Analysis on h-Harmonics and Dunkl Transforms

Advanced Courses in Mathematics CRM Barcelona

Feng Dai Yuan Xu

Analysis on h -Harmonics and Dunkl Transforms

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Carles Casacuberta

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

Feng Dai • Yuan Xu

Analysis on h-Harmonics and Dunkl Transforms Editor for this volume: Sergey Tikhonov, ICREA and CRM, Barcelona

Feng Dai Department of Mathematics and Statistical Sciences University of Alberta Edmonton, AB, Canada

Yuan Xu Department of Mathematics University of Oregon Eugene, OR, USA

ISSN 2297-0304 ISSN 2297-0312 (electronic) Advanced Courses in Mathematics - CRM Barcelona ISBN 978-3-0348-0886-6 ISBN 978-3-0348-0887-3 (eBook) DOI 10.1007/978-3-0348-0887-3 Library of Congress Control Number: 2014959869 Mathematics Subject Classification (2010): Primary: 41A10, 42B15; Secondary: 42B25, 42B08, 41A17 Springer Basel Heidelberg New York Dordrecht London © Springer Basel 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

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Contents Preface

vii

1

Introduction: Spherical Harmonics and Fourier Transform 1.1 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Dunkl Operators Associated with Reflection Groups 2.1 Weight functions invariant under a reflection group 2.2 Dunkl operators . . . . . . . . . . . . . . . . . . 2.3 Intertwining operator . . . . . . . . . . . . . . . . 2.4 Notes and further results . . . . . . . . . . . . . .

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h-Harmonics and Analysis on the Sphere 3.1 Dunkl h-harmonics . . . . . . . . . . . . . . . . 3.2 Projection operator and intertwining operator . . 3.3 Convolution operators and orthogonal expansions 3.4 Maximal functions . . . . . . . . . . . . . . . . 3.5 Convolution and maximal function . . . . . . . . 3.6 Notes and further results . . . . . . . . . . . . .

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Littlewood–Paley Theory and the Multiplier Theorem 4.1 Vector-valued inequalities for self-adjoint operators 4.2 The Littlewood–Paley–Stein function . . . . . . . 4.3 The Littlewood–Paley theory on the sphere . . . . 4.3.1 A crucial lemma . . . . . . . . . . . . . . 4.3.2 Proof of Theorem 4.3.3 . . . . . . . . . . . 4.4 The Marcinkiewicz type multiplier theorem . . . . 4.5 A Littlewood–Paley inequality . . . . . . . . . . . 4.6 Notes and further results . . . . . . . . . . . . . .

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1 1 5 7 7 10 12 14

35 35 37 39 40 42 45 47 50

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vi 5

6

7

Contents Sharp Jackson and Sharp Marchaud Inequalities 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 5.2 Moduli of smoothness and best approximation . . . 5.3 Weighted Sobolev spaces and K-functionals . . . . 5.4 The sharp Marchaud inequality . . . . . . . . . . . 5.5 The sharp Jackson inequality . . . . . . . . . . . . 5.6 Optimality of the power in the Marchaud inequality 5.7 Notes and further results . . . . . . . . . . . . . .

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51 51 52 54 56 59 61 62

Dunkl Transform 6.1 Dunkl transform: L2 theory . . . . . . . . . . . . . . 6.2 Dunkl transform: L1 theory . . . . . . . . . . . . . . 6.3 Generalized translation operator . . . . . . . . . . . 6.3.1 Translation operator on radial functions . . . 6.3.2 Translation operator for G = Zd2 . . . . . . . 6.4 Generalized convolution and summability . . . . . . 6.4.1 Convolution with radial functions . . . . . . 6.4.2 Summability of the inverse Dunkl transform . 6.4.3 Convolution operator for Zd2 . . . . . . . . . 6.5 Maximal function . . . . . . . . . . . . . . . . . . . 6.5.1 Boundedness of maximal function . . . . . . 6.5.2 Convolution versus maximal function for Zd2 6.6 Notes and further results . . . . . . . . . . . . . . .

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65 65 72 76 77 80 82 82 84 86 87 87 90 94

Multiplier Theorems for the Dunkl Transform 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Proof of Theorem 7.1.1: part I . . . . . . . . . . . . . . . . . . . . . 7.3 Proof of Theorem 7.1.1: part II . . . . . . . . . . . . . . . . . . . . . 7.4 Proof of Theorem 7.1.1: part III . . . . . . . . . . . . . . . . . . . . 7.5 H¨ormander’s multiplier theorem and the Littlewood–Paley inequality . 7.6 Convergence of the Bochner–Riesz means . . . . . . . . . . . . . . . 7.7 Notes and further results . . . . . . . . . . . . . . . . . . . . . . . .

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95 95 96 101 105 106 108 109

Bibliography

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Index

117

Preface These lecture notes were written as an introduction to Dunkl harmonics and Dunkl transforms, which are extensions of ordinary spherical harmonics and Fourier transforms with the usual Lebesgue measure replaced by weighted measures. The theory was initiated by C. Dunkl and subsequently developed by many authors in the past two decades. In this theory, the role of orthogonal groups, which provide the underline 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 commuting first order differential and difference operators, and the Lebesgue measure is replaced by 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 the case of h-harmonics, which are the analogues of ordinary spherical harmonics. There are still many problems to be solved and the theory is still at its infancy, especially in the case of 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 on reflection groups are kept to a bare minimum. In fact, even assuming the group is Zd2 , which requires essentially no prior knowledge of reflection groups, a reader can still gain access to the essence of the theory and to many highly non-trivial results, where the weight function hκ is simply d

hκ (x) = ∏ |xi |κi ,

κi ≥ 0, 1 ≤ i ≤ d,

i=1

the surface measure dσ on the sphere Sd−1 is replaced by h2κ dσ , and the Lebesgue measure dx on Rd is replaced by h2κ dx. To motivate the weighted results, we give a brief recount of basics of ordinary spherical harmonics and the Fourier transform in the first chapter, which can be skipped altogether. The Dunkl operators and the intertwining operator between partial derivatives and the Dunkl operators, are introduced and discussed in the second chapter. The intertwining operator plays a key role in the theory as it appears in the concise formula for the reproducing kernel of the h-spherical harmonics and in the definition of the Dunkl transform. The next three chapters are devoted to analysis on the sphere. The third chapter is an 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, and is 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 the embedding theory of function spaces. The final two chapters are devoted to the Dunkl transform. The sixth chapter is an introduction to Dunkl transforms, where the basic results are developed in detail. The Littlewood–Paley theory and a multiplier theorem are established in the seventh chapter, using a transference between h-harmonic expansions on the sphere and the Dunkl transform in Rd . The topics reflect the authors’ choice. There are many results for Dunkl transforms on the real line (where the measure is |x|κ dx) that we did not discuss, since the setting on the real line is closely related to the Hankel transforms and often cannot even be extended to the Zd2 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 intertwining operator, the case Zd2 has seen far more, and deeper, results, especially in the case of analysis on the sphere such as those for Ces`aro means. We chose the Littlewood–Paley theory and the multiplier theorem, as this part is relatively complete and the results are related in the two settings, the sphere and the Euclidean space. These lecture notes were written for the advanced courses in the program Approximation Theory and Fourier Analysis at the Centre de Recerca Matem`atica, Barcelona. We are grateful to the CRM for the warm hospitality during our two months stay, to the participants in our lectures, and thank especially the organizer of the program, Sergey Tikhonov from CRM, for his great help. We gratefully acknowledge the support received from NSERC Canada under grant RGPIN 311678-2010 (F.D.), from National Science Foundation under grant DMS-1106113 (Y.X.), and from the Simons Foundation (# 209057 to Y. X.). Edmonton, Alberta, and Eugene, Oregon September, 2014

Feng Dai 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 spherical harmonics and the Fourier transform, in which the underlying rotation group is replaced by a finite reflection group. This chapter serves as an introduction, in which we briefly recall classical results on the spherical harmonics and the Fourier transform. Since all results are classical, no proof will be given.

1.1

Spherical harmonics

First we introduce several notations that will be used throughout these lecture notes. d is denoted by For x ∈ Rd , we write x = (x1 , . . . , xd ). The inner product  of x, y ∈ R d d−1 := {x ∈ Rd : x, y := ∑i=1 xi yi and the norm of x is denoted by x := x, x. Let S d x = 1} denote the unit sphere of R , and let N0 denote the set of nonnegative integers. α For α = (α1 , . . . , αd ) ∈ Nd0 , a monomial xα is a product xα = x1α1 · · · xd d , which has degree |α| := α1 + · · · + αd . A homogeneous polynomial P of degree n is a linear combination of monomials of 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 Pnd 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     n+d −1 n+d d d dim Pn = and dim Πn = . n n

© Springer Basel 2015 F. Dai, Y. Xu, Analysis on h-Harmonics and Dunkl Transforms, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0887-3_1

1

2

Chapter 1. Introduction: Spherical Harmonics and Fourier Transform

A harmonic polynomial is a homogeneous polynomial that satisfies the Laplace equation. Let ∂i be the partial derivative in the i-th variable and Δ the Laplacian operator Δ = ∂12 + · · · + ∂d2 . Definition 1.1.1. For n = 0, 1, 2, . . ., let Hnd be the linear space of real harmonic polynomials, homogeneous of degree n, on Rd , that is,   Hnd = P ∈ Pnd : ΔP = 0 . Spherical harmonics are the restrictions of elements in Hnd on the unit sphere Sd−1 . If Y ∈ Hnd , then Y (x) = xnY (x ), where x = xx and x ∈ Sd−1 . We will also call Hnd the space of spherical harmonics. Spherical harmonics of different degrees are orthogonal with respect to the inner product  1 f (x)g(x)dσ (x),  f , gSd−1 := ωd Sd−1 where dσ is the surface area measure on Sd−1 , and ωd denotes the surface area of Sd−1 , ωd :=

 Sd−1

dσ =

2π d/2 . Γ(d/2)

Theorem 1.1.2. If Yn ∈ Hnd , Ym ∈ Hmd , and n = m, then Yn ,Ym Sd−1 = 0. For n = 0, 1, 2, . . ., Pnd admits the decomposition Pnd =



d x2 j Hn−2 j.

0≤ j≤n/2

In other words, for each P ∈ Pnd , there is a unique decomposition P(x) =

∑

x2 j Pn−2 j (x)

with

d Pn−2 j ∈ Hn−2 j.

0≤ j≤n/2

From the orthogonal decomposition, one immediately deduces the following: Corollary 1.1.3. For n = 0, 1, 2, . . . d = dim Hnd = dim Pnd − dim Pn−2

    n+d −1 n+d −3 − , n n−2

d = 0 for n = 0, 1. with the convention dim Pn−2

In the spherical-polar coordinates x = rξ , r > 0, ξ ∈ Sd−1 , the Laplace operator is written 1 ∂2 d −1 ∂ + Δ0 , Δ= 2 + ∂r r ∂ r r2

1.1. Spherical harmonics

3

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

d−1

∂2

d−1 d−1

i=1

i

i=1 j=1

∂2

d−1

∂

∑ ∂ ξ 2 − ∑ ∑ ξi ξ j ∂ ξi ∂ ξ j − (d − 1) ∑ ξi ∂ ξi . i=1

Using this expression of Δ, ΔY = 0 for Y being a homogeneous polynomial leads to the following result. Theorem 1.1.4. The spherical harmonics are eigenfunctions of Δ0 : Δ0Y (ξ ) = −n(n + d − 2)Y (ξ ),

∀Y ∈ Hnd ,

ξ ∈ Sd−1 .

In spherical coordinates, an orthogonal basis of Hnd can be given explicitly. Let {Yα } be an orthonormal basis of Hnd , that is, Yα ,Yβ Sd−1 = δα,β . A function f in L2 (Sd−1 ) can be expanded in a Fourier series f (x) = ∑ cα Yα (x),

cα =

where

1 ωd

 Sd−1

f (y)Yα (y)dσ (y).

It is often more convenient to consider the orthogonal expansions in terms of the spaces Hnd . Collecting terms of spherical harmonics of the same degree, the Fourier series takes the form f (x) =

∞

∑ projn f (x),

n=0

where projn f is the orthogonal projection of f onto the space Hnd and satisfies 1 projn f (x) = ωd



f (y)Zn (x, y)dσ (y), x ∈ Sd−1 ,

Sd−1

in which Zn (·, ·), called the reproducing kernel of Hnd , is given by Zn (x, y) =

dim Hnd

∑

Yk (x)Yk (y),

x, y ∈ Sd−1 .

k=1

Since the space of spherical polynomials is dense in C(Sd−1 ) by the Weierstrass theorem and, as a consequence, dense in L2 (Sd−1 ), the following theorem is a standard Hilbert 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 ) =

∞

∑ Hnd ,

n=0

f=

∞

∑ projn f

n=0

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 =

∞

∑  projn f 22 .

n=0

4

Chapter 1. Introduction: Spherical Harmonics and Fourier Transform

Much of the analysis on the sphere beyond the L2 setting depends on the knowledge of the kernel Zn . It is known that this kernel is uniquely determined by its reproducing property  1 Zn (x, y)p(y)dσ (y) = p(x), ∀p ∈ Hnd , x ∈ Sd−1 ωd Sd−1 and the requirement that Zn (x, ·) is an element of Hnd for each fixed x. In particular, Zn is independent of the particular choice of bases of Hnd . The space Hnd is invariant under the action of the orthogonal group O(d) and the surface measure is also invariant, so that the 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 distance d(x, y) = arccos x, y. Hence, Pn (x, y) = Fn (x, y), which is often called a zonal harmonic as it is harmonic and depends only on x, y. It turns out that the function Fn has a concise formula in terms of the Gegenbauer polynomial Cnλ of degree n, defined by  n 1−n  1 (λ )n 2n n −2, 2 λ x 2 F1 ; , (1.1.1) Cn (x) := 1 − n − λ x2 n! for λ > 0 and n ∈ N0 , where 2 F1 is the hypergeometric function. Theorem 1.1.6. For n ∈ N0 and x, y ∈ Sd−1 , d ≥ 3, Zn (x, y) =

n+λ λ Cn (x, y), λ

λ=

d −2 . 2

(1.1.2)

The Gegenbauer polynomials are also called ultra-spherical polynomials. They are orthogonal with respect to the weight function 1

wλ (t) := (1 − t 2 )λ − 2 ,

t ∈ [−1, 1].

Let cλ be the normalization constant of wλ , cλ = 1/  1

cλ

−1

Cnλ (x)Cmλ (x)wλ (x)dx =

1

−1 wλ (t)dt.

Then

(2λ )n λ δn,m . (n + λ ) n!

These classical polynomials have been extensively studied. For their essential properties, see [53]. In particular, there is a generating function ∞ n+λ λ 1 Cn (t)rn , = ∑ 2 λ +1 (1 − 2rt + r ) n=0 λ

0 ≤ r < 1,

λ > 0.

The concise formula for Zn (x, y) is one of the most useful ingredients for analysis on the sphere. For example, it leads to the following definition of a convolution on the sphere: 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 definition: 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 , t ∈ [−1, 1], (1 − 2rt + r2 )d/2

for 0 ≤ r < 1. The Poisson kernel and Poisson integral satisfy the following properties: n (1) for x, y ∈ Sd−1 , Pr (x, y) = ∑∞ n=0 Zn (x, y)r ; n (2) Pr f = ∑∞ n=0 r projn f ;

(3) Pr (x, y) ≥ 0 and ωd−1

 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 of mathematics. We outlined the essential structure for analysis on the sphere. For proofs and further results we refer to [16, 40, 52] and the discussion at the end of Chapter 1 in [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−ix,y dy, x ∈ Rd .

For f ∈ L1 (Rd ), f ∈ C0 (Rd ). The basic properties of the Fourier transform are summarized 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−ix,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 ).

6

Chapter 1. Introduction: Spherical Harmonics and Fourier Transform

(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) = H d−2 f0 (x) is again a radial function, 2 where Hα denotes the Hankel transform defined by Hα g(s) =

 ∞

1 Γ(α + 1)

g(r) 0

Jα (rs) 2α+1 r dr, (rs)α

in which Jα denotes the Bessel function of the first kind. The usual proof of (i) uses convolution defined by 1 f ∗ g(x) = (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). 2

normalize Φ so Let Φ be a nice function, say Φ(x) = e−x or e−x /2 , and let φ := Φ; −d that Rd φ (x) = 1. For ε > 0, define φε (x) := ε φ (x/ε). It is easy to see that

( f ∗ φε )(x) =

1 (2π)d/2



Rd

Φ(εy) f (y)eix,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 harmonics. Let Y ∈ Hnd . Define n+ d−2 2

φm (Y ; x) = Lm

(x2 )Y (x)e−x

2 /2

,

x ∈ Rd ,

where Lnα denotes the Laguerre polynomial of degree n with index α, normalized so that    ∞ n+α 1 α α α −x L (x)Lm (x)x e dx = δm,n . α Γ(α + 1) 0 n If {Yk,n : 1 ≤ k ≤ dim Hnd } denotes an orthonormal basis of Hnd , then it is easy to verify, using spherical polar coordinates and the orthogonality of Lnα , that {φm (Yk,n ; x) : m, n ≥ 0, 1 ≤ k ≤ dim Hnd } is an orthogonal basis of L2 (Rd ). Theorem 1.2.2. For m, n = 0, 1, 2, . . ., Y ∈ Hnd and x ∈ Rd , n+2m φm (Y ; x). φ m (Y )(x) = (−i)

There are many books on Fourier transforms. For the basics that we need here and the 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 harmonic analysis. Since our purpose is to study harmonic analysis in weighted spaces, we start with the definition of a family of weight functions invariant under a reflection group in Section 2.1. Dunkl operators are a family of commuting first-order differential and difference operators associated with a reflection group, and are introduced in Section 2.2. The intertwining operator between the Dunkl operators and ordinary derivatives is discussed in Section 2.3. For readers who are primarily interested in analysis, the prerequisites on reflection groups are reduced to a minimum. In fact, all essential ideas are presented in the case of G = Zd2 , 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 d

hκ (x) := ∏ |xi |κi ,

x ∈ Rd ,

(2.1.1)

i=1

for κi ≥ 0, 1 ≤ i ≤ d, and x = (x1 , . . . , xd ). Obviously, they are invariant under sign changes, that is, invariant under the group Zd2 . This is a special case of weight functions invariant under reflection groups. To define the general weight functions, we first need to recall basic facts on reflection groups. Readers who are not interested in reflection groups can skip to the end of the section and keep in mind the functions hκ in (2.1.1) and Zd2 in the 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

© Springer Basel 2015 F. Dai, Y. Xu, Analysis on h-Harmonics and Dunkl Transforms, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0887-3_2

7

8

Chapter 2. Dunkl Operators Associated with Reflection Groups

respect to the hyperplane v⊥ perpendicular to v, xσv := x − 2(x, v/v2 )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 called reduced. 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 roots R+ := {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 is exactly {σu : u ∈ R+ }. For a given root system R, a multiplicity function v → κv : R → R≥0 is 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 define a weight function hκ by hκ (x) :=

∏ |x, v|κv ,

x ∈ Rd .

