The Range of the Spectral Projection Associated with the Dunkl Laplacian

Ben Said, Salem; Mejjaoli, Hatem

doi:https://doi.org/10.1155/2020/7803719

Journal of Function Spaces

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Research Article | Open Access

Volume 2020 | Article ID 7803719 | https://doi.org/10.1155/2020/7803719

The Range of the Spectral Projection Associated with the Dunkl Laplacian

Salem Ben Said¹and Hatem Mejjaoli²

Academic Editor: Mitsuru Sugimoto

Received24 Oct 2019

Accepted27 Apr 2020

Published24 Jul 2020

Abstract

For denote by the “projection” of a function in into the eigenspaces of the Dunkl Laplacian corresponding to the eigenvalue The parameter comes from Dunkl’s theory of differential-difference operators. We shall characterize the range of on the space of functions supported inside the closed ball As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.

1. Introduction

Analysis of the Dunkl Laplacian operator on commenced in the early 90’s, inspired by numerous results in the Euclidean setting, as well as some extensions of this to flat symmetric spaces. Here, the parameter comes from Dunkl’s theory of differential-difference operators [1]. In recent years, there have been increasing interests in the study of problems involving the Dunkl Laplacian and have received a lot of attention, see for instance [1–15]. The purpose of this paper is to study a family of eigenfunctions for the Dunkl Laplacian derived through the use of the inversion formula for the Dunkl transform. Our main result may be interpreted as a contribution to the spectral theory of the Dunkl Laplacian.

To state our main result, we need to introduce some notation. Writing the inversion formula for the Dunkl transform in polar coordinates, we obtain where are “projections” of into the eigenspaces of corresponding to the eigenvalue (see (35)). We may also write the projection operators as Dunkl-convolution with a normalized Bessel function of the first kind (see (43)). In this paper, we discuss on how properties of are related to properties of the eigenfunctions Essentially, we prove a theorem characterizing for with involving analytic continuation to and growth estimates of type, for all and for all multi-index where is a positive continuous increasing function on (see Theorem 9). Several contributions have been dedicated to this subject, see for instance [16–22].

As an application of the main result, we prove a spectral version of the complex Paley-Wiener theorem for the Dunkl transform given in [23]. More precisely, we characterize the set of functions defined on for which there exists a compactly supported smooth function with support in so that (see Theorem 10).

2. Background

For , we let denote the usual Euclidean inner product of and the Euclidean norm. Let be the unit sphere in We denote by as the Lebesgue surface measure on

For a nonzero vector define the reflection by

A root system is a finite set of nonzero vectors in such that implies If, in addition, and for some scalar implies then is called reduced. Henceforth, we will assume that is a reduced root system. Fix a set of positive roots so that

The finite reflection group generated by the root system is the subgroup of the orthogonal group generated by the reflections

For a given root system a multiplicity function is a nonnegative -invariant function.

Given a reduced root system on and a multiplicity function we define the weight function by

Then, is a positively homogeneous -invariant function of degree where

The main ingredient of the Dunkl theory is a family of commuting first-order differential-difference operators, (called the Dunkl operators [1]), defined by where is the ordinary partial derivative with respect to The Dunkl operators are akin to the partial derivatives and they can be used to define the Dunkl Laplacian , which plays the role similar to that of the ordinary Laplacian , where is an orthonormal basis of The above explicit expression of has been proved in [24].

For arbitrary finite reflection group and for any nonnegative multiplicity function there is a unique linear operator on the space of algebraic polynomials on that intertwines between the Dunkl operators and the partial derivatives,

It has been proved in [25] that has a Laplace type representation which allows to extend to larger function spaces: with a unique probability measure In fact, induces a homeomorphism of and also that of see [23, 26].

