ ce−c2ρ ,

where c is independent of x, y and N.

Assertion 2. For any fixed x,y ∈ Rd and ρ > 0,

limN→∞

bN(ρ,x,y) = cm(ρ)Vκ

[Jλκ

(ρu(x,y, ·))(

ρu(x,y, ·))λκ

](y), (7.3.6)

where u(x,y,ξ ) =√‖x‖2 +‖y‖2 −2〈x,ξ 〉, and c is a constant depending only on d and

κ .

For the moment, we take the above two assertions for granted, and proceed withthe proof of Theorem 7.1.1. By Assertion 1 and Holder’s inequality, we can apply thedominated convergence theorem to the integrals in (7.3.5), and obtain

limN→∞

IN =∣∣∣∫

Rd

[∫Rd

(∫ ∞

0lim

N→∞bN(ρ,x,y)ρ2λκ+1 dρ

)f (y)g(x)h2

κ(x)h2κ(y)dx

]dy

∣∣∣,which, using Assertion 2, equals

c∣∣∣∫

Rd

[∫Rd

(∫ ∞

0m(ρ)Vκ

[Jλκ

(ρu(x,y, ·))(

ρu(x,y, ·))λκ

](y)ρ2λκ+1 dρ

)f (y)g(x)h2

κ(x)h2κ(y)dx

]dy

∣∣∣= c

∣∣∣∫Rd

(∫Rd

K(x,y) f (y)g(x)h2κ(x)h

2κ(y)dx

)dy

∣∣∣ = cI,

where the second step uses (7.2.11). Thus, we have shown the desired relation limN→∞ IN =cI, assuming Assertions 1 and 2.

Now we return to the proofs of Assertions 1 and 2. We start with Assertion 1.Assume that n

N ≤ ρ < n+1N for some n ∈ Z+. Then |m( n

N )| ≤ c1e−c2nN ≤ ce−c2ρ , and∫ n+1

NnN

t2λκ+1 dt ≥ cN−1ρ2λκ+1. Hence,

|bN(ρ,x,y)|= N−2λκ−2∣∣∣m(

nN)Pκ ′

n

(ψ(

xN),ψ(

yN))∣∣∣(∫ n+1

N

nN

t2λκ+1 dt

)−1

≤ cN−2λκ−1ρ−2λκ−1e−c2ρ n+λκ +1/2λκ +1/2

∣∣∣Vκ ′[Cλκ+1/2

n (〈ψ(xN), ·〉)

](ψ(

yN))∣∣∣

≤ c(Nρ)−2λκ−1e−c2ρ n2λκ+1 ≤ ce−c2ρ ,

where we used (3.2.3) in the second step, and the positivity of Vκ and the estimate|Cλκ+1/2

n (t)| ≤ cn2λκ in the third step. This proves Assertion 1.


Next, we show Assertion 2. A straightforward calculation shows that for nN ≤ ρ ≤

n+1N and ρ > 0, (∫ n+1

N

nN

t2λκ+1 dt

)−1

=N

ρ2λκ+1 (1+oρ(1)), as N → ∞.

This implies that for nN ≤ ρ ≤ n+1

N and ρ > 0,

bN(ρ,x,y) = m(ρ)n2λκ+1

(Nρ)2λκ+1 n−2λκ−1Pκ ′n

(ψ

( xN

),ψ

( yN

))(1+oρ(1))

= cm(ρ)n−2λκVκ ′[Cλκ+1/2

n

(〈ψ

( xN

), ·〉

)]( y‖y‖ sin

‖y‖N

,cos‖y‖N

)+oρ(1),

where the continuity of m is used in the first step, and n−2λκ−1∣∣∣Pκ ′

n(ψ( x

N ),ψ( yN )

)∣∣∣ ≤ c

and limN→∞

n2λκ+1

(Nρ)2λκ+1 = 1 in the last step. Thus, using Lemma 7.2.1 and (7.2.9), we obtain

bN(ρ,x,y) = cm(ρ)n−2λκ∫Rd

Cλκ+1/2n

(1‖x‖ sin

‖x‖N

d

∑j=1

x jξ j + cos‖y‖N

cos‖x‖N

)×dμκ

y‖y‖ sin ‖y‖

N(ξ )+oρ(1)