(2.1.2)

v∈R+

Then hκ is invariant under the reflection group G generated by R. It is a homogeneous function of degree (2.1.3) γκ := ∑ κv . v∈R+

Zd2 , γκ

= |κ| = κ1 + · · · + κd . For hκ in (2.1.1) associated with 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 this group, so that the weight function is hκ (x) =

∏

|xi − x j |κ ,

κ ≥ 0,

x ∈ Rd .

(2.1.4)

1≤i< j≤d

This reflection group is of the type Ad−1 and it is the same as the symmetric, or permutation, 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 d

hκ (x) = ∏ |xi |κ0 i=1

∏

|xi2 − x2j |κ1 ,

κ0 , κ1 ≥ 0,

x ∈ Rd .

(2.1.5)

1≤i< j≤d

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 permutations of 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 L p spaces with these reflection 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 the surface area and ωdκ denote the normalization constant of hκ : ωd :=

 Sd−1

dσ =

2π d/2 Γ(d/2)

and ωdκ :=

 Sd−1

h2κ (y)dσ .

The closed form of ωdκ is known for every reflection group (cf. [26]). For 1 ≤ p ≤ ∞ we denote by L p (h2κ ) the space of functions defined on Sd−1 with finite norm 1 

1/p | f (y)| p h2κ (y)dσ (y) , 1 ≤ p < ∞,  f κ,p := κ ωd Sd−1 and for p = ∞ we assume that L∞ is replaced by C(Sd−1 ), the space of continuous functions on Sd−1 with the usual uniform norm  f ∞ . Let Pnd 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 /2

x ∈ Rd ,

dx,

and let ch denote the normalization constant  −1  1 2 −x2 /2 h (x)e dx . ch := (2π)d/2 Rd κ

(2.1.6)

The inner product with respect to dμ is closely related to the one with respect to dx when 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 ∈ Pnd , 

ch

Rd

p(x)q(x)h2κ (x)e−x

where λκ =

d−2 2

2 /2

dx = (2π)d/2 2n (λκ + 1)n

1 ωdκ

 Sd−1

p(ξ )q(ξ )h2κ (ξ )dσ (ξ ),

+ γκ .

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

2 g(x)h2κ (x)e−x /2 dx Rd

=

 ∞

r

2γκ +2n+d−1 −r2 /2

e

0

= 2n+λκ Γ(λκ + n + 1)



dr

 Sd−1

Sd−1

g(ξ )h2κ (ξ )dσ (ξ )

(2.1.7)

g(ξ )h2κ (ξ )dσ (ξ ),

from which the result follows.



It is worth noting that, setting n = 0 in (2.1.7), c−1 h =

κ 1 λκ κ λκ Γ(λκ + 1) ωd 2 Γ(λ + 1)ω = 2 . κ d Γ(d/2) ωd (2π)d/2

(2.1.8)

10

2.2

Chapter 2. Dunkl Operators Associated with Reflection Groups

Dunkl operators

The main ingredient of the theory of h-harmonics is a family of first-order differentialdifference operators, Di , called the Dunkl operators. Definition 2.2.1. Let R+ be a positive root system and κv be a multiplicity function from R+ to R≥0 . For 1 ≤ i ≤ d, define Di f (x) := ∂i f (x) +

∑

v∈R+

κv

f (x) − f (xσv ) v, ei , x, v

1 ≤ i ≤ d.

(2.2.1)

d , so that the D are indeed first-order differentialIt is easy to verify that Di Pnd ⊂ Pn−1 i d difference operators. In the case of Z2 , the Dunkl operators take on the form

f (x) − f (xσi ) xi

Di f (x) = ∂i f (x) + κi

(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: Di D j = D j Di ,

1 ≤ i, j ≤ d.

Proof. The proof of the general case is rather involved. We give the proof only for the case of Zd2 , for which a straightforward computation shows that, for i = j, κj κi (∂ j f (x) − ∂ j f (xσi )) + (∂i f (x) − ∂i f (xσ j )) xi xj κi κ j ( f (x) − f (xσ j ) − f (xσi ) + f (xσ j σi )), + xi x j

Di D j f (x) = ∂i ∂ j f (x) +

from which Di D j = D j Di 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 := D12 + · · · + Dd2 .

(2.2.3)

This is a second-order differential-difference operator and it reduces to the usual Laplacian Δ when all κi = 0. It has the following explicit formula related to the weight function hκ 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 hκ

and

Eh f := −2

∑

v∈R+

κv

f (x) − f (xσv ) v, x2

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

v2 ,

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 ) , xj

1 ≤ j ≤ d,

so that D j = ∂ j + κ j E j in the case of G = Zd2 . A straightforward computation shows that E 2j = 0 and κj κj D 2j = ∂ j2 f + κ j ∂ j E j + κ j E j ∂ j = ∂ j2 + 2 ∂ j − E j . xj xj Summing over j we obtain d κj κj ∂ j f − ∑ E j. j=1 x j j=1 x j d

Dh = Δ f + 2 ∑

In this case the sum over R+ means the sum over 1 ≤ j ≤ d, so that the sum over E j gives Eh and the differential part is Dh , which can be written in the stated expression in terms  of hκ by a simple verification. Later we will need to perform an integration by parts for the Dunkl operator, at least over the space of polynomials. For this to make sense, we consider the integral with respect to the measure dμ := h2κ (x)e−x

2 /2

x ∈ Rd .

dx,

Theorem 2.2.4. The adjoint Di∗ acting on L2 (Rd ; dμ) is given by Di∗ p(x) = xi p(x) − Di p(x),

p ∈ Πd .

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

  2 ∂i p(x) q(x)h2κ (x)e−x /2 dx = −



Rd

+



Rd

Rd

  2 p(x) ∂i q(x) h2κ (x)e−x /2 dx

 2 p(x)q(x) − 2hκ (x)∂i hκ (x) + h2κ (x)xi e−x /2 dx. 

For a fixed root v,  Rd

p(x) − p(xσv ) q(x)dμ = x, v =





Rd



Rd

p(x)q(x) p(xσv )q(x) dμ − dμ x, v x, v Rd  p(x)q(x) p(x)q(xσv ) dμ + dμ, x, v x, v Rd

where in the second integral we have replaced x by xσv which changes x, v to xσv , v = −x, v and leaves h2κ invariant. Note also that hκ (x)∂i hκ (x) =

∑

v∈R+

κv

vi 2 h (x). x, v κ

12

Chapter 2. Dunkl Operators Associated with Reflection Groups

Combining these ingredients, we obtain      p(x) xi q(x) − ∂i q(x) Di p(x)q(x)dμ = Rd

+

∑

Rd

   κv vi p(x) − 2q(x) + q(x) + q(xσv ) x, v dμ,



v∈R+

  where the term inside the square brackets is exactly p(x) xi q(x) − Di q(x) .

2.3



Intertwining operator

There is a linear operator that intertwines between the Dunkl operators and the partial derivatives, which plays an important role in harmonic analysis. Definition 2.3.1. Let Di be the Dunkl operators associated with a given positive root system and a multiplicity function κ. A linear operator Vκ on the space Πd of algebraic polynomials on Rd is called an intertwining operator if it satisfies DiVκ = Vκ ∂i ,

1 ≤ i ≤ d,

Vκ 1 = 1,

Vκ Pn ⊂ Pn , n ∈ N0 .

(2.3.1)

Strictly speaking, (2.3.1) is not the definition of Vκ , but rather the property that we most often use. Indeed, the existence of such a Vκ is by no means automatic. The operator Vκ was introduced in [26], where it was defined inductively on homogeneous polynomials. The definition is extended from homogeneous polynomials to the space A(Bd ) defined below. ∞ d For f ∈ Πd , let  · A := ∑∞ n=0  f n S , where f = ∑n=0 f n with f n ∈ Pn and  f S = d d d supx∈Sd−1 | f (x)|. Let A(B ) be the closure of Π in A-norm. Then A(B ) is a commutative Banach 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 proposition holds (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 our purpose, 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

d

f (x1t1 , . . . , xd td ) ∏(1 + ti )(1 − ti2 )κi −1 dti , i=1

(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, then the formula holds under the limit  1

lim cμ

μ→0

−1

f (t)(1 − t 2 )μ−1 dt =

f (1) + f (−1) . 2

(2.3.3)

Proof. The integrals are normalized so that Vκ 1 = 1. Recall that D j = ∂ j + κ j E j . Taking derivatives we get ∂ jVκ f (x) = cκ



d

[−1,1]d

∂ j f (x1t1 , . . . , xd td )t j ∏(1 + ti )(1 − ti2 )κi −1 dti . i=1

Taking into account the parity of the integrand, integration by parts shows that κ j E jVκ f (x) =

κj cκ xj

= cκ

 [−1,1]d

 [−1,1]d

f (x1t1 , . . . , xd td )2t j



∏(1 + ti )



i = j

2 κi −1 (1 − t ) dti ∏ i d

i=1

d

∂ j f (x1t1 , . . . , xd td )(1 − t j ) ∏(1 + ti )(1 − ti2 )κi −1 dti . i=1

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 dihedral group D4 , an explicit formula for Vκ is not known. In general, however, we have the following theorem of R¨osler, which, in particular, asserts that Vκ is nonnegative. Theorem 2.3.4. For each x ∈ Rd , there exists a unique probability measure μxκ on the Borel σ -algebra of Rd such that for all algebraic polynomials f on Rd , Vκ f (x) =

 Rd

f (ξ )dμxκ (ξ ).

(2.3.4)

Furthermore, the measures μxκ are compactly supported in the convex hull C(x) := conv{xg : g ∈ G} of the orbit of x under G, and satisfy κ κ (E) = μxκ (r−1 E), and μxg (E) = μxκ (Eg−1 ) μrx

(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 the measure μxκ depends on x. The most useful part of the result is that Vκ is a nonnegative operator. By means of (2.3.4), Vκ can be extended to an operator in the space C(Rd ) of continuous functions on Rd , which we will denote by Vκ again. Definition 2.3.6. For x, y ∈ Rd , define (x)  x,y 

E(x, y) := Vκ

e

14

Chapter 2. Dunkl Operators Associated with Reflection Groups

and, for n = 0, 1, 2, . . ., define En (x, y) :=

1 (x) V (x, yn ), n! κ

where the superscript x means that Vκ acts on the x variable. Definition 2.3.7. For p, q ∈ Pnd , define p, qD := 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, ·), pD = p(x),

∀p ∈ Pnd .

Proof. The first two properties follow from the inductive definition of Vκ , see [26]. For (x) the third property, we note that if p ∈ Pnd , then p(x) = (x, ∂ (y) n /n!)p(y). Applying Vκ (x) leads to Vκ p(x) = En (x, ∂ (y) )p(y). The left-hand side is independent of y so, applying (x) (y) V (y) to both sides, we get Vκ p(x) = En (x, D (y) )Vκ p(y). Thus, the desired identity holds d  for all Vκ p with p ∈ Pn , 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 inner products in Section 2.3 were studied in [26]. For a complete proof of the existence of the intertwining operator and its basic properties, see [29, Chapter 6]. The positivity of the intertwining operator was proved in [43]. The explicit formula for Vκ in the case of Zd2 was given in [69]. For the symmetric group S3 with hκ (x) = |(x1 − x2 )(x2 − x3 )(x3 − x1 )|κ for x ∈ S2 and the dihedral group I(4) with hκ (x) = |x1 x2 |κ0 |x12 − x22 |κ1 , some explicit integral formulas for Vκ are given in [28] and [70], respectively. But neither of them is in a strong enough form for carrying out the harmonic analysis that will be developed in latter chapters.

Chapter 3

h-Harmonics and Analysis on the Sphere Dunkl h-harmonics are defined as homogeneous polynomials satisfying the Dunkl Laplacian equation. They are defined and studied in Section 3.1. Projection operators and orthogonal expansions in spherical h-harmonics are studied in Section 3.2, which includes a concise expression for the reproducing kernel of the spherical h-harmonics. This expression is an analog of the zonal harmonics, which suggests a definition of a convolution operator, defined in Section 3.3 and it helps us to study various summability methods for spherical h-harmonic expansions. Maximal functions are introduced in Section 3.4 and proved to be of strong type (p, p) and weak type (1, 1). Finally, the relation between convolution 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 ∈ Pnd 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 Hnd (h2κ ) denote the linear space of h-harmonic polynomials of degree n. Elements of Hnd (h2κ ) are homogeneous polynomials so that they are uniquely determined by their restrictions to the unit sphere Sd−1 . The restrictions of h-harmonics to the sphere are spherical h-harmonics, analogues to spherical harmonics. We shall not distinguish between Ynh ∈ Hnd (h2κ ) and its restriction to the sphere. Let hκ be the weight function defined in (2.1.2). The inner product in L2 (h2κ , Sd−1 ) is denoted by  1 f (x)g(x)h2κ (x)dσ (x). (3.1.1)  f , gκ := κ ωd Sd−1

© Springer Basel 2015 F. Dai, Y. Xu, Analysis on h-Harmonics and Dunkl Transforms, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0887-3_3

15

16

Chapter 3. h-Harmonics and Analysis on the Sphere

Theorem 3.1.2. With respect to ·, ·κ , spherical h-harmonics of different degree are orthogonal. More precisely, if f ∈ Hnd (h2κ ), g ∈ Hmd (h2κ ) and n = m, then  f , gκ = 0. Proof. As in the classical proof for ordinary harmonics, this follows from an analog of Green’s formula stated for the differentiation part Dh of Δh :  Sd−1

∂f 2 gh dσ = ∂n κ

 Bd

(gDh f + ∇ f , ∇g)h2κ dx,

where ∂ f /∂ n denotes the normal derivative of f . Consequently, since homogeneous of degree n, and Δh f = 0, Δh g = 0, (n − m)

 Sd−1

f gh2κ dσ =

 Bd

=−

(gDh f − f Dh g)h2κ dx = −

 1

r2γκ +n+m+d−5 dr

0

 Sd−1

 Bd

∂f ∂n

= n f for f

(gEh f − f Eh g)h2κ dx

(gEh f − f Eh g)h2κ dσ ,

using the spherical polar coordinates x = rξ , r > 0 and ξ ∈ Sd−1 . The last integral is zero  since the difference part Eh of Δh is self-adjoint with respect to ·, ·κ . Theorem 3.1.3. For n = 0, 1, 2, . . ., Pnd admits the decomposition 

Pnd =

d 2 x2 j Hn−2 j (hκ ).

(3.1.2)

0≤ j≤n/2

Furthermore, for n = 0, 1, 2, . . .,  d = dim Hnd (h2κ ) = dim Pnd − dim Pn−2

   n+d −1 n+d −3 − . d −1 d −1

(3.1.3)

Proof. Briefly, the proof follows by induction, using the orthogonality of Hnd (h2κ ) and d .  the fact that Δh maps Pnd onto Pn−2 From (2.3.1) it follows immediately that ΔhVκ = Vκ Δ and, consequently, if P is an ordinary harmonic polynomial, then Vκ P is an h-harmonic. In terms of the spherical polar coordinates x = rξ , the Dunkl Laplacian Δh admits a decomposition as in the case of ordinary Laplace operator. Let us define λk := γk +

d −2 d −2 = ∑ κv + . 2 2 v∈R+

(3.1.4)

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

1 2λκ + 1 d d2 + Δh,0 , + dr2 r dr r2

(3.1.5)

3.1. Dunkl h-harmonics

17

where Δh,0 f =

1 (ξ ) [Δ0 ( f hκ ) − f Δ0 hκ ] − Eh f , hκ (ξ )

Δ0 denotes the usual Laplace–Beltrami operator, and Eh ξ variable.

(3.1.6)

means that Eh is acting on the

Proof. By the decomposition Δh = Dh + Eh , we can apply the decomposition of the ordinary Laplacian Δ to the differential part Dh , which gives the part of Δh,0 expressed in terms of the classical Laplace–Beltrami operator Δ0 . The difference part follows readily  from 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,0Ynh (ξ ) = −n(n + 2λκ )Ynh (ξ ),

∀Ynh ∈ Hnd (h2κ ),

ξ ∈ Sd−1 .

(3.1.7)

Proof. Since Ynh is a homogeneous polynomial of degree n, Ynh (x) = rnYnh (ξ ). Apply ing (3.1.5) to Ynh , equation (3.1.7) follows from ΔhYnh = 0. The following theorem gives an orthogonal basis for Hnd (h2κ ). Theorem 3.1.6. For α ∈ Nd0 , n = |α|, define pα (x) :=

(−1)n x2|α|+2λκ D α {x}−2λk , 2n (λκ )n

(3.1.8)

α

where D α := Dd d · · · D1α1 . Then 1. pα ∈ Hnd (h2κ ) and pα is a monic spherical h-harmonic of the form pα (x) = xα + x2 qα (x),

d qα ∈ Pn−2 ;

(3.1.9)

2. pα satisfies the recurrence relation pα+ei (x) = xi pα (x) −

1 x2 Di pα (x); 2n + 2λκ

3. {pα : |α| = n, αd = 0 or 1} is a basis of Hnd (h2κ ). The proof of this theorem is more or less a straightforward computation. Indeed, for g ∈ Pnd and ρ ∈ R, the explicit expressions for Di and Δh can be used to show that Di (xρ g) = ρxi xρ−2 g + xρ Di g, ρ

ρ−2

Δh (x g) = ρ(2n + 2λk + ρ)x

(3.1.10) ρ

g + x Δh g.

(3.1.11)

18

Chapter 3. h-Harmonics and Analysis on the Sphere

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 that Δh pα (x) = 0, so that pα ∈ Hnd (h2κ ). However, for a generic reflection group, it is not clear how to evaluate the norm of pα , that is, an explicit formula for the norm of pα is not known. As a consequence, we cannot provide an explicit orthonormal basis from {pα }. In fact, no orthonormal basis for Hnd (h2κ ) is explicitly known beyond the case of the group Zd2 . 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 Zd2 , an orthonormal basis can be explicitly given in spherical coordinates, as h2κ in this case is a simple product. Let projκn denote the orthogonal projection operator projκn : L2 (Sd−1 ; h2κ ) → Hnd (h2κ ). By the orthogonal decomposition of homogeneous polynomials, p ∈ Pnd can be written d ; we have, by definition, that as p(x) = pn + x2 qn , where pn ∈ Hnd (h2κ ) and qn ∈ Pn−2 κ pn = projn p. In this regard, by (3.1.9), the polynomial pα in (3.1.8) is the orthogonal projection of xα , pα (x) = projκn qα (x),

qα (x) = xα with |α| = n.

(3.1.12)

Proposition 3.1.7. Let p ∈ Pnd . Then projκn p =

n/2

∑

j=0

1 4 j j!(1 − n − λ

κ)j

x2 j Δhj p.

(3.1.13)

Proof. It suffices to prove equation (3.1.13) for p = qα . Since projκn qα = pα , we need to show that pα equals the right-hand side of (3.1.13) with p = qα . This can be established by induction 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 [n/2]   D α x−2λ = (−1)n 2n (λ )n x−2λ −2n ∑ j=0

4 j j!(−λ

1 x2 j Δhj qα (x), − n + 1) j

where λ = λκ . Applying here Di and using the first identity in (3.1.10) with g = Δhj {xα }, we conclude that   Di D α x−2λ = (−1)n 2n (λ )n (−2λ − 2n)x−2λ −2n−2 ×

[(n+1)/2]

∑

j=0

1 4 j j!(−λ

− n) j

  x2 j xi Δhj {xα } + 2 jΔhj−1 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 on j, that j = 1, 2, 3, . . . . Δhj {xi f (x)} = xi Δhj f (x) + 2 jDi Δhj−1 f (x),

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 constant multiple of the projection of xi qα (x), and the right-hand side is a constant multiple of the right-hand side of (3.1.13) with p(x) = xi qα (x), which completes the induction.  There is another inner product on the space of homogeneous polynomials Pnd that will be useful in our study below. Theorem 3.1.8. For p, q ∈ Pnd , p, qD = En (D (x) , D (y) )p(x)q(y) = q, pD . Proof. By Proposition 2.3.8, p(x) = En (x, D (y) )p(y). The operators D (x) and D (y) commute and thus p, qD = En (D (x) , D (y) )p(y)q(x) = En (D (y) , D (x) )p(y)q(x). 

The last expression equals q, pD . The pairing ·, ·D is related to ·, ·κ when p ∈ Pnd and q ∈ Hnd (h2κ ). Theorem 3.1.9. If p ∈ Pnd and q ∈ Hnd (h2κ ), then p, qD = 2n (λκ + 1)n p, qκ .

(3.1.15)

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

 R

d

Rd

p(D)q(x)h2κ (x)e−x

2 /2

dx

  2 q(x) p(D ∗ )1 h2κ (x)e−x /2 dx.

Repeatedly applying the formula for the adjoint operator Di∗ g(x) = xi g(x) − Di g(x), and using the fact that the degree of Di g 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 ∈ Hnd (h2κ ), it follows from the 2 spherical polar integral that Rd q(x)s(x)h2κ (x)e−x /2 dx = 0. Consequently, we conclude that  2 q(x)p(x)h2κ (x)e−x /2 dx. p, qh = ch Rd

Now, (3.1.15) follows from Proposition 2.1.1.



In the case where both p, q ∈ Pnd , the inner product p, qD also has an integral expression: Theorem 3.1.10. For p, q ∈ Pnd , p, qD = ch

 Rd



e−Δh /2 p(x)



2 e−Δh /2 q(x) h2κ (x)e−x /2 dx.

(3.1.16)

20

Chapter 3. h-Harmonics and Analysis on the Sphere

Proof. First of all, decomposing p ∈ Pnd as p(x) = ∑0≤ j≤n/2 x2 j pn−2 j , where pn−2 j ∈ d (h2 ), and decomposing q similarly, by the definition of Δ and p Hn−2 n−2 j , qn−2 j h , we h j κ conclude that p, qh =

n/2 n/2

n/2 n/2

j=0 i=0

j=0 i=0

∑ ∑ pn−2 j , qn−2 j h = ∑ ∑

  Δih pn−2i (D) x2 j qn−2 j (x) .

By the identity (3.1.11),   Δih x2 j qn−2 j (x) = 4i (− j)i (−n − λ + j)i x2 j−2i qn−2 j (x), which is zero if i > j. If i < j, then x2 j qn−2 j (x), x2i pn−2i (x)h = 0 by the same argument and the fact that p, q = q, p. Hence, the only remaining terms are those with j = i, which are given by 4 j j!(−n − λ + j) j pn−2 j (D)qn−2 j (x). Therefore, n/2

∑

p, qh =

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

j=0

Thus, we only need to prove (3.1.16) for polynomials of the form p(x) = x2 j pm (x) and d (h2 ). For such p, q, by (3.1.11) and (3.1.15), q(x) = x2 j qm (x) with pm , qm ∈ Hn−2 j κ p, qD = 4 j j!(m + λκ + 1) j pm , qm D = 4 j j!(m + λκ + 1) j 2m (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   κ (x2 /2)pm (x), e−Δh /2 x2 j pm (x) = (−1) j j!2 j Ln+λ j

(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

κ [Lm+λ (x2 /2)]2 pm (x)qm (x)h2κ (x)e−x j

= 2m+λκ

Γ(m + j + λκ + 1) j!

 Sd−1

2 /2

dx

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 }, adn := dim Hnd (h2κ ), be an orthonormal basis of Hnd (h2κ ). For f ∈ L2 (Sd−1 , h2κ ), the usual Hilbert space theory shows that f can be expanded in spherical h-harmonics as f=

∞

adn

∑∑

n=0 ν=0

f ν,nYν,n ,

f ν,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 Hnd (h2κ ), projκn : L2 (Sd−1 ; h2κ ) → Hnd (h2κ ), the expansion in h-harmonics can be rewritten as f=

∞

∑ projκn f .

n=0

In particular, the projection operator can be expressed as an integral, projκn f (x) =

1 ωdκ

 Sd−1

f (y)Znκ (x, y)h2κ (y)dσ (y), x ∈ Sd−1 ,

(3.2.1)

where Znκ (·, ·) is the kernel function defined by Znκ (x, y) =

adn

∑ Yν,n (x)Yν,n (y),

ν=0

x, y ∈ Sd−1 .

This kernel, however, is independent of the choice of particular basis of Hnd (h2κ ). Indeed, it is the reproducing kernel of Hnd (h2κ ), i.e., 1 ωd



Sd−1

Znκ (x, y)p(y)h2κ (y)dσ (y) = p(x),

∀p ∈ Hnd (h2κ ),

x ∈ Sd−1 .

(3.2.2)

In terms of the intertwining operator Vκ , the reproducing kernel Znκ has a concise expression given in terms of the Gegenbauer polynomial Cnλ : Theorem 3.2.1. Let λκ = γk + d−2 2 . For y ≤ x = 1,    n + λκ y Znκ (x, y) = yn Vκ Cnλκ ·, (x). λκ y

(3.2.3)

Proof. From Proposition 2.3.8 (ii), it follows that for every p ∈ Hnd (h2κ ), p(x) = En (x, ·), pD =  projκn (En (x, ·)), pD

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

Since Znκ (·, ·) is uniquely determined by the reproducing property, this shows that Znκ (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   λκ + 1 n 2n−2 j κ  x2 j y2 j En−2 j (x, y). Zn (x, y) = ∑ 0≤ j≤n/2 1 − n − λκ ) j j! When x = 1, we can write the right-hand side as Vκ (Gn (·, y/y))(x), where Gn is a hypergeometric function 2 F1 , which turns out to be a constant multiple of the Gegenbauer polynomial. 

22

Chapter 3. h-Harmonics and Analysis on the Sphere

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 Ynh ∈ Hnd (h2κ ), 1 ωdκ



Sd−1

Vκ [ f (x, ·)](y)Ynh (y)h2κ (y)dσ (y) = Λn ( f )Ynh (x),

x ∈ Sd−1 ,

(3.2.4)

where Λn ( f ) is a constant defined by Λn ( f ) = cλκ

 1 −1

f (t)

Cnλκ (t) Cnλκ (1)

1

(1 − t 2 )λk − 2 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 Gegenbauer polynomials m k + λ κ λκ Ck (t), f (t) = ∑ Λk λκ k=0 where Λκ are determined by the orthogonality of Gegenbauer polynomials, Λk =

cλκ

 1

Ckλκ (1) −1

f (t)Ckλκ (t)(1 − t 2 )λκ − 2 dt, 1

1 2 λ − 2 dt. Using (3.2.3) and the reproducing property of Z κ (x, y) it and c−1 n λ = −1 (1 − t ) follows that, for n ≤ m, 1

1 ωdκ



Sd−1

Vκ [ f (x, ·)](y)Ynh (y)h2κ (y)dσ (y) = ΛnYnh (x),

x ∈ Sd−1 .

Since Λn /ωdκ = 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 1 ωdκ



Sd−1

Vk f (y)h2κ (y)dσ (y) = aκ



Bd

f (x)(1 − x2 )|κ|−1 dx.

(3.2.5)

In particular, if f (y) = g(x, y) with g : R → R, then 1 ωdκ

 S

Vk [g(x, ·)](y)h2κ (y)dσ (y) = cλκ d−1

 1 −1

1

g(xt)(1 − t 2 )λκ − 2 dt.

(3.2.6)

Proof. Applying the Funk–Hecke formula (3.2.4) with n = 0 to the function ξ → g(xξ ), ξ ∈ Sd−1 , gives (3.2.6). It is known that every polynomial f can be written as a linear sum of 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 side of (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 gL1 (h2κ ;Sd−1 ) ≤ bκ

 Bd

|g(x)|(1 − x2 )|κ|−1 dx.

Since the right-hand side is a constant multiple of the norm of L1 (W ; Bd ), where W (x) := (1−x2 )|κ|−1 , this allows us to extend Vκ to a positive, bounded operator from L1 (W ; Bd ) to L1 (h2κ ; 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 intertwining operator Vκ . Property (3.2.5) of Vκ is highly non-trivial, as can be seen in the special case of Zd2 , where Vκ is given explicitly by formula (2.3.2), and it is highly useful, as the development below will show.

3.3

Convolution operators and orthogonal expansions

The expression of Znκ (·, ·) 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 ωdκ

 Sd−1

f (y)Vκ [g(·, y)](x)h2κ (y)dσ (y).

(3.3.1)

Denote the norm of the space L p (wλ ; [−1, 1]) by  · λ ,p , and the norm of the space d−1 ) by  ·  d−1 ) for p = ∞. The convolution ∗ κ,S κ,p for 1 ≤ p < ∞ and by C(S κ satisfies Young’s inequality:

L p (h

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]), (3.3.2)  f ∗κ gκ,p ≤  f κ,q gλκ ,r . In particular, for 1 ≤ p ≤ ∞,  f ∗ gκ,p ≤  f κ,p gλκ ,1

and

 f ∗ gκ,p ≤  f κ,1 gλκ ,p .

(3.3.3)

Proof. The standard proof of Young’s inequality applies in this setting. By Minkowski’s inequality, it suffices to show that G(x, ·)κ,r ≤ gλκ ,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,∞ ≤ gλκ ,∞ and we deduce by (3.2.6) that G(x, ·)κ,1 ≤

1 ωdκ

 Sd−1

Vκ [|g(x, · )|] (y)h2κ (y)dσ = cλκ

 1 −1

|g(t)|wλ (t)dt = gλκ ,1 .

The log-convexity of the L p -norm implies then G(x, ·)κ,r ≤ gλκ ,r .



24

Chapter 3. h-Harmonics and Analysis on the Sphere By (3.2.1) and (3.2.3), the projection projκn is a convolution operator projκn f = f ∗κ Znκ ,

Znκ (t) :=

n + λκ λκ Cn (t). λκ

(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) = g λn κ projκn f ,

n = 0, 1, 2 . . . ,

(3.3.5)

where g λn κ is the Fourier coefficient of g in the Gegenbauer polynomial, g λn κ = cλκ

 1 −1

g(t)

Cnλκ (t) Cnλκ (1)

1

(1 − t 2 )λκ − 2 dt.

Proof. By (3.2.1) and the Funk–Hecke formula in Theorem 3.2.2, projκn ( f ∗κ g)(x) = 1 = κ ωd = g λn κ

 Sd−1

1 ωdκ



1 ωdκ 

 Sd−1

1 f (y) ωdκ

Sd−1



S

( f ∗ g)(ξ )Znκ (x, ξ )h2κ (ξ )dσ (ξ )

g(ξ , y)Znκ (x, ξ )h2κ (ξ )dσ (ξ ) d−1

 h2κ (y)dσ (y)

f (y)Znκ (x, y)h2κ (y)dσ (y) = g λn κ projκn f (x), 

which is what we needed to prove. L2

Since the convergence of the h-harmonic series does not go beyond norm in general, it is necessary to consider summability methods. We consider the Ces`aro means of the h-harmonic series. First, we give the definition of the Ces`aro means for a sequence of complex numbers. Definition 3.3.4. The Ces`aro (C, δ )-means of a given sequence {an }∞ n=0 of complex numbers are defined by n Aδ n− j sδn := ∑ δ a j , n = 0, 1, . . . , (3.3.6) A n j=0 where the coefficients Aδj are defined by (1 − t)−1−δ =

∞

∑ Aδn t n ,

t ∈ (−1, 1).

n=0

For convenience, we also define Aδj = 0 for j < 0. The following useful properties follow easily from the definition: Aδj − Aδj−1 = Aδj −1 , |Aδj | ∼

δ

n

n

j=0

j=0

∑ Aδj = Anδ +1 , ∑ Aδn− j Aαj = Anα+δ +1 ,

( j + 1) , whenever j + δ + 1 > 0.

3.3. Convolution operators and orthogonal expansions

25

  Let us denote by Snδ h2κ ; f the Ces`aro means of the h-harmonic series, that is,   1 Snδ h2κ ; f := δ An

n

∑ Aδn− j projκj f ,

(3.3.7)

j=0

where Sn0 (h2κ ; f ) is the n-th partial sum. By (3.3.4), the Ces`aro means are convolution operators,     Snδ h2κ ; f = f ∗κ Knδ h2κ , where the kernel is defined by   1 Knδ h2κ ;t := δ An

n

∑ Aδn−k

k=0

  k + λκ λκ Ck (t) = knδ wλκ ; 1,t , λκ

(3.3.8)

in which knδ (wλκ ; ·, ·) is the kernel of the (C, δ )-means of the Fourier orthogonal series in the Gegenbauer polynomials. Theorem 3.3.5. The Ces`aro means of the spherical h-harmonic series satisfy: 1. if δ ≥ 2λk + 1, then Snδ (h2κ ) is a nonnegative operator; 2. if δ > λκ , then sup Snδ (h2κ ; g)κ,p ≤ cgκ,p ,

1 ≤ p ≤ ∞.

(3.3.9)

n≥0

In particular, Snδ (h2κ ; f ) converges to f in L p (h2κ ; Sd−1 ) for 1 ≤ p ≤ ∞. Proof. By (3.3.8), the non-negativity of Snδ (h2κ ) follows from that of the Gegenbauer kernel knδ (wλ ; 1,t), which is a classical result. In order to prove the convergence, it is sufficient to show that Snδ (h2k )κ,p is bounded. By Young’s inequality (3.3.3), Snδ (h2κ , f )κ,p ≤  f κ,p knδ (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 ingredients are (3.2.6) and the positivity of Vκ (namely, Theorem 2.3.4). Equality (3.2.6) removes Vκ and allows us to reduce the problem to the Gegenbauer series. This is similar to the result 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 > λκ is not sharp in general. In fact, taking the average of Vκ by (3.2.6) erases the information on the reflection group. For a generic reflection group, we know very little about Vκ . In the case of Zd2 , however, Vκ is given by the explicit formula in (2.3.2), which allows us to obtain much deeper results on the convergence of the Ces`aro means. These studies require sharp pointwise estimates of the kernel functions and, sometimes, asymptotics of the kernel functions,

26

Chapter 3. h-Harmonics and Analysis on the Sphere

which in turn require long estimations and detailed analysis. We shall state the results without proof. Recall that, for the hκ defined in (2.1.1) associated with Zd2 , λκ = |κ| +

d −2 . 2

Theorem 3.3.6. Let hκ be as in (2.1.1). Let δ > −1 and define σκ := λk − min κi = |κ| + 1≤i≤d

Then for p = 1 and p = ∞,  projn (h2κ )κ,p ∼ nσκ

and

d −2 − min κi . 1≤i≤d 2

⎧ ⎪ ⎨1, Snδ (h2κ )κ,p ∼ log n, ⎪ ⎩ −δ +σκ n ,

δ > σκ δ = σκ −1 < δ < σκ .

In particular, the (C, δ )-means Snδ (h2κ ; f ) converge to f in L1 (h2κ ; Sd−1 ) or in C(Sd−1 ) if and only if δ > σκ . The zero set of the weight function hκ (x) serves as boundary on the sphere, away from which we have better convergence behavior. Indeed, for hκ in (2.1.1), the zero set is the collection of great circles defined by the intersection of Sd−1 with the coordinate planes. Let us define d−1 Sd−1 \ int := S

d 

{x ∈ Sd−1 : xi = 0},

i=1

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−2 2 , then and the convergence is uniform over each Snδ (h2κ ; f ) converges to f for every x ∈ Sd−1 int d−1 compact subset of Sint . By the Riesz interpolation theorem, Theorem 3.3.6 also implies that Snδ (h2κ ; f ) converges to f in the L p norm if δ > σκ , for all 1 < p < ∞. However, for each fixed p, this order is not sharp. The sharp results for L p convergence are given in the following two theorems: Theorem 3.3.8. Suppose that f ∈ L p (Sd−1 ; h2κ ), 1 ≤ p ≤ ∞, | 1p − 12 | ≥ δ > δκ (p) := max{(2σκ + 1)| 1p − 12 | − 12 , 0}. Then Snδ (h2κ ; f ) converges to f in L p (h2κ ; Sd−1 ) and sup Snδ (h2κ ; f )κ,p ≤ c f κ,p .

n∈N

1 2σκ +2

and (3.3.10)

3.4. Maximal functions

27

Theorem 3.3.9. Assume 1 ≤ p ≤ ∞ and 0 < δ ≤ δκ (p). Then there exists a function f ∈ L p (Sd−1 ; h2κ ) such that Snδ (h2κ ; f ) diverges in L p (Sd−1 ; h2κ ). The proofs of these two theorems are much more involved and require heavy machinery, such as the Fefferman–Stein inequality and Stein’s analytic interpolation theorem.

3.4

Maximal functions

For x ∈ Sd−1 and 0 ≤ θ ≤ π, we define b(x, θ ) := {y ∈ Bd : x, y ≥ cos θ }. Let χE denote the characteristic function of the set E. Definition 3.4.1. For f ∈ L1 (Sd−1 ; h2κ ), define the maximal function

2 Sd−1 | f (y)|Vκ [χb(x,θ ) ](y)hκ (y)dσ (y) 2 0 0 and f ∈ L1 (X, μ), where c is 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 Prκ f (ξ ) :=

1 ωdκ

 Sd−1

f (y)Prκ (ξ , y)h2κ (y)dσ (y),

ξ ∈ Sd−1 ,

(3.4.3)

where the kernel Prκ (x, ·) is given by Prκ (x, y)

 := Vκ

 1 − r2 (x), (1 − 2r·, y + r2 )λκ +1

(3.4.4)

for 0 < r < 1. Using the generating function of the Gegenbauer polynomials, the Poisson kernel is the generating function of the spherical h-harmonics, as seen in the first item of the following lemma, from which the other two items follow directly. Lemma 3.4.5. For 0 < r < 1, the Poisson kernel satisfies the following properties:   n n+λκ λκ (1) for x, y ∈ Sd−1 , Prκ (x, y) = ∑∞ n=0 r λκ Vκ Cn (x, ·) (y); κ n (2) Prκ f = ∑∞ n=0 r projn f ;

(3) Prκ (x, y) ≥ 0 and

1 ωdκ



κ 2 Sd−1 Pr (x, y)hκ (y)dσ (y) =

1.