= cm(ρ)n−2λκ∫‖ξ‖≤‖y‖

Cλκ+1/2n (cosθN(x,y,ξ ))dμκ

y (ξ )+oρ(1), (7.3.7)

where θN(x,y,ξ ) ∈ [0,π] satisfies

cosθN(x,y,ξ ) =

(1

‖x‖‖y‖d

∑j=1

x jξ j

)sin

‖x‖N

sin‖y‖N

+ cos‖x‖N

cos‖y‖N

.

Since

cosθN(x,y,ξ ) = 1− 12N2

(‖x‖2 +‖y‖2 −2

d

∑j=1

x jξ j

)+O‖x‖,‖y‖(N−4)

= 1− 12N2 u(x,y,ξ )2 +O‖x‖,‖y‖(N−4),

it follows that

θN(x,y,ξ ) = 2arcsin(

12N

√u(x,y,ξ )2 +O‖x‖,‖y‖(N−2)

)=

1N

√u(x,y,ξ )2 +O‖x‖,‖y‖(N−2)+O‖x‖,‖y‖(N−2)

=ρu(x,y,ξ )+o‖x‖,‖y‖,ρ(1)

n,

7.4. Proof of Theorem 7.1.1: part III 105

where the last step uses the uniform continuity of the function t ∈ [0,M] → √t for any

M > 0, and the relation limN→∞n

Nρ = 1.Thus, by (7.3.7) and (7.2.12), we have

limN→∞

bN(ρ,x,y)

= cm(ρ) limN→∞

∫‖ξ‖≤‖y‖

n−2λκCλκ+1/2n

(cos

ρu(x,y,ξ )+ox,y,ρ(1)n

)dμκ

y (ξ )

= cm(ρ)∫‖ξ‖≤‖y‖

(ρu(x,y,ξ ))−λκ Jλκ (ρu(x,y,ξ ))dμκy (ξ )

= cm(ρ)Vκ

[(ρu(x,y, ·))−λκ Jλκ (ρu(x,y, ·))

](y),

where we used the fact that ‖Cλκ+1/2n ‖∞ ≤ cn2λκ , the bounded convergence theorem, and

(7.2.12) in the last step. This proves Assertion 2.Thus, we have shown the theorem under the additional assumption that |m(t)| ≤

c1e−c2t .

7.4 Proof of Theorem 7.1.1: part III

In this section, we shall show how to prove Theorem 7.1.1 without the additional as-sumption that |m(t)| ≤ c1e−c2t . To this end, let mδ (t) = m(t)e−δ t for δ > 0, and defineTmδ : L2(Rd ,h2

κ)→ L2(Rd ;h2κ) by

Fκ(Tmδ f )(ξ ) = mδ (ξ )Fκ f (ξ ), f ∈ L2(Rd ;h2κ).

By Lemma 3.4.5, for a given ε > 0, f �→ ∑∞n=0 e−nε projκn f is a positive operator on

Lp(Sd ;h2κ) that satisfies

supε>0

∥∥∥ ∞

∑n=0

e−nε projκn f∥∥∥

Lp(Sd ;h2κ )≤ ‖ f‖Lp(Sd ;h2

κ ).

Thus, applying Theorem 7.1.1 for the already proven case, we have

supδ>0

∥∥∥Tmδ f∥∥∥

Lp(Rd ;h2κ )≤ cA‖ f‖Lp(Rd ;h2

κ ). (7.4.1)

On the other hand, in view of the definition we can decompose the operator Tmδ as

Tmδ f = Pδ (T f ), (7.4.2)

where Fκ(T f )(ξ ) = m(‖ξ‖)Fκ f (ξ ) and Fκ(Pδ f )(ξ ) = e−δ‖ξ‖Fκ f (ξ ). The functionPδ f is called the Poisson integral of f , and it can be expressed as a generalized convolu-tion

Pδ f (x) := ( f ∗κ Pδ )(x)

with

Pδ (x) := 2γκ+d2

Γ(γκ +d+1

2 )√π

δ

(δ 2 +‖x‖2)γκ+d+1

2.

By Lemma 3.4.7, it follows that

limδ→0+

Pδ f (x) = f (x), a.e. x ∈ Rd

for any f ∈ Lq(Rd ;h2κ) with 1 ≤ q < ∞. Since m is bounded, T f ∈ L2(Rd ;h2

κ) for f ∈L2(Rd ;h2

κ). Thus, for any f ∈ S , using (7.4.2),

limδ→0+

Tmδ f (x) = limδ→0+

Pδ (T f )(x) = T f (x), a.e. x ∈ Rd , (7.4.3)

which combined with (7.4.1) and the Fatou theorem implies the desired estimate

‖T f‖Lp(Rd ;h2κ )≤ cA‖ f‖Lp(Rd ;h2

κ ).

This completes the proof of the theorem. �

7.5 Hormander’s multiplier theorem and the

Littlewood–Paley inequality

As a first application of Theorem 7.1.1, we shall prove the following Hormander typemultiplier theorem for the Dunkl transform:

Theorem 7.5.1. Let m : (0,∞) → R be a bounded function satisfying ‖m‖∞ ≤ A andHormander’s condition

1R

∫ 2R

R|m(r)(t)|dt ≤ AR−r, for all R > 0, (7.5.1)

where r is the smallest integer greater than or equal to λκ +3/2. Let Tm be the operatoron L2(Rd ;h2

κ) defined by

Fκ(Tm f )(ξ ) = m(‖ξ‖)Fκ f (ξ ), ξ ∈ Rd .

Then‖Tm f‖κ,p ≤CpA‖ f‖κ,p

for all 1 0 and �= 0,1, . . .. Then

|�rμ�|= εr∣∣∣∫

[0,1]rm(r)(εt1 + · · ·+ εtr + ε�

)dt1 · · ·dtr

∣∣∣≤

∫[0,ε]r

|m(r)(t1 + · · ·+ tr + ε�)|dt1 · · ·dtr ≤ εr−1

∫ ε(r+�)

ε�|m(r)(t)|dt.

7.5. Hormander’s multiplier theorem and the Littlewood–Paley inequality 107

This implies that, for 2 j ≥ r,

2 j(r−1)2 j+1

∑l=2 j

|Δrμl | ≤ 2 j(r−1)εr−12 j+1

∑l=2 j

∫ ε(r+�)

ε�|m(r)(t)|dt

≤ (r−1)2 j(r−1)εr−1∫ ε(2 j+1+r)

2 jε|m(r)(t)|dt

≤ 2 j(r−1)(r−1)εr−1∫ 2 j+2ε

2 jε|m(r)(t)|dt ≤ crA,

where the last step uses (7.5.1). On the other hand, however, for 2 j ≤ r, we have

2 j(r−1)2 j+1

∑l=2 j

|Δrμl | ≤ cr maxj

|μ j| ≤ crA.

Thus, using Theorem 4.4.2, we deduce

supε>0

∥∥∥ ∞

∑n=0

m(εn)projκ′

n f∥∥∥

Lp(Sd ;h2κ ′ )

≤ c‖ f‖Lp(Sd ;h2κ ′ ).

The desired conclusion then follows by Theorem 7.1.1. �Remark 7.5.2. Hormander’s condition is normally stated in the form(

1R

∫ 2R

R|m(r)(t)|2 dt

) 12

≤ AR−r, for all R > 0; (7.5.2)

see, for instance, [31, Theorem 5.2.7]. Clearly, the condition (7.5.1) in Theorem 7.5.1is weaker than (7.5.2). On the other hand, however, Theorem 7.5.1 is applicable only toradial multipliers m(‖ · ‖).Corollary 7.5.3. Let Φ be an even C∞-function that is supported in the set {x ∈R : 9

10 ≤|x| ≤ 21

10} and satisfies either

∑j∈Z

Φ(2− jξ ) = 1, ξ ∈ R\{0},

or∑j∈Z

|Φ(2− jξ )|2 = 1, ξ ∈ R\{0}.