Put T t = Prκ with r = e−t . Using the above lemma, it is easy to see that T t is a diffusion semi-group. We will need another semi-group, which is the discrete analog of the heat operator: Htκ f := f ∗κ qtκ ,

qtκ (s) :=

∞

∑ e−n(n+2λκ )t

n=0

n + λκ λκ Cn (s). λκ

(3.4.5)

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

3.4. Maximal functions

29

Proof. The kernel qtκ is known to be nonnegative, from which it immediately follows that Htκ are positive and that qtκ λκ ,1 = 1, by the orthogonality of the Gegenbauer polynomials. Hence, by Young’s inequality, Htκ f κ,p ≤  f k,p . The other requirements in Definition 3.4.2 can be verified directly.  Lemma 3.4.7. The Poisson and the heat semi-groups are connected by Peκ−t f (x) = where

 ∞ 0

φt (s)Hsκ f (x)ds,

(3.4.6)

 2 t −3/2 − 2√t s −λκ √s √ φt (s) := s e . 2 π

Furthermore, if f (x) ≥ 0 for all x, then P∗κ f (x) := sup Prκ f (x) ≤ c sup 00 s 0 Hu f (x)du is bounded on L p (h2κ , Sd−1 ) for 1 < p ≤ ∞ and of week type (1, 1). Therefore, it is sufficient to prove (3.4.6) and (3.4.7). First we prove (3.4.6). We use the well-known identity ([49, p.46]) 1 e−v = √ π

 ∞ −u 2 e √ e−v /4u du, 0

v > 0,

u

(3.4.8)

with v = (n + λκ )t. Making the change of variable s = t 2 /4u, we obtain 1 e−nt = eλκ t √ π t = √ 2 π =

 ∞ 0

 ∞ −u λκ2 t 2 n(n+2λκ )t 2 e √ e− 4u e− 4u du

 ∞

0

u

 2 t −λ √s − 2√ κ s ds

e−n(n+2λκ )s s−3/2 e

0

e−n(n+2λκ )s φt (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   ∞  s κ κ Pe−t f (x) = − Hu f (x)du φt (s)ds 0 0  ∞  s 1 κ Hu f (x)du s|φt (s)|ds, ≤ sup 0 s>0 s 0

30

Chapter 3. h-Harmonics and Analysis on the Sphere

where the derivative of φt (s) is taken with respect to s. Furthermore, since Prκ f = f ∗k pκr and |pκr (t)| ≤ c for 0 < r ≤ e−1 , it follows that 1 s→∞ s

sup Prκ f (x) ≤ c f 1,κ = c lim

0 αt , where t2  ∼ t 2, 0 ≤ t ≤ 1. αt := 3 + 9 + 4λκ2t 2 Since the integral of φt (s) over [0, ∞) is 1 and φt (s) ≥ 0, integration by parts gives  ∞ 0

s|φt (s)|ds = 2αt φt (αt ) −

 αt 0

φt (s)ds +

≤ 2αt φt (αt ) + 1 = √

 ∞ αt

φt (s)ds

(t−2λκ αt )2 t e− 4αt + 1 ≤ c, παt



as desired.

We are now ready to prove the main result on the maximal function. To state the weak type inequality, we define, for any measurable subset E of Sd−1 , the measure with respect to h2κ as  measκ E :=

E

h2κ (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 > 0. α

(3.4.9)

Furthermore, if f ∈ L p (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 1 − r2 pκr (cos θ ) =  λκ +1 (1 − r)2 + 4r sin2 θ2 1 − r2

≥ c (1 − r)−(2λκ +1) . ((1 − r)2 + rθ 2 )λκ +1   For j ≥ 0 define r j := 1 − 2− j θ and set B j := y ∈ Bd : 2− j−1 θ ≤ d(x, y) ≤ 2− j θ . The lower bound of pκr proved above shows that ≥c

χB j (y) ≤ c (2− j θ )2λk +1 pκr j (x, y),

3.5. Convolution and maximal function

31

which immediately implies that χb(x,θ ) (y) ≤

∞

∞

j=0

j=0

∑ χB j (y) ≤ c θ 2λk +1 ∑ 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)h2κ (y)dσ (y) 2λκ +1

≤ cθ

∞

∑2

− j(2λκ +1)

 Sd−1

j=0 ∞

  | f (y)|Vκ pr j (x, y) (y)h2κ (y)dσ (y)

= c θ 2λκ +1 ∑ 2− j(2λκ +1) Prκj (| f |; x) j=0

2λκ +1

≤ cθ

sup Prκ (| f |; x).

0 2λ + 1, then the pointwise estimate of the kernel knδ (wλ ;t, 1) shows that |Knδ (h2κ ; cos θ )| is bounded by a single term and the same proof yields |Snδ f (x)| ≤ cM f (x). 

3.6

Notes and further results

The first studies on h-harmonic expansions appeared in [69] (which contains (2.3.2) and a proof for the formula for Znκ , in the case of Zd2 , by summing over a specific orthonormal 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 unit ball). The proof of (3.2.5) in Theorem 3.2.3 was given in [16]. The Funk–Hecke formula (3.2.4) was established in [71]. The results on the Ces`aro summability for G = Zd2 were established in [13, 14, 37], which reduces to the classical results on the spherical harmonics [3, 5, 48] when κ = 0. The convolution and the translation operators were defined in [72], and used to study weighted best approximation on the sphere. The maximal function Mκ f was defined in [73], the results in Section 3.5 were established in [12]. In the case of G = Zd2 and hκ (x) = ∏di=1 |xi |κi , we can consider the weighted Hardy–Littlewood maximal function defined by

2 c(x,θ ) | f (y)|hκ (y)dσ (y) , 2 0 2. We break the first sum on the ∞ j j+1 − 1 right side of (5.4.5) into two parts: ∑m−4 j=0 and ∑ j=m−3 . Observe that if 2 ≤ i ≤ 2 and 0 ≤ j ≤ m − 4, then Siδ ((−Δh,0 )1/2 F) = L2κ j+2 (Siδ ((−Δh,0 )1/2 F)) = Siδ (L2κ j+2 ((−Δh,0 )1/2 F)) = Siδ ((−Δh,0 )(α+1)/2 (L2κ j+2 ( f ))).

(5.4.6)

Thus, if δ > λκ ,

1 m−4 2 j+1 −1 q − j(1+q) ∑2 ∑ Siδ ((−Δh,0 )1/2 F)qκ,p j=0

i=2 j

" ≤c

m−4

∑

" ≤c

! !q ! !(−Δh,0 )(α+1)/2 (L2κ j+2 ( f ))!

− jq !

2

∑2

q

κ,p

j=0

m−4

#1

# jαq

Kα+1 ( f , 2− j )qκ,p

1 q

,

(5.4.7)

j=0

where the second step uses (5.4.6) and the Ces`aro (C, δ )-summability of h-harmonic expansions for δ > λκ , and the last step uses (5.3.6). However, on the other hand, for δ > λκ ,

∞

∑

2− j(1+q)

j=m−3

2 j+1 −1

∑j

Siδ ((−Δh,0 )1/2 F)qκ,p

i=2

1 q

" ≤c

∞

∑

#1 2− jq

q

(−Δh,0 )1/2 Fκ,p

j=m−3

≤ c2−m (−Δh,0 )1/2 Fκ,p = c2−m (−Δh,0 )(α+1)/2 (L2κm ( 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).

58

Chapter 5. Sharp Jackson and Sharp Marchaud Inequalities It remains to prove claim (5.4.4). Note that ∞ ∞ ∂ κ Pr (F) = ∑ iri−1 projκi (F) = (1 − r)δ +1 ∑ ∂r i=1 i=1

"

#

i

∑ jAδi− j projκj (F)

ri−1 ,

j=1

δ k −1−δ in the last step. Thus, where we used the identity ∑∞ k=0 Ak r = (1 − r) ∞ ∂ κ Pr (F) ≤ (1 − r)δ +1 ∑ Aδi |Siδ ((−Δh,0 )1/2 F)|ri−1 ∂r i=1

≤ c(1 − r)1+δ

∞

∑ 2 jδ r2

j −1

j=0

2 j+1 −1

∑j

|Siδ ((−Δh,0 )1/2 F)|,

i=2

which, using the Cauchy–Schwartz inequality, implies ∞ 2  2 −1 2 ∞ δ 2 −1 j ∂ κ Pr (F) ≤ c(1 − r)2+2δ ∑ 2 jδ r2 −1 ∑ |Siδ ((−Δh,0 )1/2 F)| ∑2 r ∂r j=0 =0 i=2 j j+1

j+1  ∞   2 −1 2 ∞  j = c ∑ 2 jδ ∑ |Siδ ((−Δh,0 )1/2 F)| ∑ 2δ r2 +2 −2 (1 − r)2+2δ .

=0

i=2 j

j=0

(5.4.9) On the other hand, a straightforward calculation shows that ∞

 1

=0

0

∑ 2δ

∞



r2 +2

= c ∑ 2δ

j −2

 1

=0 ∞

0



r2 +2

j −2

(1 − r)3+2δ dr

Γ(2 + 2 j − 1)Γ(4 + 2δ ) ≤ c ∑ 2δ (2 + 2 j )−4−2δ Γ(2 + 2 j + 3 + 2δ ) =0

=0 − j(δ +4)

≤ c2

∞

(1 − r)2+2δ r| log r| dr ≤ c ∑ 2δ

.

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

g(F) ≤ c

∞

∑ 2−4 j

2

j=0

j+1 −1

∑

|Siδ ((−Δh,0 )1/2 F)|

2 12

,

(5.4.10)

i=2 j

which implies claim (5.4.4) for p > 2. Finally, for 1 < p ≤ 2, we use (5.4.10) and H¨older’s inequality to obtain #p " j+1 ∞

|g(F)| p ≤ c ∑ 2−2 j p

2

j=0

−1

∑j

|Siδ ((−Δh,0 )1/2 F)|

i=2

∞

2 j+1 −1

j=0

i=2

≤ c ∑ 2− j(p+1)

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

∑j

|Siδ ((−Δh,0 )1/2 F)| p , 

5.5. The sharp Jackson inequality

5.5

59

The sharp Jackson inequality

The main goal in this section is to prove the following sharp Jackson inequality. Theorem 5.5.1. For f ∈ L p (h2κ ; Sd−1 ), 1 < p < ∞, r > 0, and s = max(p, 2), tr



∑

jsr−1 E j ( f )sκ,p

1/s

≤ cω r ( f ,t)κ,p .

(5.5.1)

1≤ j≤1/t

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 ∈ L p (h2κ ; Sd−1 ), 1 < p < ∞, and s = max(p, 2), tr



1/2

t

ω r+1 ( f , u)sκ,p 1/s du ≤ cω r ( f ,t)κ,p . urs+1

Proof of Theorem 5.5.1. Obviously, it suffices to show that 2−nr



n

∑ 2 jrs E2 j ( f )sκ,p

1/s

≤ cKr ( f , 2−n )κ,p .

(5.5.2)

j=1

Setting gn = L2κn−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

∑ 2 jrs E2 j ( f )sκ,p

j=1

≤ 2−nr



n

1/s

∑ 2 jrs E2 j ( f − gn )sκ,p

1/s

+ 2−nr



j=1

n

∑ 2 jrs E2 j (gn )sκ,p

j=1

n

1/s 2r −nr jrs s ≤ rs  f − g  + 2 2 E (g ) . j n κ,p n ∑ κ,p 2 (2 − 1)1/s j=1 Therefore, for the proof of (5.5.2), it remains to show that 2−nr



n

∑ 2 jrs E2 j (gn )sκ,p

j=1

1/s

  ≤ cKr f , 2−n κ,p ,

1/s

60

Chapter 5. Sharp Jackson and Sharp Marchaud Inequalities

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

∑ 2 jrs E2 j (gn )sκ,p ≤ c(−Δh,0 )r/2 gn sκ,p .

(5.5.3)

j=1

For the proof of (5.5.3), we write for j ≤ n, ! ! E2 j+1 (gn )κ,p ≤ gn − L2 j gn κ,p = L2n gn − L2 j gn κ,p = !

n+1

∑

= j+2

! ! κθ , gn !

κ,p

.

Note that L2κi (L2κn f − L2κ j f ) = 0 for i < j ≤ n, and κθ ,i (L2κn gn − L2κ j gn ) = 0 for i > n + 1. Thus, applying Theorem 5.3.2 to the function L2κn gn − L2κ j gn , we deduce that ! ! L2κn gn − L2κ j gn κ,p ∼ !

n+1

∑

(κθ , gn )2

= j+1

1/2 ! ! !

κ,p

.

Thus, proving (5.5.3) reduces to showing that ! jrs ! 2 ∑ !

n+1 j=0

n+1

∑

(κθ , gn )2

= j+1

1/2 !s ! !

κ,p

≤ c(−Δh,0 )r/2 gn 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 + gq ≤  | f | + |g| q for the quasi-norm  · q when q ≤ 1, and obtain ! jr2 ! 2 ∑ !

! ! (κθ , gn )2 !

n+1

n+1

j=1

= j+1

∑

! n+1 ! ≤ ! ∑ 2 jr2

p/2

j=1

! n+1

1/2 !2 ! ! ≤ c! ∑ (κθ , gn )2 2r2 !

κ,p

=2

n+1

∑

! ! (κθ , gn )2 !

= j+1

p/2

≤ c(−Δh,0 )r/2 gn 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 



n+1

Sd−1 j=0

≤

n+1

∑ 2 jrp ∑



n+1

= j+1

(κθ , gn )2 n+1

p/2

=



j=0

n+1

j=0

=1

∑ 2 jrp ∑ χE ( j, )|κθ , gn |2

Sd−1

∑ |κθ , gn |2 ∑ 2 jrp χE ( j, )

Sd−1 =1

n+1

2 p p

2

p/2 2p 2p

! n+1

1/2 ! p ! ! ≤ c! ∑ |κθ , (gn )|2 2r2 ! , =2

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

5.6

61

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 inequality (5.5.1) and the sharp Marchaud inequality (5.4.1). More precisely, we have Theorem 5.6.1. For 1 < p < ∞,

1/s n−r ∑nk=1 krs−1 Ek ( f )sκ,p

' max{p, 2} = min s > 0 : sup sup

ω r ( f , n−1 )κ,p

f n∈N

'

min{p, 2} = max q > 0 : sup sup n

f

n−r

0,

Fκ qtκ (x) = e−tx . 2

(6.1.4)

As an analog of the Fourier transform, we show that the Dunkl transform is an isometry of L2 (Rd , h2κ ). First we give an orthogonal basis for L2 (h2κ ). Let Y ∈ Hnd (h2κ ). Define 2 n+λκ (x2 )Y (x)e−x /2 , x ∈ Rd . (6.1.5) φm (Y ; x) = Lm 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 = δmk δnl

(λκ + 1)n+m ωd 2λκ +1 m ωdκ

 Sd−1

[Yn (x)]2 h2κ (x)dσ .

Proof. Using spherical polar coordinates, the first integral equals  ∞

ch

0

n+λκ 2 l+λκ 2 −r n+l+2γκ +d−1 Lm (r )Lk (r )e r dr 2

21−d/2 Γ(d/2)

 Sd−1

YnYl h2κ 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)δmk Γ(n + λk + 1 + m)/m!, which gives the constant in the formula by (2.1.8).  In particular, if {Yk,n } denotes an orthogonal basis of Hnd (h2κ ), then {φm (Yk,n ; x)} forms an orthogonal basis of L2 (Rd , h2κ ).

68

Chapter 6. Dunkl Transform

Theorem 6.1.6. For m, n = 0, 1, 2, . . ., Y ∈ Hnd (h2κ ) and y ∈ Rd , (Fκ φm (Y ))(y) = (−i)n+2m φm (Y ; y). Proof. By (6.1.2) and (3.1.17), 

bh

Rd

n+λκ Lm (x2 /2)p(x)E(x, y)h2κ (x)e−x

2 /2

dx

= (−1)m (m!2m )−1 eν(y)/2 ν(y)m p(y).

We change the argument in the Laguerre polynomial by using the identity α (t) = Lm

m

∑ 2j

j=0

(α + 1)m (−1)m− j α L (t/2), (α + 1) j (m − j)! j

t ∈ R,

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

bh

Rd

n+λk Lm (x2 )p(x)e−x

2 /2

E(x, y)h2κ (x) dx

(n + λκ + 1)m m!

= eν(y)/2 p(y)(−1)m

m

(−m) j

∑ (n + λκ + 1) j

j=0

(−ν(y)) j . j!

We now replace y by −i y for y ∈ Rd , so that ν(y) becomes −y2 and p(y) becomes (−1)m (−i)n p(y), and the sum yields a Laguerre polynomial. Consequently, the integral 2 n+λ  equals (−1)m (−i)n p(y)Lm k (y2 )e−y /2 . d 2 2 d 2 Since the functions {φm (Y ) : Y ∈ H √ n (hk ), m ≥ 0} constitutes a basis of L (R , hκ ) and the eigenvalues are powers of i = −1, this proves the isometry properties, which are 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 function is also radial, and it can be expressed in terms of the Bessel function Jα (t), defined by 

1 (t/2)α eits (1 − s2 )α−1/2 ds π Γ(α + 1/2) −1 t α ∞ t 2n (−1)n . = ∑ 2 n=0 n!Γ(n + α + 1) 2

Jα (t) := √

(6.1.7)

6.1. Dunkl transform: L2 theory

69

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 ωd rλκ

 ∞ 0

f0 (s)Jλκ (rs)sλκ +1 ds.

Proof. Using polar coordinates and (3.2.6), we obtain Fκ f (y) = bh

 ∞ 0

f0 (s)sd−1+2γκ

= bh cλκ ωdκ

 ∞ 0

 Sd−1

f0 (s)s2λκ +1

E(sx , y)h2κ (x )dσ (x )ds

 1 −1

1

eisyt (1 − t 2 )λκ − 2 dt ds,

from which the stated result follows by the definition of Jα (t) and, using (2.1.8), putting constants together.  As a consequence, we see that the Dunkl transform of f0 (x) is a Hankel transform in x. In general, the Hankel transform Hα is defined on the positive reals R+ . For α > −1/2,  ∞ 1 Jα (rs) 2α+1 Hα f (s) := f (r) r dr. (6.1.8) Γ(α + 1) 0 (rs)α The inverse Hankel transform is given by f (r) =

1 Γ(α + 1)

 ∞ 0

Hα f (s)

Jα (rs) 2α+1 s ds, (rs)α

(6.1.9)

which holds under mild conditions on f ; for example, it holds if f is√piecewise continuous and 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|) − i sign(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 −1

e−isxy (1 + s)(1 − s2 )κ−1 ds,

so that, integrating by parts, E(x, −iy) = cκ

 1 −1

e−isxy (1 − s2 )κ−1 ds −

ix cκ 2κ

 1 −1

e−isxy (1 − s2 )κ ds

 −κ+1/2   = Γ(κ + 1/2) |xy|/2 Jκ−1/2 (|x|) − i sign(x)Jκ+1/2 (|x|) . The following proposition is useful for dealing with the Dunkl transform of functions that involve h-harmonics.