Let � j be an operator defined by

Fκ(� j f )(ξ ) = Φ(2− j‖ξ‖)Fκ f (ξ ), ξ ∈ Rd .

Then we have‖ f‖κ,p ∼κ,p ‖(∑

j∈Z|� j f |2) 1

2 ‖κ,p

for all f ∈ Lp(Rd ;h2κ) and 1 < p < ∞.

Proof. Corollary 7.5.3 follows directly from Theorem 7.5.1. Since the proof runs alongthe same line as that of Theorem 4.5.2, we omit the details. �


7.6 Convergence of the Bochner–Riesz means

Recall that the Bochner–Riesz means of order δ >−1 for the Dunkl transform SδR f (x)≡

SδR(h

2κ ; f )(x) are defined by (6.4.4). According to Theorem 6.4.7, if δ > λκ +

12 := d−1

2 +γκ and 1 ≤ p ≤ ∞, then

supR>0

‖SδR(h

2κ ; f )‖κ,p ≤ c‖ f‖κ,p. (7.6.1)

Our next result concerns the critical indices for the validity of (7.6.1) in the case ofG = Zd

2:

Theorem 7.6.1. Suppose that G = Zd2 , f ∈ Lp(Rd ;h2

κ), 1 ≤ p ≤ ∞, and | 1p − 1

2 | ≥ 12λκ+3 .

Then (7.6.1) holds if and only if

δ > δκ(p) := max{(2λκ +2)

∣∣∣ 1p− 1

2

∣∣∣− 12,0

}. (7.6.2)

Proof. We start with the proof of the sufficiency. Assume that κ := (κ1, . . . ,κd) andhκ(x) := ∏d

j=1 |x j|κ j . Let κ ′ = (κ,0) and hκ ′(x,xd+1) = hκ(x) for x ∈ Rd and xd+1 ∈ R.Set m(t) = (1− t2)δ

+. By the equivalence of the Riesz and the Cesaro summability meth-ods of order δ ≥ 0 (see [30]), we deduce from Theorem 3.3.8 that

supε>0

∥∥∥ ∞

∑n=0

m(εn)projκ′

n f∥∥∥

Lp(Sd ;h2κ ′ )

≤ c‖ f‖Lp(Sd ;h2κ ′ )

whenever | 1p − 1

2 | ≥ 12σκ ′+2 and δ > δκ ′(p), where σκ ′ = λκ + 1

2 and δκ ′(p) = δκ(p).Thus, invoking Theorem 7.1.1, we conclude that for δ > δκ(p),

‖Sδ1 (h

2κ ; f )‖κ,p ≤ c‖ f‖κ,p.

The estimate (7.6.1) then follows by dilation. This proves the sufficiency.The necessity part of the theorem follows from the corresponding result for the

Hankel transform. To see this, let f (x) = f0(‖x‖) be a radial function in Lp(Rd ,h2κ).

Using (6.4.4) and Theorem 6.2.11 (vii), we have

SδR(h

2κ ; f )(x) =

∫ R

0

(1− r2

R2

)δ

Hλκ f0(r)r2λκ+1[∫

Sd−1Eκ(ix,ry′)h2

κ(y′)dσ(y′)

]dr.

However, by [60, Proposition 2.3] applied to n = 0 and g = 1, we have∫Sd−1

Eκ(ix,ry′)h2κ(y

′)dσ(y′) = c(

r‖x‖2

)−λκ

Jλκ (r‖x‖).

It follows that

SδR(h

2κ ; f )(x) = c

∫ R

0

(1− r2

R2

)δ

Hλκ f0(r)(

r‖x‖2

)−λκ

Jλκ (r‖x‖)r2λκ+1 dr

= cSδR f0(‖x‖),

7.7. Notes and further results 109

where SδR denotes the Bockner–Riesz mean of order δ for the Hankel transform Hλκ .

However, it is known (see [66]) that SδR, 0 < δ < λκ +

12 , is bounded on Lp((0,∞), t2λκ+1)

if and only if2λκ +2

λκ +δ +3/2< p <

2λκ +2λκ −δ +1/2

. (7.6.3)

Thus, to complete the Proof of the necessity part of the theorem, by (7.6.3), we just needto observe that if f (x) = f0(‖x‖) is a radial function in Lp(Rd ;h2

κ), Then

‖ f‖κ,p = c‖ f0‖Lp(R;|x|2λκ+1). �


Most of the results in this chapter were proved in [10]. In the case of ordinary Fouriertransform and spherical harmonics, Theorem 7.1.1 is due to Bonami and Clerc [3, Theo-rem 1.1].

In the unweighted case, for the classical Fourier transform, Theorem 7.6.1 is wellknown, and in fact, it is a consequence of the following Tomas–Stein restriction theorem(see, for instance, [31, Section 10.4]):

‖ f‖L2(Sd−1) ≤ cp‖ f‖Lp(Rd), 1 ≤ p ≤ 2d +2d +3

, (7.7.1)

where f denotes the usual Fourier transform of f . In the weighted case, while estimatessimilar to (7.7.1) can be proved for the Dunkl transform Fκ f (see [38, Theorem 4.1]),they do not seem to be enough for the proof of Theorem 7.6.1. A similar fact was indicatedin [14] for the case of the Cesaro means for h-harmonic expansions on the unit sphere,where global estimates for the projection operators have to be replaced with more delicatelocal estimates, which are significantly more difficult to prove.

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Index

h-Laplace–Beltrami operator, 16fractional power, 54

h-harmonic expansionLittlewood–Paley inequality, 47multiplier theorem, 45

h-harmonics, 15

Cesaro means, 24convolution, 6

ordinary, 6weighted on Rd , 74weighted on the sphere, 23

diffusion semi-group, 27Dunkl Laplacian, 10Dunkl operators, 10Dunkl transform, 65

multiplier theorem, 106Bochner–Riesz means, 86inversion formula, 75maximal function, 87Paley–Wiener theorem, 70Plancherel theorem, 68summability, 82

Fourier transform, 5Funk–Hecke formula, 22

Gegenbauer polynomials, 4generalized translation operator

for G = Zd2, 80

on radial functions, 77on the sphere, 31weight on Rd , 72

Hankel transform, 6, 76

Heat kernel transform, 85homogeneous polynomials, 1Hopf–Dunford–Schwartz theorem, 28

intertwining operator, 12explicit formula for Zd

2, 12positivity, 13

Laplace operator, 2Laplace–Beltrami operator, 3Littlewood–Paley inequality

weighted on Rd , 107on the sphere, 55

Littlewood–Paley theoryg-functions, 38Littlewood–Paley–Stein function, 37

maximal functionon Rd , 88on the sphere, 27

multiplicity function, 8

Poisson integral, 5weighted, 28weighted on Rd , 85

projection operator, 21

reflection, 8reflection groups, 8root system, 8

sphere Sd−1, 1spherical h-harmonics, 15

Cesaro summability, 25expansion, 21Lebesgue constant, 26


in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-08 -3, , Advanced Courses


87

117

118 Index

projection, 21spherical convolution, 4spherical harmonics, 2

expansions, 3eigenfunctions of Δ0, 3orthogonality, 2reproducing kernel, 3space H d

n , 2surface area ωd , 2

transference theorem, 96

weight functioninvariant under Zd

2, 7reflection invariant, 8

weighted K-functionaldirect and inverse theorem, 56on the sphere, 55realization, 55

weighted exponentials, 13weighted moduli of smoothness

direct and inverse theorem, 54on the sphere, 53sharp Jackson inequality, 59sharp Marchaud inequality, 56

weighted Sobolev space, 54

Young’s inequality, 23

analysis on h-harmonics and dunkl transforms

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