70

Chapter 6. Dunkl Transform

Proposition 6.1.9. Let f ∈ Hnd (h2κ ), n = 0, 1, 2, . . .. If y ∈ Rd , ρ > 0, then the function g(y) =

ωd ωdκ

 Sd−1

f (x)E(x, −i yρ)h2κ (x)dσ (x)

satisfies Δh g = −ρ 2 g and g(y) = (−i)n Γ(λκ + 1)

ρy −λκ y f (ρy). J 2 y n+λκ

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 /2

dx = 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 /2

dx =

ν(y)m f (y). 2m m!

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 ωdκ

 Sd−1 ∞

=

f (x)E(x, y)h2κ (x)dσ (x)

1 ∑ 2n+m (d/2 + γκ )n+m ch m=0

 Rd

f (x)En+2m (x, y)h2κ (x)e−x

2 /2

dx.

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

integral of a series ∑∞ n=0 Δh En (x, −i y) converges absolutely. Indeed (y)

(y)

Δh E(x, −iρy) = Δh E(−iρx, y) =

N

∑ (−iρx j )2 E(−iρx, y) = −ρ 2 x2 E(x, −iρy).

j=1

But x2 = 1 on Sd−1 , and so Δh g = −ρ 2 g.



Let us denote by {Y j,n : 1 ≤ j ≤ dim Hnd (h2κ )} an orthonormal basis of Hnd (h2κ ). We can prove a Paley-Wiener theorem for the Dunkl transform. Let S denote the space of Schwartz class of functions on Rd .

6.1. Dunkl transform: L2 theory

71

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)Y j,n (x)h2κ (x)dσ (x)

extends to an entire function of ρ ∈ C satisfying the estimate |Fj,n (ρ)| ≤ c j,n eR Im ρ . Proof. By the definition of Fκ f and Proposition 6.1.9, 

Sd−1

Fκ f (ρx)Y j,n (x)h2κ (x)dσ (x)  

=c E(y, −iρx)Y j,n (x)h2κ (x)dσ (x) f (y)h2κ (y)dy Rd

=c



Rd

= cρ n

Sd−1

Jλk +n (ρy) 2 h (y)dy (ρy)λk κ Jλ +n (rρ) f j,n (r) k λ +n r2λκ +2n+1 dr, (rρ) k

f (y)Y j,n (y )

 ∞ 0

where c is a constant and f j,n (r) =

 Sd−1

f (ry )Y j,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 theorem then 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 if Fκ f extends to an entire function of ζ ∈ Cd which satisfies |Fκ f (ζ )| ≤ c eR Im ζ  . Proof. The direct part follows from the fact that E(x, −iζ ) is entire and |E(x, −iζ )| ≤ c ex· Im ζ  . For the converse we look at 

Sd−1

Fκ f (ρy )Y j,n (y )h2κ (y )dσ (y ),

ρ ∈ C.

This is certainly entire and, from the proof of the previous theorem, has a zero of order n at the origin. Hence, ρ −n

 Sd−1

Fκ f (ρy )Y j,n (y )h2κ (y )dσ (y )

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

72

6.2

Chapter 6. Dunkl Transform

Dunkl transform: L1 theory

Let S denote the space of Schwartz functions on Rd . The inversion formula of the Dunkl transform holds for f ∈ S . Theorem 6.2.1. Let f ∈ S . Then for y ∈ Rd , Fκ D j f (y) = i y j Fκ f (y) for j = 1, . . . , d. Furthermore, if g j (x) = x j f (x), then Fκ g j (y) = iD j Fκ f (y), y ∈ Rd . The operator −iD j is densely defined on L2 (Rd , h2κ ) 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 )gh2κ dx = −



Rd

f (D j g)h2κ dx,

j = 1, . . . , d.

For fixed y ∈ Rd , put g(x) = E(x, −i y) in the above identity. Then D j g(x) = −i y j E(x, −i y) and Fκ D j f (y) = (−1)(−i y j )Fκ f (y). The multiplication operator defined by M j f (y) = y j f (y), j = 1, . . . , d, is densely defined and self-adjoint on L2 (Rd , h2κ ). Furthermore, −iD j is the inverse image of M j under the Dunkl transform.  In particular, this shows that if f ∈ S , then Fκ f ∈ S . The assumption f ∈ S can of course be substantially relaxed. Theorem 6.2.2. If f ∈ L1 (Rd ; h2κ ), then f ∈ C0 (Rd ). Proof. The space S is dense in L1 (Rd ; h2κ ). For each f ∈ L1 (Rd ; h2κ ), there are functions fn ∈ S such that  f − fn κ,1 → 0. Since Fκ fn ∈ S ⊂ C0 (Rd ) and Fκ fn converges uniformly to Fκ f by Fκ f ∞ ≤  f κ,1 , it follows that Fκ f ∈ C0 (Rd ).  We want to show that if both f and Fκ f are in L1 (R d ; h2κ ), then the inversion theorem holds. For this purpose, we use a generalized convolution operator. Recall that the 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−ix,y Fκ f (x). The translation operator works for the Lebesgue measure since it leaves L1 (Rd ) invariant. It is not obvious what operation is translation invariant for L1 (Rd , h2κ ). We define it in the Dunkl transform side. Definition 6.2.3. Let y ∈ Rd be given. The generalized translation operator f → τy f is defined on L2 (Rd ; h2κ ) 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 properties of the usual translation operator is obvious; for example, boundedness in L p , translation invariance, or positivity. Some of these properties will be studied below and in the next section. 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)

6.2. Dunkl transform: L1 theory

73

Proposition 6.2.4. For t > 0 and x ∈ Rd , τy e−tx = e−t(x 2

2 +y2 )

√ √ E( 2tx, 2ty).

Proof. This follows immediately from (6.1.1), (6.1.4) and (6.2.2).

(6.2.3) 

Proposition 6.2.5. Assume that f , g ∈ S . Then 

(1)

Rd

τy f (ξ )g(ξ )h2κ (ξ )dξ =



Rd

f (ξ )τ−y g(ξ )h2κ (ξ )dξ .

(2) τy f (x) = τ−x f (−y). Proof. The property (2) follows from the definition, since E(λ x, ξ ) = E(x, λ ξ ) for any λ ∈ C. If f , g ∈ S , then both integrals in (1) are well defined. From the definition,     2 2 τy f (ξ )g(ξ )hκ (ξ )dξ = bh E(ix, ξ )E(−iy, ξ )Fκ f (ξ )hκ (ξ )dξ g(x)h2κ (x)dx Rd

=

Rd



Rd

Rd

Fκ f (ξ )Fκ g(−ξ )E(−iy, ξ )h2κ (ξ )dξ

by the inversion theorem applied to g. We also have    2 2 f (ξ )τ−y g(ξ )hκ (ξ )dξ = bh E(ix, ξ )E(iy, ξ )Fκ g(ξ )hκ (ξ )dξ f (x)h2κ (x)dx



Rd

= =

Rd



Rd



Rd

Rd

Fκ f (−ξ )Fκ g(ξ )E(iy, ξ )h2κ (ξ )dξ Fκ f (ξ )Fκ g(−ξ )E(−iy, ξ )h2κ (ξ )dξ

by the inversion theorem applied to f . This proves (1).



{x ∈ Rd

: x ≤ R}. Then τy f is supported

Proposition 6.2.6. Let f ∈ S be supported in in {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 ) extends to Cd as an entire function of type R + y. Hence, the stated result follows from Corollary 6.1.11.  Theorem 6.2.7. If f ∈ C0∞ (Rd ) is supported in x ≤ R, then τy f − f  p ≤ c f y(R + y)

2λκ +2 p

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 .

74

Chapter 6. Dunkl Transform

Using the mean value theorem and estimates on the derivatives of E(x, iξ ), we obtain the estimate  ξ  |Fκ f (ξ )|h2κ (ξ )dξ . τy f − f ∞ ≤ cy Rd

As f is supported in x ≤ R and τy f is supported in x ≤ (R + y), we can restrict the integration domain above to x ≤ (R + y) and conclude, accordingly, that τy f − f  p ≤ c f y(R + y)

2λκ +2 p

, 

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)τx g$(y)h2κ (y)dy,

where g$(y) := g(−y). Since τx g$ ∈ 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 qtκ (x) = (2t)−λk +1 e−tx denotes the heat kernel. Changing variable shows that bh Rd qtκ (x)h2κ (x)dx = 1. 2

Lemma 6.2.9. For f ∈ L1 (Rd ; h2κ ), lim  f ∗κ qtκ − f κ,1 = 0.

t→0+

Proof. By (1) in Proposition 6.2.5 with f = qtκ and g = 1, bh Since τu qtκ ≥ 0 by (6.2.3), it follows then that



κ 2 Rd τx qt (y)hκ (y)dy

= 1.

 f ∗ qtκ κ,1 ≤  f κ,1 . For a given ε > 0 we choose g ∈ S such that g − f κ,1 < ε/3. The triangle inequality then leads to 2  f ∗κ qtκ − f κ,1 ≤ ε + g ∗κ qtκ − gκ,1 . 3 Since g ∈ S , it follows that g ∗κ qtκ (x) =



Rd

2 κ g(y)τx (q* t )(y)hκ (y)dy =



Rd

τ−x g(y)qtκ (−y)h2κ (y)dy.

We also know that τ−x g(y) = τ−y g(x). Therefore, g ∗κ qtκ (x) =



Rd

τy g(x)qtκ (y)h2κ (y)dy.

6.2. Dunkl transform: L1 theory

75

In view of this, 

g ∗κ qtκ (x) − g(x) =

Rd

(τy g(x) − g(x)) qtκ (y)h2κ (y)dy,

which implies then g ∗κ qtκ − gκ,1 ≤

 Rd

τy g − gκ,1 |qtκ (y)|h2κ (y)dy.

If g is supported in x ≤ R, then the estimate in Theorem 6.2.7 gives g ∗κ qtκ − gκ,1 ≤ c



y (R + y)2λκ +2 |qtκ (y)|h2κ (y)dy

Rd



≤ ct

y (R + ty)2λk +1 e−y h2κ (y)dy, 2

Rd

which can be made smaller than ε/3 by choosing ε small. This completes the proof of the lemma.  Theorem 6.2.10. If both f and Fκ f ∈ L1 (Rd , h2κ ), then for almost all x ∈ Rd , f (x) = bh



Rd

Fκ f (y)E(ix, y)h2κ (y)dy.

Proof. Since Fκ qtκ (x) = e−x , for f ∈ S we have 2



f ∗ qtκ (x) = bh

Fκ f (y)e−ty E(ix, y)h2κ (y)dy. 2

Rd

This extends to f ∈ L1 (h2κ ; Rd ) since the convolution operator extends to L1 (h2κ ; Rd ) as a bounded operator by  f ∗κ qtκ κ,1 ≤  f κ,1 . Letting t → 0+ , applying Lemma 6.2.9 to the left-hand side and the dominant convergence theorem to the right-hand side, we see that the inversion formula follows almost everywhere.  For the convenience of applications, we summarize some of the most useful properties 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 Fκ f (ξ ) = 0. ξ →∞

(ii) The Dunkl transform Fκ is an isomorphism of the Schwartz class S (Rd ) onto itself, and Fκ2 f (x) = f (−x). (iii) The Dunkl transform Fκ on S (Rd ) extends uniquely to an isometric isomorphism on L2 (Rd ; h2κ ), 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 .

76

Chapter 6. Dunkl Transform

(v) If f , g ∈ L2 (Rd ; h2κ ), then 

Fκ f (x)g(x) h2κ (x) dx Rd

=

 Rd

f (x)Fκ g(x) h2κ (x) dx.

(vi) Given ε > 0, let fε (x) = ε −2−2γκ f (ε −1 x). 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 1 Hα g(s) = Γ(α + 1)

 ∞

g(r) 0

Jα (rs) 2α+1 r dr, (rs)α

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 define a class of functions on which (6.2.1) holds pointwisely: Aκ (Rd ) := { f ∈ L1 (Rd ; h2κ ) : 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 ; h2κ ) is bounded. Then 

(1)

Rd

τy f (ξ )g(ξ )h2κ (ξ )dξ =



f (ξ )τ−y g(ξ )h2κ (ξ )dξ .

Rd

(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 ; h2κ ) ∩ L∞ . Since g ∈ L2 (Rd ; h2κ ), τy g is defined as an L2 function. As f is in L2 (Rd ; h2κ ) and bounded, both integrals are finite. The relation 

f (ξ )Fκ g(ξ )h2κ (ξ )dξ Rd

=

 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



Rd

τy f (−x)g(−x)h2κ (x)dx 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 expression. Hence (1) is proved. 

6.3. Generalized translation operator

6.3.1

77

Translation operator on radial functions

A function is called radial, if it depends only on x. For such a function, there is an explicit 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 ,   + y x 2 2  . x + y − 2x yx , · τy f (x) = Vκ f0 where x = y x Proof. Since f is radial, its Dunkl transform is also a radial function, which we denote by Fκ f0 (r). Using (6.2.1), the property (iv) of Theorem 6.2.11, and the spherical-polar coordinates ξ = rξ  , we get τy f (x) = bh

 ∞

r2λκ +1

0

 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 Hnd (h2κ ). Hence, Proposition 6.1.9 shows that ωd ωdκ

 Sd−1

E(x, −irξ  )Pn (h2κ ; y , ξ  )h2κ (ξ  )dσ (ξ  ) = (−i)n 2λκ (rx)−λκ Jn+λκ (rx)Pn (h2κ ; y , x ),

where x = x/x, which implies that in L2 (Sd−1 , h2κ ), E(x, −irξ  ) = cκ,d

∞

∑ (−i)n (rx)−λκ Jn+λκ (rx)Pn (h2κ ; ξ  , x ).

n=0

Replacing ξ  by −ξ  gives the expansion of E(x, irξ  ). Hence, using the reproducing property of Pn (h2κ ; ·, ·), we get ωd ωdκ

 Sd−1

E(x, −irξ  )E(y, irξ  )h2κ (ξ  )dσ (ξ  ) ∞

Jn+λκ (rx) Jn+λκ (ry) Pn (h2κ ; y , x ) λκ λκ (rx) (ry) n=0 , ∞ J n+λκ (rx) Jn+λκ (ry) n + λκ λκ   = cκ,d Vκ ∑ Cn (x , ·) (y ). λκ λκ λ (rx) (ry) κ n=0 = cκ,d

∑

By the addition formula for Bessel functions ([2, p. 215]), the last expression is equal to ,  Jλκ (r |x2 + y2 − 2x yx , ·) (y ), cVκ λ r κ (x2 + y2 − 2x yx , ·)λκ /2

78

Chapter 6. Dunkl Transform

where c is a constant. Consequently, we conclude that   ∞ Jλ (rz(x, y, ·)) τy f (x) = cVκ r2λκ +1 κ F f (r)dr (y ) κ 0 (rz(x, y, ·))λκ 0   = cVκ Hλκ Fκ f0 (z(x, y, ·)) (y ),  where z(x, y, ·) = x2 + y2 − 2x yx , · 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 of the Hankel transform (6.1.9) that τy f (x) = cVκ [ f0 (z(x, y, ·))](y ). The constant c can be determined 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 ; h2κ ) and   τy f (x)h2κ (x)dx = f (x)h2κ (x)dx. (6.3.1) Rd

Rd

Proof. As f is radial, the explicit formula in Proposition 6.3.2 shows that τy f ≥ 0 since 2 Vκ is a positive operator. Taking g(x) = e−tx and making use of (6.2.3) we obtain from (1) of Proposition 6.2.5 that  Rd

τy f (x)e−tx h2κ (x)dx = 2

 Rd

f (x)e−t(x

2 +y2 )

√ √ E( 2tx, 2ty)h2κ (x)dx.

As |E(x, y)| ≤ ex y , we can take limit as t → 0 to get 

lim

t→0 Rd

τy f (x)e−tx h2κ (x)dx = 2

 Rd

f (x)h2κ (x)dx.

Since τy f ≥ 0, the monotone convergence theorem applied to the integral on the left completes 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 qtκ is also radial and nonnegative. Since f is both in L1 (Rd ; h2κ ) and L2 (Rd ; h2κ ), f ∗κ qtκ ∈ L1 (Rd , h2κ ) because qtκ ∈ Aκ (Rd ) and, by the Plancherel theorem and H¨older’s inequality, Fκ f ∗κ qtκ κ,1 = Fκ f · Fκ qtκ κ,1 ≤  f κ,2 qtκ κ,2 . Hence, f ∗κ qtκ ∈ Aκ (Rd ). Thus, by Theorem 6.3.3, τy ( f ∗κ qtκ )(x) ≥ 0. Since f ∈ L2 (Rd ; h2κ ), it is easy to see that  f ∗κ qtκ − f κ,2 → 0. Since τy is bounded on L2 (Rd ; h2κ ), we have τy ( f ∗κ qt ) → τy f in L2 (Rd ; h2κ ) as t → 0. By passing to a subsequence if necessary, we can assume that the convergence is also almost everywhere. This gives us lim τy ( f ∗κ qt )(x) = τy g(x) ≥ 0

t→0

6.3. Generalized translation operator

79

for almost every x. Since τy f is nonnegative, we let t → 0 and can apply the monotone convergence theorem to   √ √ 2 2 2 τy f (x)e−tx h2κ (x)dx = f (x)e−t(x +y ) E( 2tx, 2ty)h2κ (x)dx Rd

Rd



and get (6.3.1). p Let Lrad (Rd ; h2κ ) denote the space of all radial functions in L p (Rd ; h2κ ).

Theorem 6.3.5. The generalized translation operator τy can be extended to all radial p (Rd , h2κ ) → L p (Rd ; h2κ ) is a bounded functions in L p (Rd ; h2κ ), 1 ≤ p ≤ 2, and τy : Lrad operator. 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 ≤ p (Rd ; h2κ ). This proves the theorem.   f κ,p for all 1 ≤ p ≤ 2 and all f ∈ Lrad 1 (Rd ; h2 ), Theorem 6.3.6. For every f ∈ Lrad κ



τy f (x)h2κ (x)dx Rd



=

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 ; h2κ ). Since   τy fn (x)g(x)h2κ (x)dx = fn (x)τ−y g(x)h2κ (x)dx for every g ∈ Aκ

Rd d (R ) we



Rd

Rd

get, taking limit as n tends to infinity,

τy f (x)g(x)h2κ (x)dx =



f (x)τ−y g(x)h2κ (x)dx.

Rd

Now take g(x) = e−tx and take the limit as t goes to 0. Since τy f ∈ L1 (Rd ; h2κ ), the dominated convergence theorem shows that 2

 Rd

for f ∈ L1 (Rd ; h2κ ).

τy f (x)h2κ (x)dx =

 Rd

f (x)h2κ (x)dx 

For non-radial functions, say in the Schwartz class, it is known that τy is not positive when the group G is either Zd2 or the symmetric group, and this should be the case for all other reflection groups. It remains an open problem if τy f can be defined for all f ∈ L1 (Rd ; h2κ ) when the group G is not Zd2 . The case G = Zd2 is discussed in the next subsection.

80

Chapter 6. Dunkl Transform

6.3.2

Translation operator for G = Zd2

Recall that the weight function hκ , invariant under the group Zd2 , takes the form d

hκ (x) = ∏ |xi |κi ,

κi ≥ 0.

i=1

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) =

1 2

 1

+

−1

f



 1 1

2

−1

x2 + y2 − 2xyt

1+ 

x−y

Φκ (t)dt

x2 + y2 − 2xyt 



x−y f − x2 + y2 − 2xyt 1 −  Φκ (t)dt, x2 + y2 − 2xyt

(6.3.2)

where Φκ (t) = bκ (1 + t)(1 − t 2 )κ−1 . Proof. In this case, f radial means that f is an even function. Using the explicit formula of Vκ in (2.3.2), the formula in Theorem 6.3.2 shows that if f is even, then  1 

f x2 + y2 − 2xyt Φκ (t)dt. τy f (x) = −1

Making use of the fact that the derivative of an even function is odd, we derive a formula for τy f  using the fact that Dτy = τy D. In this simple case the Dunkl operator D is given by f (x) − f (−x) D f (x) = f  (x) + κ . x On the one hand, since a radial function is invariant under the difference part, we have Dτy f (x) = τy D f (x) = τy f  (x). On the other hand, for f even, a simple computation shows that  

τy f (x) − τy f (−x) 1 1 t d  2 = f x + y2 − 2xys ds Φκ (t)dt x x −1 −t ds

  1  t f 

x2 + y2 − 2xys  ds Φκ (t)dt = −y −1 −t x2 + y2 − 2xys

  1  t f 

x2 + y2 − 2xys  = −2ybκ ds t(1 − t 2 )κ−1 dt 0 −t x2 + y2 − 2xys

  1 f

x2 + y2 − 2xys  1  t(1 − t 2 )κ−1 dt ds = −ybκ −1 |s| x2 + y2 − 2xys

6.3. Generalized translation operator

=−

  y 1 f

κ

−1

81

 x2 + y2 − 2xys  (1 − s)Φκ (s)ds. x2 + y2 − 2xys

Together with the formula for τy f  (x), this leads to  1 

x−y f x2 + y2 − 2xys  Dτy f (x) = Φκ (s)ds. 2 −1 x + y2 − 2xys Consequently, replacing f  by any odd function fo , we conclude that  1 

x−y fo x2 + y2 − 2xys  Φκ (s)ds. τy fo (x) = −1 x2 + y2 − 2xys Any function f can be written as f = fe + fo , where fe (x) = ( f (x) + f (−x))/2 is the even part and fo (x) = ( f (x) − f (−x))/2 is the odd part, from which the stated formula follows.  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) = ∏di=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 4κ2 x1 y1 + x2 y2 . 2κ1 2κ2

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 operator in the case of Zd2 . Theorem 6.3.9. For each y ∈ Rd , the generalized translation operator τy is a bounded operator on L p (Rd , h2κ ). 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 case d = 1. We have  1 

x−y Φκ (t)dt f x2 + y2 − 2xyt 1 +  −1 x2 + y2 − 2xyt  1 

|x − y| ≤  f ∞ + cκ f x2 + y2 − 2xyt  (1 + t)(1 − t 2 )κ−1 dt. −1 x2 + y2 − 2xyt

82

Chapter 6. Dunkl Transform

Since (x − y)(1 + t) = (x − yt) − (y − xt), it follows that |x − y|  (1 + t) ≤ 2. 2 x + y2 − 2xyt Consequently, the above integral is bounded by 3 f ∞ . Hence, by the explicit formula of τy 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)h2κ (x)dx is a linear functional L on C0 (Rd ). Using the property (1) of Proposition 6.3.1 we get 

g(x)τy f (x)h2κ (x)dx Rd

=

 Rd

τ−y g(x) f (x)h2κ (x)dx

≤ τy g∞  f κ,1 ≤ 3g∞  f κ,1 , where we have used the fact that τy g∞ ≤ 3g∞ . Hence, L is a bounded linear functional on C0 (Rd ). By the Riesz representation theorem, τy f (x)dx is the unique regular measure providing 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

6.4.1

and

f ∗κ g = g ∗κ f .

(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 gκ,2 ≤ Fκ g∞  f k,2 ≤ gκ,1  f k,2 . Since we do not know if the generalized translation operator is bounded in L p (Rd ; h2κ ), the usual proof of Young’s inequality does not apply. For convolution with radial functions we 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 L p (Rd ; h2κ ), 1 ≤ p ≤ ∞, as a bounded operator. In particular,  f ∗κ gκ,p ≤ gκ,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 |τy g| ≤ τy |g|, which shows that  Rd

Therefore,

|τy g(x)|h2κ (x)dx ≤

 Rd

 Rd

|g(x)|h2κ (x)dx.

| f ∗κ g(x)|h2κ (x)dx ≤  f κ,1 gκ,1 .

We also have  f ∗κ g∞ ≤  f ∞ gκ,1 . By the Riesz–Thorin interpolation theorem, we  obtain  f ∗κ gκ,p ≤ gκ,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)h2κ (x)dx = 1. Then for f ∈ L p (Rd ; h2κ ), 1 ≤ p < ∞, and f ∈ C0 (Rd ), p = ∞, lim  f ∗κ φε − f κ,p = 0. ε→0

Proof. For a given η > 0 we choose g ∈ C0∞ such that g − f κ,p < η/3. The triangle inequality and (6.4.2) lead to 2  f ∗κ φε − f κ,p ≤ η + g ∗κ φε − gκ,p . 3 Since φ is radial, we can choose a radial function ψ ∈ C0∞ such that φ − ψκ,1 ≤ (12gκ,p )−1 η. If we let a = ch

Rd

ψ(y)h2κ (y)dy, then, by the triangle inequality and (6.4.2),

g ∗κ φε − gκ,p ≤ gκ,p φ − ψκ,1 + g ∗κ ψε − agκ,p + |a − 1|gκ,p ≤ η/6 + g ∗κ ψε − agκ,p , since gκ,p φ − ψκ,1 ≤ |a − 1| = ch

η 12

and

 Rd

(φε (x) − ψε (x)) h2κ (x)dx ≤ (12gκ,p )−1 η.

Thus, 5  f ∗κ φε − f κ,p ≤ η + g ∗κ ψε − agκ,p . 6

84

Chapter 6. Dunkl Transform

Hence it suffices to show that g ∗κ ψε − agκ,p ≤ η/6. Now g ∈ Aκ (Rd ), hence g ∗κ φε (x) =

 Rd

g(y)τx φ$ε (y)h2κ (y)dy =

 Rd

τ−x g(y)φε (−y)h2κ (y)dy.

We also know that τ−x g(y) = τ−y g(x), as g ∈ C0∞ . Therefore, 

g ∗κ φε (x) =

τy g(x)φε (y)h2κ (y)dy.

Rd

In view of this g ∗κ ψε (x) − ag(x) =

 Rd

(τy g(x) − g(x)) ψε (y)h2κ (y)dy,

which gives, by Minkowski’s integral inequality, g ∗κ ψε − agκ,p ≤

 Rd

τy g − gκ,p ψε (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 in Theorem 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 from Plancherel’s theorem. Let us study the convergence of Tε f as ε → 0. Note that T0 f = f by the inversion formula for the Dunkl transform. If Tε f can be extended to all f ∈ L p (Rd ; h2κ ) and if Tε f → f in L p (Rd ; h2κ ), we say that 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, then Tε 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 ) extend to L2 (Rd ; h2κ ) 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 = ch

 Rd



Rd

Fκ τ−x f (y)Φ(−εy)h2κ (y)dy τ−x f (ξ )ch

= ch ε −(d+2γκ )



Rd



Rd

Φ(−εy)E(y, −iξ )h2κ (y)dyh2κ (ξ )dξ

τ−x f (ξ )Fκ Φ(−ε −1 ξ )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 ; h2κ ) is bounded and Φ(0) = 1. For f ∈ L p (Rd ; h2κ ), Tε f converges to f in L p (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 L p (Rd ; h2κ ), Tε f can be extended to L p (Rd ; h2κ ). The convergence of Tε f to f now follows from Theorem 6.4.2.  By choosing specific radial functions Φ, we consider several examples of summability methods. Heat kernel transform. We consider f ∗κ qtκ , where qtκ 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 ∈ L p (Rd ; h2κ ), 1 ≤ p < ∞ or f ∈ C0 (Rd ), p = ∞. 1. The heat transform Ht f (x) := ( f ∗κ qtκ )(x) = ch

 Rd

f (y)τy qtκ (x)h2κ (y)dy,

t > 0,

converges to f in L p (Rd ; h2κ ) as t → 0. 2. Define H0 f (x) = f (x). Then the function Ht f (x) solves the initial value problem Δh u(x,t) = ∂t u(x,t),

u(x, 0) = f (x),

(x,t) ∈ Rd × [0, ∞).

Proof. By (6.1.4), Fκ qtκ = e−tx , so that (1) follows by setting Φ(x) = e−x and ε = √ t. Using the spherical-polar form of Δh in (3.1.5), it is easy to see that qtκ satisfies the  heat equation Δh u(x,t) = ∂t u(x,t), so that (2) holds. 2

2

Poisson integral. The Poisson kernel is defined by P(x, ε) := cd,κ

ε

3 ,

(ε 2 + x2 )λk + 2

cd,κ = 2λκ +1

Γ(λκ + 32 ) √ . π

(6.4.3)

86

Chapter 6. Dunkl Transform

Theorem 6.4.6. Suppose f ∈ L p (Rd ; h2κ ), 1 ≤ p < ∞, or f ∈ C0 (Rd ), p = ∞. Then the Poisson integral f ∗κ Pε converges to f in L p (Rd ; h2κ ). Proof. Set Φ(x) = e−x . Then P(x, 1) = Fκ Φ(x) and P(x, ε) = ε −λκ −2 P(x/ε, 1). Indeed, this can be proved as for the ordinary Fourier transform using the integral relation between 2 2 2 e−t and e−t in (3.4.8) and the fact that the Dunkl transform of e−x /4 is e−x , as shown by (6.1.4). Since Φ(0) = 1, it readily follows that ch Rd P(x, ε)h2κ (x)dx = 1. Thus, the convergence is established by Theorem 6.4.4.  Bochner–Riesz means. Here we consider ' (1 − x2 )δ , x ≤ 1, Φ(x) = 0, otherwise, where δ > 0. As in the case of the ordinary Fourier transform, we take ε = 1/R where R > 0. Then the Bochner–Riesz means of order δ are defined by SRδ f (x) := ch

 y≤R

 1−

y R

δ

Fκ f (y)E(ix, y)h2κ (y)dy.

Theorem 6.4.7. If f ∈ L p (Rd ; h2κ ), 1 ≤ p ≤ ∞ and δ > SRδ f − f κ,p → 0,

d−1 2

(6.4.4)

+ γκ , then

as R → ∞.

Proof. The proof follows as in the case of ordinary Fourier transform [52, p. 171]. From Theorem 6.1.8 and the properties of the Bessel function, we have Fκ Φ(x) = 2λκ x−λκ −δ −1 Jλκ +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) parameters in the weight function. If all parameters are zero, then hκ (x) ≡ 1 and we are back to the classical Fourier transform, for which the index (d − 1)/2 is the critical index for the Bochner–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 L p (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 , h2κ ) and g ∈ Lr (Rd , h2κ ). Then  f ∗κ gκ,p ≤ c f κ,q gκ,r .

6.5. Maximal function

87

The boundedness of τy allows us to remove the assumption that φ is radial in Theorem 6.4.2 when G = Zd2 . Theorem 6.4.9. Let φ ∈ L1 (Rd , h2κ ) with if 1 ≤ p < ∞, or f ∈ C0 (Rd ) if p = ∞,



Rd

φ (x)h2κ (x)dx = 1. Then for f ∈ L p (Rd , h2κ )

lim  f ∗κ φε − f κ,p = 0,

ε→0

1 ≤ p ≤ ∞.

Proof. For f ∈ L p (Rd , h2κ ) we write f = f1 + f2 , where f1 is in C0∞ 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 as ε → 0 by Theorem 6.2.7. This proves that τy f (x) − f (x)κ,p → 0 as y → 0. We have then 

ch

Rd

| f ∗κ gε (x) − f (x)| p h2κ (x)dx

= ch ≤ ch = ch





ch

Rd



Rd



Rd

p

Rd

(τy f (x) − f (x))gε (y)h2κ (y)dy h2κ (x)dx

p τy f − f κ,p |gε (x)|h2κ (x)dx p τεy f − f κ,p |g(x)|h2κ (x)dx,

which goes to zero as ε → 0.



All results on the summability for the inverse Dunkl transform in the previous section are proved for a generic reflection group, hence they all hold for the case of Zd2 .

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 its characteristic function. Definition 6.5.1. For f ∈ L1 (Rd ; h2κ ), we define the maximal function Mκ f by

Mκ f (x) := sup r>0

Rd

f (y)τx χBr (y)h2κ (y)dy . 2 Br hκ (y)dy

88

Chapter 6. Dunkl Transform

The function χBr is radial. If ϕ ∈ C0∞ (Rd ) is a radial function such that χBr (x) ≤ ϕ(x), then, by Theorem 6.3.3, τy χBr (x) ≤ τy ϕ(x). As τy ϕ is bounded, τy χBr is bounded and compactly supported so that it belongs to L1 (Rd ; h2κ ). Hence, Mκ f is well defined for f ∈ L1 (Rd ; h2κ ). Using spherical polar coordinates, we also have  Br

h2κ (y)dy =

 r



sd−1+2γκ ds

Sd−1

0

h2κ (ξ )dσ (ξ ) =

ωdκ r2λκ +2 . 2λκ + 2

By definition, we can also write Mκ f as Mκ f (x) = sup r>0

1 dκ

r2λκ +2

| f ∗κ χBr (x)|,

dκ :=

ωdκ . 2λκ + 2

Since τy χBr ≥ 0, we have Mκ f (x) ≤ Mκ | f |(x). Theorem 6.5.2. The maximal function is bounded on L p (Rd ; h2κ ) for 1 < p ≤ ∞; moreover it is of weak type (1, 1), that is, for f ∈ L1 (Rd ; h2κ ) and α > 0, 

E(a)

h2κ (x)dx ≤

c  f κ,1 , α

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 the Poisson kernel defined in (6.4.3), and let σ := 2λκ + 3. For j ≥ 0, define Br, j := {x : 2− j−1 r ≤ x ≤ 2− j r}. Then χBr, j (y) = (2− j r)σ (2− j r)−σ χBr, j (y) ≤ c(2− j r)σ −1

2− j r χB (y) ((2− j r)2 + y2 )σ /2 r, j

≤ c(2− j r)σ −1 P2− j r (y), where c is a constant independent of r and j. Since both χBr and Pε are bounded, integrable radial functions, it follows from Theorem 6.3.3 that τx χBr, j (y) ≤ c(2− j r)σ −1 τx P2− j r (y). This shows that for any positive integer m  Rd

m

∞

f (y) ∑ τx χBr, j (y)h2κ (y)dy ≤ c ∑ (2− j r)σ −1 j=0

j=0

 Rd

f (y)τx P2− j r (y)h2κ (y)dy

≤ c rd+2γκ sup f ∗κ Pt (x). t>0

1 (Rd ; h2 ) As ∑mj=0 χBr, j (y) converges to χBr (y) in L1 (Rd ; h2κ ), the boundedness of τx on Lrad κ m 1 d 2 shows that ∑ j=0 τx χBr, j (y) converges to τx χBr (y) in L (R ; hκ ). By passing to a subsequence if necessary, we can assume that ∑mj=0 τx χBr, j (y) converges to τx χBr (y) for almost

6.5. Maximal function

89

every y. Thus all the functions involved are uniformly bounded by τx χBr (y). This shows  that ∑mj=0 τx χBr, j (y) converges to τx χBr (y) in L p (Rd ; h2κ ) and hence 

m

lim

m→∞ Rd

f (y) ∑ τx χBr, j (y)h2κ (y)dy =



j=0

Rd

f (y)τx χBr (y)h2κ (y)dy.

Thus we have proved f ∗κ χBr (x) ≤ crd+2γκ sup f ∗κ Pt (x), t>0

cP∗ f (x),

where P∗ f (x) = supt>0 f ∗κ Pt (x) is the which gives the inequality Mκ f (x) ≤ maximal function associated to the Poisson semi-group. Therefore, it is enough to prove the boundedness of P∗ f . By looking at the Dunkl transforms of the Poisson kernel and the heat kernel we conclude, as in the proof of Lemma 3.4.7, that t f ∗κ Pt (x) = √ 2π

 ∞ 0

( f ∗κ qs )(x)e−t

2 /2s

s−3/2 ds,

which allows us to conclude that P∗ f (x) ≤ c sup t>0

1 t

 t 0

Qs (| 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 on  L p (Rd ; h2κ ) for 1 < p ≤ ∞, and the weak type (1, 1). The maximal function can be used to study almost everywhere convergence of the convolution 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 . Then sup | f ∗κ φε (x)| ≤ cMκ f (x). ε>0

Consequently, f ∗κ φε (x) → f (x) for almost every x as ε goes to 0, for all f in L p (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) ≤ c

m

∑

(1 + 2 j )−2λκ −3 ε −2λκ −2 .

j=−m

90

Chapter 6. Dunkl Transform

This shows that   f (y)τx φε (y) Rd



m

∑

j=−m

χε2 j ≤y≤ε2 j+1 (y)h2κ (y)dy m

∑

≤ cε −2λκ −2

(1 + 2 j )−2λκ −3 (ε2 j )2λκ +2 Mκ f (x) ≤ cMκ f (x).

j=−m

Since |φ (y)| ≤ c(1 + y)−2λκ −3 ≤ cP1 (y) it follows that |τx φ (y)| ≤ cτx P1 (y) is bounded. Arguing as in the previous theorem, we can show that the left-hand side of the above inequality converges to f ∗κ φε (x). Thus we obtain sup | f ∗κ φε (x)| ≤ cMκ f (x), ε>0

from which the proof of the almost everywhere convergence follows via the standard argument. 

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 ωdκ

 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 sphericalpolar coordinates, the generalized convolution f ∗κ g can be expressed in terms of the spherical mean: ( f ∗κ g)(x) = ch = ch =

 d R∞

0

ch ωdκ

τy f (x)g(y)h2κ (y)dy 

r2λκ +1 g0 (r) τry f (x)h2κ (y )dy dr

 ∞ 0

Sd−1

Sr f (x)g0 (r)r2λκ +1 dr.

The spherical mean operator is bounded. Theorem 6.5.4. Let G = Zd2 . For f ∈ L p (Rd , h2κ ), Sr f κ,p ≤ c f κ,p ,

1 ≤ p ≤ ∞.

Furthermore, Sr f − f κ,p → 0 as r → 0+ . Proof. Using H¨older’s inequality, |Sr f (x)| p ≤

1 ωdκ

 Sd−1

|τry f (x)| p h2κ (y)dσ (y).

6.5. Maximal function

91

Hence, a simple computation shows that 

ch

Rd

|Sr f (x)| p h2κ (x)dx ≤ ch =



1 ωdκ

Rd



1 ωdκ

Sd−1

 Sd−1

|τry f (x)| p h2κ (y)dω(y) h2κ (x)dx

p τry f κ,p h2κ (y)dσ (y)

≤ c f κ,p . Furthermore, p Sr f − f κ,p ≤



1 ωdκ

Sd−1

p τry f − f κ,p h2κ (y)dσ (y),

which goes to zero as r → 0, since τry f − f κ,p → 0.



Zd2 .

As-

Theorem 6.5.5. Set G = sume that φ0 is differentiable,

Let φ (x) = φ0 (x) ∈ L1 (Rd ; h2κ ) be a radial function. 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)h2κ (x)dx = 1, then

1. for 1 ≤ p ≤ ∞, f ∗κ φε converges to f as ε → 0 in L p (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 r 2λ +1 t κ St f (x)dt Mκ f (x) = sup 0 r 2λ +1 . r>0

t

0

κ

dt

Since |Mκ f (x)| ≤ cMκ | f |(x), we can assume f (x) ≥ 0. The assumption on φ0 shows that lim φ0 (r)

r→∞

 r 0

St f (x)t 2λκ +1 dt = lim φ0 (r) r→∞

= lim φ0 (r) r→∞

 Rd



Rd

τy f (x)h2κ (y)dy f (y)h2κ (y)dy = 0.

Hence, using the spherical-polar coordinates and integrating by parts, we get  ∞

φ0 (r)r2λκ +1 Sr f (x)dr  ∞  r

St f (x)t 2λκ +1 dt φ  (r)dr, =−

( f ∗κ φ )(x) =

0

0

0

which implies that |( f ∗κ φ )(x)| ≤ cMκ f (x)

 ∞ 0

r2λκ +2 |φ0 (r)|dr.

Boundedness of the last integral proves the maximal inequality.



92

Chapter 6. Dunkl Transform

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 the maximal function, even though τy f may not be nonnegative. Lemma 6.5.6. If f ∈ L1 (Rd , h2κ ) is a nonnegative function, then

2 Br τy f (x)hκ (y)dy . 2 r>0 Br hκ (y)dy

Mκ f (x) = sup

In particular, if f and g are two nonnegative functions, then Mκ f + Mκ g = Mκ ( f + g). Proof. Since τy χBr (x) is nonnegative, we have that ( f ∗κ χBr )(x) =

 Rd

f (y)τy χBr (x)h2κ (y)dy

is nonnegative if f is nonnegative. Hence, we can drop the absolute value symbol in the definition of Mκ f .  Theorem 6.5.7. Set G = Zd2 . Let φ ∈ L1 (Rd , h2κ ) 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)h2κ (x)dx = 1, then, for f ∈ L1 (Rd , h2κ ), ( 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 explicit formula for τy f . Let us first consider the case d = 1. Since ψ is an even function, τy ψ is given by  1 

τy f (x) = f x2 + y2 − 2xyt Φκ (t)dt, −1

according to (6.3.2). Since (x − y)(1 + t) = (x − yt) − (y − xt), we have |x − y|  (1 + t) ≤ 2. 2 x + y2 − 2xyt Consequently, by the explicit formula for τy f , see (6.3.2), the inequality |φ (x)| ≤ ψ(x) implies |τy φ (x)| ≤ τy ψ(x) + 2τy,1 ψ(x), where τy,1 ψ is defined by τy,1 ψ(x) = bκ

 1 −1

f



x2 + y2 − 2xyt (1 − t 2 )κ−1 dt.

6.5. Maximal function

93

Note that τy,1 ψ differs from τy ψ by the factor 1 + t in the weight function. The change of variables t → −t and y → −y in the integrals shows that  R

f (y)τy,1 ψ(x)h2κ (y)dy =

 R

F(y)τy ψ(x)h2κ (y)dy,

where F(y) = ( f (y) + f (−y))/2. It follows that |( f ∗κ φ )(x)| = ch

 R

f (y)τy φ (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 j e j (that is, multiplying the j-th component of x by δ j gives xδ j ). For 1 ≤ j ≤ d we define

∑

Fj1 ,..., jk = 2−k

f (xδ j1 · · · δ jk ).

(δ j1 ,...,δ jk )∈Zk2

In particular, Fj (x) = (F(x) + F(xδ j ))/2,

Fj1 , j2 (x) = (F(x) + F(xδ j1 ) + F(xδ j2 ) + F(xδ j1 δ j2 ))/4,

and the last sum is over Zd2 , F1,...,d (x) = 2−d ∑σ ∈Zd f (xσ ). Following the proof in the case 2 d = 1 it is not hard to see that d

|( f ∗κ φ )(x)| ≤ ( f ∗κ ψ)(x) + 2 ∑ (Fj ∗κ ψ)(x) + 4 j=1

∑

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 d

|( f ∗κ φ )(x)| ≤ Mκ f (x) + 2 ∑ Mκ Fj (x) + 4 j=1

∑

j1 = j2

Mκ Fj1 , j2 (x)

+ · · · + 2d Mκ F1,...,d (x). Since all Fj are clearly nonnegative, by Lemma 6.5.6, the last expression can be written 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}

h2κ (y)dy ≤ c

Hκ,1  f κ,1 ≤ cd . a a

Hence, f ∗κ φ is of weak type (1, 1), from which the almost everywhere convergence follows as usual. 

94

6.6

Chapter 6. Dunkl Transform

Notes and further results

The Dunkl transform was introduced in [27], where the L2 isometry was established as we have seen in Section 6.1. The Dunkl transform was studied in [34], where the bound for E(x, y) was established for general parameters κ with Re κ ≥ 0 and without the positivity of Vκ when κ is real and nonnegative, and where the main results of the L1 theory (Theorem 6.2.10) were also established. Our development in Section 6.2, based on the convolution operator, follows closely the approach of the classical harmonic analysis (see, for example, [52]). The generalized translation operator was studied in [39, 63] for smooth functions, and the starting point in [63] was the expression (x)

(y) 

τy f (x) = Vκ ⊗Vκ

 (Vκ−1 f )(x + y) ,

where Vκ−1 denotes the inverse of Vκ , which satisfies Vκ−1 f (x) = e−y,D f (x)|y=0 . This expression by itself, however, does not provide much useful information on τy f . The formula 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 in Section 6.3 follows from the latter paper. Our Section 6.4, convolution and summability also follows the treatment in [59]. Some of the summability method, such as the heat transform and Poisson transforms were studied earlier in [42, 45]. The maximal functions were 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 more general 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 transforms. For example, in the distributional sense Fκ



P(x) x2λκ +2+n−α

α = dn,κ

P(x) , xn+α

α dn,κ = i−n

2λκ +1−α Γ( n+α 2 ) n−α , Γ(λκ + 1 + 2 )

where P ∈ Hnd (h2κ ) and 0 < Re{α} < 2λκ + 2, which allows one to define analogues of the Riesz potentials and Bessel potentials for the Dunkl transforms and to study their boundedness in L p spaces [60]. For the latter purpose, however, we need the boundedness of the translation operator, which however is known, as shown in Section 6.3, to hold, only for G = Zd2 or radial functions. Thus, the boundedness can be established at this point only for G = Zd2 ; 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 it agrees with the Hankel transform. For general functions, the difference part comes in and needs to be dealt with. Nevertheless, the structure is relatively simple and many tools in harmonic analysis are accessible. There are numerous papers on Dunkl transforms on the real 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 a transference theorem between the L p multiplier of h-harmonic expansions on Sd and that of the Dunkl transform. This theorem is stated together with some related definitions and notations in Section 7.1. The proof of this transference theorem is, however, rather long, 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 the Dunkl transform on Rd from the corresponding results for the h-harmonic expansions on Sd . This is done in the last two sections, 7.5 and 7.6. More precisely, in Section 7.5, the transference theorem combined with Theorem 4.4.2, Theorem 4.5.2 is used to establish a H¨ormander type multiplier theorem and the Littlewood–Paley inequality for the Dunkl transform on Rd . In Section 7.5, we apply the transference theorem and Theorem 3.3.6 to deduce the convergence of the Bochner–Riesz means of order above the critical index in the weighted L p spaces for the group G = Zd2 .

7.1

Introduction

Let hκ denote the weight function on Rd defined by (2.1.2), invariant under a finite reflection group G generated by a reduced root system R in Rd . Throughout this chapter, the root system R is normalized so that α, α = 2 for all α ∈ R, and κ denotes a nonnegative multiplicative function on R. For each g ∈ G, we denote by g the reflection on Rd+1 given by 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 system R := {(α, 0) : α ∈ R}. Let κ  denote the nonnegative multiplicity function on R given by κ  (α, 0) = κα for α ∈ R. We denote by Vκ  the intertwining operator on C(Rd+1 )

© Springer Basel 2015 F. Dai, Y. Xu, Analysis on h-Harmonics and Dunkl Transforms, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0887-3_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 Hnd+1 (h2κ  ) de notes the space of h-harmonics of degree n on the sphere Sd , and projκn : L2 (Sd ; h2κ  ) → Hnd+1 (h2κ  ) denotes the orthogonal projection onto the space Hnd+1 (h2κ  ). We will keep the notations G , κ  , hκ  and Vκ  throughout this chapter. Also, recall that for a given 1 ≤ p ≤ ∞, L p (Rd ; h2κ ) denotes the weighted Lebesgue space on Rd endowed with the norm  1 p p 2 | f (y)| hκ (y) dy ,  f κ,p := Rd

with the usual change when p = ∞. One of the main goals in this chapter is to show the following transference theorem between the L p multiplier of h-harmonic expansions on Sd and that of the Dunkl transform. 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 ; h2κ  ) given by 



projκn (Uε f ) = m(εn) projκn f , n = 0, 1, . . . .

(7.1.1)

sup Uε f L p (Sd ;h2 ) ≤ A f L p (Sd ;h2 ) , ∀ f ∈ C(Sd )

(7.1.2)

Assume that ε>0

κ

κ

for some 1 ≤ p ≤ ∞. Then the function m( · ) defines an L p (Rd ; h2κ ) multiplier; that is, Tm f L p (Rd ;h2κ ) ≤ cd,κ A f L p (Rd ;h2κ ) , ∀ 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 in Sections 7.2, 7.3 and 7.4, respectively. The first part contains several technical lemmas that are crucial for the proof. The second part proves the conclusion under the additional assumption that |m(t)| ≤ c1 e−c2 t for all t > 0 and some c1 , c2 > 0. The third part shows how 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. The first 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 ) dμxκ (ξ ),

(7.2.1)

where dμxκ 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. To show the first equality, we set Vκ,1 f (x, xd+1 ) = Vκ [ f (·, xd+1 )](x) for f ∈ C(Rd+1 ) and x ∈ Rd . Since Vκ  is a linear operator uniquely determined by (2.3.1), it suffices to show that the following conditions are satisfied: Vκ,1 (Pnd+1 ) ⊂ Pnd+1 , 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 introduced in Definition 2.2.1 to emphasize their dependence on the multiplicative function κ. Indeed, these conditions can be easily verified using the properties of Vκ in (2.3.1), and the following identities, which follow directly from (2.2.1):   Dκ  ,i g(x, xd+1 ) = Dκ,i g(·, xd+1 ) (x), 1 ≤ i ≤ d, Dκ  ,d+1 g(x, xd+1 ) = ∂d+1 g(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 = xξ ∈ Rd and ξ ∈ Sd−1 .

Given N ≥ 1, we denote by NSd := {x ∈ Rd+1 : x = N} the sphere of radius N in Rd+1 , and define the mapping ψN : Rd → NSd by  x  x x , N cos ψN (x) := Nψ = Nξ sin (7.2.2) N N N with x = xξ ∈ Rd and ξ ∈ Sd−1 . Lemma 7.2.2. If f : NSd → R is supported in the set {x ∈ NSd : arccos(N −1 xd+1 ) ≤ 1}, then       2 sin(x/N) 2λκ +1 2 −2λκ −2 f (Nx)hκ  (x) dσ (x) = N f ψN (x) hκ (x) dx, x/N Sd B(0,N) where B(0, N) = {y ∈ Rd : y ≤ N}, and λκ =

d−2 2

+ |κ|.

Proof. First, using the polar coordinates transformation (ξ , θ ) ∈ Sd−1 × [0, π] → x := (ξ sin θ , cos θ ) ∈ Sd ,

98

Chapter 7. Multiplier Theorems for the Dunkl Transform

and the fact that dσ (x) = sind−1 θ dθ dσ (ξ ), we obtain  Sd

f (Nx)h2κ  (x) dσ (x)  π   = f (Nξ sin θ , N cos θ )h2κ  (ξ sin θ , cos θ ) dσ (ξ ) (sin θ )d−1 dθ 0 Sd−1  1    sin θ d−1+2γκ = f (Nξ sin θ , N cos θ )h2κ (θ ξ ) dσ (ξ ) θ d−1 dθ , θ 0 Sd−1

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 −1 xd+1 ) ≤ 1}. Using the usual spherical coordinates transformation in Rd , the last double integral equals      Ny sin y sin y 2λκ +1 , N cos y h2κ (y) f dy y y y≤1      sin(x/N) 2λκ +1 x x 2 x −d−2γκ =N sin , N cos f N dx hκ (x) x N N x/N x≤N    sin(x/N) 2λκ +1 −2λκ −2 2 =N f (ψN x)hκ (x) dx, x/N B(0,N) where the first step uses the homogeneity of the weight hκ and the change of variables y = x/N. This proves the desired formula.  Remark 7.2.3. It is easily seen that the restriction ψN

B(0,N) arccos(N −1 x

of the mapping ψN on

: B(0, N) is a bijection from B(0, N) to {x ∈ d+1 ) ≤ 1}. Thus, given a function f : B(0, N) → R, there exists a unique function fN supported in {x ∈ NSd : arccos(N −1 xd+1 ) ≤ 1} such that NSd

fN (ψN x) = 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 ,

 y  + x2 + y2 − 2yx, · τy f (x) = Vκ f0 . (7.2.5) y

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 |g j (t) − f0 (t)| ≤ 2− j

|t|≤2 j+1

"

2j

#− 1 s2λκ +1 ds

0

2

.

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 } is a sequence of radial Schwartz functions on Rd satisfying lim

sup

j→∞ 2− j ≤|t|≤2 j

| f j,0 (t) − f0 (t)| = 0

(7.2.6)

and lim  f j − f L2 (Rd ;h2κ ) = 0.

j→∞

(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

+ κ x2 + y2 − 2yx, ξ  dμy/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/ log 2] + 1. Since (x − y)2 ≤ x2 + y2 − 2y |x, ξ | ≤ (x + y)2 for all ξ  ≤ 1, it follows by (7.2.6) that +

+ x2 + y2 − 2yx, ξ  = f0 x2 + y2 − 2yx, ξ  lim f j,0 j→∞

uniformly for x ∈ An (y) and ξ  ≤ 1. This, together with (7.2.8) and Theorem 2.3.4 implies that 

+ κ lim τy ( f j )(x) = f0 x2 + y2 − 2yx, ξ  dμy/y (ξ ) j→∞ ξ ≤1 + 

 y x2 + y2 − 2yx, · =Vκ f0 y for every x ∈ An (y) \ {0} and n ≥ n0 (y). On the other hand, however, by (7.2.7), we have lim τy ( f j ) − τy f κ,2 = 0

j→∞

100

Chapter 7. Multiplier Theorems for the Dunkl Transform

for all y ∈ Rd . Thus,  +

 y , x2 + y2 − 2yx, · τy ( f )(x) = Vκ f0 y for almost every x ∈ An (y) and all n ≥ n0 (y). Finally, observing that the set Rd \

∞ 

An (y) = {x ∈ Rd : x = y}

n=n0 (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 dμxκ , Vκ F(rx) =

 Rd

F(rξ ) dμxκ (ξ ), for all F ∈ C(Rd ), x ∈ Rd , and r > 0.

Thus, (7.2.5) can be rewritten more symmetrically as

  + x2 + y2 − 2x, · (y). τy f (x) = Vκ f0

(7.2.9)

(7.2.10)

Lemma 7.2.6. Let Φ ∈ L1 (R, |x|2λκ +1 ) be an even, bounded function on R, and let TΦ be the operator L2 (Rd ; h2κ ) → 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

 ∞ 0

  J s x2 + y2 − 2x, ·  λκ 2λ +1 Φ(s)Vκ   λκ (y)s κ ds. 2 2 s x + y − 2x, ·

(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. Since Φ is an even function in L1 (R, |x|2λκ +1 )∩L∞ (R), it follows by the properties of the Hankel transform that g is a continuous radial function in L2 (Rd ; h2κ ) and Fκ g(ξ ) = Φ(ξ ). Thus, using (6.4.1), we have TΦ f (x) = f ∗κ g(x) =

 Rd

f (y)τy g(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 ; h2κ ), by Lemma 7.2.4 and Remark 7.2.5 it follows that  +

 x2 + y2 + 2x, · (y) K(x, y) : = τy g(x) = Vκ Hλκ Φ   ∞  J s x2 + y2 − 2x, ·  λκ 2λ +1 =c Φ(s)Vκ   λκ (y)s κ ds, 2 2 0 s x + y − 2x, · where the last step uses (2.3.4), the inequality Φ(s)

Jλκ (rs) ≤ c|Φ(s)| (rs)λκ

and Fubini’s theorem. This proves (7.2.11). The equality K(x, y) = K(y, x) follows from the fact that τx g(y) = τy g(x).  Our final lemma is a well-known result for ultraspherical polynomials (see, for instance, [53, (8.1.1), p.192]): Lemma 7.2.7. For z ∈ C and μ ≥ 0, z Γ(μ + 12 ) z −μ+ 12 μ lim k1−2μ Ck cos Jμ− 1 (z). = 2 k→∞ k Γ(2μ) 2

(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)| ≤ c1 e−c2 t for all t > 0 and some c1 , c2 > 0. By Lemma 7.2.6, the operator Tm has the integral representation  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)h2κ (y) dxdy ≤ cA 

(7.3.1)

whenever f ∈ L p (Rd ; h2κ ) and g ∈ L p (Rd ; h2κ ) both have compact supports and satisfy  f L p (Rd ;h2κ ) = gL p (Rd ;h2 ) = 1. Here and in what follows, 1p + p1 = 1. κ To this end, we choose a sufficiently large positive number N so that the supports of f and g are both contained in the ball B(0, N). By Remark 7.2.3, there exist functions fN and gN , both supported in {x ∈ NSd : arccos(N −1 xd+1 ) ≤ 1} and satisfying     fN ψN (x) = f (x), gN ψN (x) = g(x), x ∈ Rd , (7.3.2)

102

Chapter 7. Multiplier Theorems for the Dunkl Transform

where ψN is defined by (7.2.2). It is easily seen from (7.2.4) that  fN (N·)L p (Sd ;h2 ) ≤ cN −

2λκ +2 p

κ

,

gN (N·)L p (Sd ;h2 ) ≤ cN

− 2λκp+2

κ

Thus, using (3.2.1), (3.2.3), (7.1.1), and the assumption (7.1.2) with ε = IN := N 2λκ +2 ×

  ∞



∑ m(N −1 n)Pnκ (x, y)

Sd

Sd n=0

1 N,

.

we obtain

fN (Ny)gN (Nx)h2κ  (x)h2κ  (y) dσ (x) dσ (y)

≤ cA, 

where Pnκ (x, y) =

(7.3.3)

λκ +1/2 n+λκ +1/2  (x, ·)](y). λκ +1/2 Vκ [Cn

Setting

∞

∑ m(N −1 n)Pnκ

HN (x, y) = N −2λκ −2



n=0

y x ψ( ), ψ( ) , N N



and invoking (7.3.2) and Lemma 7.2.2, we obtain 

IN =





HN (x, y) f (y)g(x)h2κ (x)h2κ (y) Rd

Rd



×

sin(y/N) y/N

2λκ +1

 dx dy .

sin(x/N) x/N

2λκ +1 (7.3.4)

On the other hand, setting bN (ρ, x, y) = N

−2λκ −2

y n κ x )P ), ψ( ) m( ψ( ∑ N n N N n=0 ∞

we have HN (x, y) =

 ∞ 0

"

n+1 N n N

#−1 t

2λκ +1

dt

χ[ n , n+1 ) (ρ), N

N

bN (ρ, x, y)ρ 2λκ +1 dρ.

Hence, by (7.3.4), IN =



 ×

Rd

  Rd

∞

0

sin(x/N) x/N

bN (ρ, x, y)ρ 2λκ +1 dρ f (y)g(x)h2κ (x)h2κ (y)

2λκ +1 

sin(y/N) y/N

2λκ +1

(7.3.5)

 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−c2 ρ , where c is independent of x, y and N. Assertion 2. For any fixed x, y ∈ Rd and ρ > 0,  J ρu(x, y, ·)  λ lim bN (ρ, x, y) = cm(ρ)Vκ  κ λ (y), N→∞ ρu(x, y, ·) κ where u(x, y, ξ ) = κ.

(7.3.6)

 x2 + y2 − 2x, ξ , and c is a constant depending only on d and

For the moment, we take the above two assertions for granted, and proceed with the proof of Theorem 7.1.1. By Assertion 1 and H¨older’s inequality, we can apply the dominated convergence theorem to the integrals in (7.3.5), and obtain lim IN =



N→∞

 

Rd

Rd

∞

 lim bN (ρ, x, y)ρ 2λκ +1 dρ f (y)g(x)h2κ (x)h2κ (y) dx dy ,

0 N→∞

which, using Assertion 2, equals

  J ρu(x, y, ·)  λ 2λκ +1 2 2 c m(ρ)Vκ  κ dρ f (y)g(x)h (x)h (y)dx dy (y)ρ λ κ κ 0 Rd Rd ρu(x, y, ·) κ  

=c K(x, y) f (y)g(x)h2κ (x)h2κ (y) dx dy = cI, 

  

Rd

∞

Rd

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. n −c2 Nn ≤ ce−c2 ρ , and Assume that Nn ≤ ρ < n+1 N for some n ∈ Z+ . Then |m( N )| ≤ c1 e

n+1 N

n N

t 2λκ +1 dt ≥ cN −1 ρ 2λκ +1 . Hence,

|bN (ρ, x, y)| = N

−2λκ −2

x  y n m( )Pnκ ψ( ), ψ( ) N N N

≤ cN −2λκ −1 ρ −2λκ −1 e−c2 ρ

"

n+1 N n N

#−1 t

2λκ +1

dt

 y   n + λκ + 1/2 x λ +1/2 Vκ  Cn κ (ψ( ), ·) ψ( ) λκ + 1/2 N N

≤ c(Nρ)−2λκ −1 e−c2 ρ n2λκ +1 ≤ ce−c2 ρ , where we used (3.2.3) in the second step, and the positivity of Vκ and the estimate λ +1/2 |Cn κ (t)| ≤ cn2λκ in the third step. This proves Assertion 1.

104

n+1 N

Chapter 7. Multiplier Theorems for the Dunkl Transform Next, we show Assertion 2. A straightforward calculation shows that for and ρ > 0, " n+1 #−1 N N 2λκ +1 t dt = 2λ +1 (1 + oρ (1)), as N → ∞. n ρ κ N

This implies that for

n N

≤ρ ≤

n+1 N

n N

≤ρ ≤

and ρ > 0,

n2λκ +1 −2λκ −1 κ  x y

n Pn ψ ,ψ (1 + oρ (1)) N N (Nρ)2λκ +1  x  y y y λ +1/2 ψ sin , cos , · + oρ (1), = cm(ρ)n−2λκ Vκ  Cn κ N y N N   where the continuity of m is used in the first step, and n−2λκ −1 Pnκ ψ( Nx ), ψ( Ny ) ≤ c bN (ρ, x, y) = m(ρ)

n2λκ +1 = 1 in the last step. Thus, using Lemma 7.2.1 and (7.2.9), we obtain N→∞ (Nρ)2λκ +1 # "  x d x y 1 λκ +1/2 −2λκ sin Cn bN (ρ, x, y) = cm(ρ)n ∑ x j ξ j + cos N cos N x N j=1 Rd

and lim

× dμ κy = cm(ρ)n−2λκ

y



λ +1/2

ξ ≤y

where θN (x, y, ξ ) ∈ [0, π] satisfies " 1 cos θN (x, y, ξ ) = xy Since 1 cos θN (x, y, ξ ) = 1 − 2 2N

y

sin N

Cn κ

(cos θN (x, y, ξ )) dμyκ (ξ ) + oρ (1),

#

d

∑ x jξ j

(ξ ) + oρ (1)

sin

j=1

"

y x y x sin + cos cos . N N N N

d

#

x + y − 2 ∑ x j ξ j + Ox,y (N −4 ) 2

2

j=1

1 = 1 − 2 u(x, y, ξ )2 + Ox,y (N −4 ), 2N it follows that



θN (x, y, ξ ) = 2 arcsin

1 + u(x, y, ξ )2 + Ox,y (N −2 ) 2N



1+ u(x, y, ξ )2 + Ox,y (N −2 ) + Ox,y (N −2 ) N ρu(x, y, ξ ) + ox,y,ρ (1) , = n =

(7.3.7)

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] → n = 1. M > 0, and the relation limN→∞ Nρ Thus, by (7.3.7) and (7.2.12), we have lim bN (ρ, x, y)

N→∞

= cm(ρ) lim



−2λκ

N→∞ ξ ≤y

n

λ +1/2 Cn κ





ρu(x, y, ξ ) + ox,y,ρ (1) cos n



√ t for any

dμyκ (ξ )

(ρu(x, y, ξ ))−λκ Jλκ (ρu(x, y, ξ )) dμyκ (ξ )   = cm(ρ)Vκ (ρu(x, y, ·))−λκ Jλκ (ρu(x, y, ·)) (y),

= cm(ρ)

ξ ≤y

λ +1/2

where we used the fact that Cn κ ∞ ≤ 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)| ≤ c1 e−c2 t .

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 assumption that |m(t)| ≤ c1 e−c2 t . To this end, let mδ (t) = m(t)e−δt for δ > 0, and define Tmδ : L2 (Rd , h2κ ) → L2 (Rd ; h2κ ) by Fκ (Tmδ f )(ξ ) = mδ (ξ )Fκ f (ξ ), f ∈ L2 (Rd ; h2κ ). −nε projκ f is a positive operator on By Lemma 3.4.5, for a given ε > 0, f → ∑∞ n n=0 e p d 2 L (S ; hκ ) that satisfies

! !∞ ! ! sup ! ∑ e−nε projκn f ! ε>0 n=0

L p (Sd ;h2κ )

≤  f L p (Sd ;h2κ ) .

Thus, applying Theorem 7.1.1 for the already proven case, we have ! ! ! ! sup !Tmδ f ! p d 2 ≤ cA f L p (Rd ;h2κ ) . δ >0

L (R ;hκ )

(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 function Pδ f is called the Poisson integral of f , and it can be expressed as a generalized convolution Pδ f (x) := ( f ∗κ Pδ )(x)

106

Chapter 7. Multiplier Theorems for the Dunkl Transform

with d

Pδ (x) := 2γκ + 2

Γ(γκ + d+1 ) δ √ 2 . d+1 π (δ 2 + x2 )γκ + 2

By Lemma 3.4.7, it follows that lim Pδ f (x) = f (x), a.e. x ∈ Rd

δ →0+

for any f ∈ Lq (Rd ; h2κ ) with 1 ≤ q < ∞. Since m is bounded, T f ∈ L2 (Rd ; h2κ ) for f ∈ L2 (Rd ; h2κ ). Thus, for any f ∈ S , using (7.4.2), lim Tmδ f (x) = lim Pδ (T f )(x) = T f (x), a.e. x ∈ Rd ,

δ →0+

δ →0+

(7.4.3)

which combined with (7.4.1) and the Fatou theorem implies the desired estimate T f L p (Rd ;h2κ ) ≤ cA f L p (Rd ;h2κ ) . 

This completes the proof of the theorem.

7.5

H¨ormander’s multiplier theorem and the Littlewood–Paley inequality

As a first application of Theorem 7.1.1, we shall prove the following H¨ormander type multiplier theorem for the Dunkl transform: Theorem 7.5.1. Let m : (0, ∞) → R be a bounded function satisfying m∞ ≤ A and H¨ormander’s condition 1 R

 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 operator on L2 (Rd ; h2κ ) defined by Fκ (Tm f )(ξ ) = m(ξ )Fκ f (ξ ), ξ ∈ Rd . Then Tm f κ,p ≤ C p A f κ,p for all 1 < p < ∞ and f ∈

S (Rd ).

Proof. Let μ = m(ε) for ε > 0 and  = 0, 1, . . .. Then |r μ | = ε r ≤





[0,1]r

[0,ε]r

  m(r) εt1 + · · · + εtr + ε dt1 · · · dtr

   |m(r) t1 + · · · + tr + ε |dt1 · · · dtr ≤ ε r−1

ε(r+)

ε

|m(r) (t)| dt.

7.5. H¨ormander’s multiplier theorem and the Littlewood–Paley inequality

107

This implies that, for 2 j ≥ r, 2 j(r−1)

2 j+1

∑j

|Δr μl | ≤ 2 j(r−1) ε r−1

l=2

2 j+1  ε(r+)

∑j

l=2

≤ (r − 1)2 j(r−1) ε r−1 ≤ 2 j(r−1) (r − 1)ε r−1

 ε(2 j+1 +r) 2 jε  2 j+2 ε 2 jε

ε

|m(r) (t)| dt

|m(r) (t)| dt

|m(r) (t)| dt ≤ cr A,

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

|μ j | ≤ cr A. ∑j |Δr μl | ≤ cr max j

l=2

Thus, using Theorem 4.4.2, we deduce !∞ !  ! ! sup ! ∑ m(εn) projκn f !

L p (Sd ;h2κ  )

ε>0 n=0

≤ c f L p (Sd ;h2 ) . κ



The desired conclusion then follows by Theorem 7.1.1. Remark 7.5.2. H¨ormander’s condition is normally stated in the form   2R  12 1 (r) 2 |m (t)| dt ≤ AR−r , for all R > 0; R R

(7.5.2)

see, for instance, [31, Theorem 5.2.7]. Clearly, the condition (7.5.1) in Theorem 7.5.1 is weaker than (7.5.2). On the other hand, however, Theorem 7.5.1 is applicable only to radial multipliers m( · ). Corollary 7.5.3. Let Φ be an even C∞ -function that is supported in the set {x ∈ R : |x| ≤ 21 10 } and satisfies either

∑ Φ(2− j ξ ) = 1,

9 10

≤

ξ ∈ R \ {0},

j∈Z

or

∑ |Φ(2− j ξ )|2 = 1,

ξ ∈ R \ {0}.

j∈Z

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 f |2 ) 2 κ,p 1

j∈Z

for all f ∈

L p (Rd ; h2κ )

and 1 < p < ∞.

Proof. Corollary 7.5.3 follows directly from Theorem 7.5.1. Since the proof runs along the same line as that of Theorem 4.5.2, we omit the details. 

108

7.6

Chapter 7. Multiplier Theorems for the Dunkl Transform

Convergence of the Bochner–Riesz means

Recall that the Bochner–Riesz means of order δ > −1 for the Dunkl transform SRδ f (x) ≡ SRδ (h2κ ; 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 (7.6.1) sup SRδ (h2κ ; f )κ,p ≤ c f κ,p . R>0

Our next result concerns the critical indices for the validity of (7.6.1) in the case of G = Zd2 : Theorem 7.6.1. Suppose that G = Zd2 , f ∈ L p (Rd ; h2κ ), 1 ≤ p ≤ ∞, and | 1p − 12 | ≥ 2λκ1+3 . Then (7.6.1) holds if and only if  1  1 1 (7.6.2) δ > δκ (p) := max (2λκ + 2) − − , 0 . p 2 2 Proof. We start with the proof of the sufficiency. Assume that κ := (κ1 , . . . , κd ) and hκ (x) := ∏dj=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 − t 2 )δ+ . By the equivalence of the Riesz and the Ces`aro summability methods of order δ ≥ 0 (see [30]), we deduce from Theorem 3.3.8 that !∞ !  ! ! sup ! ∑ m(εn) projκn f ! p d 2 ≤ c f L p (Sd ;h2 )  L (S ;hκ  )

ε>0 n=0

κ

whenever | 1p − 12 | ≥ 2σ 1 +2 and δ > δκ  (p), where σκ  = λκ + κ Thus, invoking Theorem 7.1.1, we conclude that for δ > δκ (p),

1 2

and δκ  (p) = δκ (p).

S1δ (h2κ ; 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 L p (Rd , h2κ ). Using (6.4.4) and Theorem 6.2.11 (vii), we have δ  R   r2 Hλκ f0 (r)r2λκ +1 Eκ (ix, ry )h2κ (y ) dσ (y ) dr. 1− 2 SRδ (h2κ ; f )(x) = R 0 Sd−1 However, by [60, Proposition 2.3] applied to n = 0 and g = 1, we have    rx −λκ  2   Eκ (ix, ry )hκ (y ) dσ (y ) = c Jλκ (rx). 2 Sd−1 It follows that SRδ (h2κ ; f )(x) = c

 R 0

1−

r2 R2

= cS$Rδ f0 (x),



δ Hλκ f0 (r)

rx 2

−λκ

Jλκ (rx)r2λκ +1 dr

7.7. Notes and further results

109

where S$Rδ denotes the Bockner–Riesz mean of order δ for the Hankel transform Hλκ . However, it is known (see [66]) that S$Rδ , 0 < δ < λκ + 12 , is bounded on L p ((0, ∞),t 2λκ +1 ) if and only if 2λκ + 2 2λκ + 2 < p< . (7.6.3) λκ + δ + 3/2 λκ − δ + 1/2 Thus, to complete the Proof of the necessity part of the theorem, by (7.6.3), we just need to observe that if f (x) = f0 (x) is a radial function in L p (Rd ; h2κ ), Then  f κ,p = c f0 L p (R;|x|2λκ +1 ) .

7.7



Notes and further results

Most of the results in this chapter were proved in [10]. In the case of ordinary Fourier transform and spherical harmonics, Theorem 7.1.1 is due to Bonami and Clerc [3, Theorem 1.1]. In the unweighted case, for the classical Fourier transform, Theorem 7.6.1 is well known, 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 ) ≤ c p  f L p (Rd ) , 1 ≤ p ≤

2d + 2 , d +3

(7.7.1)

where f denotes the usual Fourier transform of f . In the weighted case, while estimates similar 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 indicated in [14] for the case of the Ces`aro means for h-harmonic expansions on the unit sphere, where global estimates for the projection operators have to be replaced with more delicate local estimates, which are significantly more difficult to prove.

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Index h-Laplace–Beltrami operator, 16 fractional power, 54 h-harmonic expansion Littlewood–Paley inequality, 47 multiplier theorem, 45 h-harmonics, 15 Ces`aro means, 24 convolution, 6 ordinary, 6 weighted on Rd , 74 weighted on the sphere, 23 diffusion semi-group, 27 Dunkl Laplacian, 10 Dunkl operators, 10 Dunkl transform, 65 multiplier theorem, 106 Bochner–Riesz means, 86 inversion formula, 75 maximal function, 87 Paley–Wiener theorem, 70 Plancherel theorem, 68 summability, 82 Fourier transform, 5 Funk–Hecke formula, 22 Gegenbauer polynomials, 4 generalized translation operator for G = Zd2 , 80 on radial functions, 77 on the sphere, 31 weight on Rd , 72 Hankel transform, 6, 76

Heat kernel transform, 85 homogeneous polynomials, 1 Hopf–Dunford–Schwartz theorem, 28 intertwining operator, 12 explicit formula for Zd2 , 12 positivity, 13 Laplace operator, 2 Laplace–Beltrami operator, 3 Littlewood–Paley inequality weighted on Rd , 107 on the sphere, 55 Littlewood–Paley theory g-functions, 38 Littlewood–Paley–Stein function, 37 maximal function on Rd , 88 on the sphere, 27 multiplicity function, 8 Poisson integral, 5 weighted, 28 weighted on Rd , 85 projection operator, 21 reflection, 8 reflection groups, 8 root system, 8 sphere Sd−1 , 1 spherical h-harmonics, 15 Ces`aro summability, 25 expansion, 21 Lebesgue constant, 26

© Springer Basel 2015 F. Dai, Y. Xu, Analysis on h-Harmonics and Dunkl Transforms, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0887-3

117

118 projection, 21 spherical convolution, 4 spherical harmonics, 2 expansions, 3 eigenfunctions of Δ0 , 3 orthogonality, 2 reproducing kernel, 3 space Hnd , 2 surface area ωd , 2 transference theorem, 96 weight function invariant under Zd2 , 7 reflection invariant, 8 weighted K-functional direct and inverse theorem, 56 on the sphere, 55 realization, 55 weighted exponentials, 13 weighted moduli of smoothness direct and inverse theorem, 54 on the sphere, 53 sharp Jackson inequality, 59 sharp Marchaud inequality, 56 weighted Sobolev space, 54 Young’s inequality, 23

Index