For define

For fixed the function is the unique real-analytic solution of with (see [13, 27]). Further, the (Dunkl) kernel has a unique holomorphic extension to and satisfies the following properties:

Fact 1 (see, for instance, [11, 25]). (1)For all and we have and (2)For every multi-index we havewhere In particular, for all
For the Dunkl transform is defined by where is the constant The Dunkl transform was introduced in [28] where the -isometry (or the Plancherel theorem) was proved, while the main results of the -theory were established in [11]. In particular, it has been proved that if and are in then for almost every It is worth mentioning that the Dunkl transform is a homeomorphism of the Schwartz space Further, according to ([29], Proposition 5.7.8), for such that with we have where and is the Hankel transform of index on given by Here, is the normalized Bessel function of the first kind defined by Let be given. For the generalized translation operator is defined by

Fact 2 (see [26]). The translation operator has the following properties: (1)For all (2)For fixed extends to a continuous linear mapping from into itself(3)If and thenThe generalized translation operator is used to define a convolution structure: For where We can also write the convolution as We refer the reader to [15] for more details on the convolution product It is worth mentioning, for the distributional version of the Dunkl transform, the translation operator and the Dunkl convolution of distributions and properties, we refer the reader to [30].
For let be the space of -harmonic polynomials of degree on , where is the Dunkl Laplacian and denotes the space of homogeneous polynomials of degree on The restriction of elements in on the unit sphere in are the so-called -spherical harmonics. We shall not distinguish between and its restriction to The space has a reproducing kernel in the sense that Here, is the constant where and are as defined in (13) and (16), respectively. According to [31], for the kernel can be written as where is the Dunkl intertwining operator (8), and is the Gegenbauer polynomial of degree for with is the Gauss hypergeometric function.
The following analogue of the Funk-Hecke formula for -spherical harmonics will be used later on; for the proof, the reader is referred to [32]. Let be a continuous function on Then, for any where is a constant defined by We summarize some basic properties of Gegenbauer polynomials in a way that we shall use later.

Fact 3 (see [[6] (1.2.10)], [[33], (3.32.3)]). For such that the following two integral formulas hold: Let denote the space of even compactly supported smooth functions with support in where The Paley-Wiener theorem for the Hankel transform (see (17)) states that maps bijectively onto the space of even entire functions satisfying, for all for some positive constant see for instance [34].
This result has been generalized by de Jeu [23] to the Dunkl transform. To state the (complex) Paley-Wiener theorem for , we introduce the following notation. For , let be the space of entire functions on with the property that for all there exists a constant such that We let denote the space of smooth compactly supported functions with support contained in the closed ball with radius and the origin as center.

Fact 4 (see [23]). The Dunkl transform is a linear isomorphism between and for all

An immediate consequence of the above Paley-Wiener theorems can be stated as follows:

Lemma 5. Let Then, for some radial function in if and only if extends to an entire function on satisfying the estimate for all

Proof. The statement follows from the fact that whenever is a radial function with (see (15)), together with the Paley-Wiener theorems stated above for the Hankel and the Dunkl transforms.

3. The Range of the Spectral Projection Associated with

Recall from (12) that the Dunkl transform of is defined by

Using polar coordinates, the Dunkl inversion formula (14) becomes where

Notice that

From (36), we may derive a second formula for Indeed, substituting (34) into (36), we obtain

According to [[35], page 2424], the inner integral is equal to where is the constant (25), is the reproducing kernel (26), and is the normalized Bessel function (18). We now use the well-known addition formula for Bessel functions (see [[36] , p. 215]): for which converges uniformly with respect to Using (38) and (26) together with the Laplace representation (9) for , we deduce that

Define then, by [[35] , p. 2429], we have

Consequently, the eigenfunction can be rewritten as

Above, we have used some of the properties of the generalized translation operator listed in Fact 2. The following statement lists the necessary conditions for Theorem 9.

Proposition 6. Assume that and let defined either by (36) or (42), with and Then the following hold: (1) is a smooth function on (2) where the upper index indicates the relevant variable(3)For be given, extends to an even entire function of (4)For every and for every multi-index there exists a constant such that(5)For any -spherical harmonic of degree and for every the mapis entire on

Proof. (1)In view of properties of the translation operator and the normalized Bessel function the first statement follows from the representation (42) of (2)The second property is immediate from (36), since (3)Now, by (42), the map is certainly extends to an even entire function on (4)Since it follows from the Paley-Wiener theorem for the Dunkl transform that extends to an entire function on satisfying the estimatefor all see Fact 4. Consequently, by (36), we obtain for all where we used the estimate for all and This finishes the proof of the estimate (44). (5)Let be a -spherical harmonic of degree By the Fubini theorem and (37), we havewhere is as in (38), We now apply the Funk-Hecke formula (28) to deduce that where, by Fact 3, we have Using the above identities, it follows that where and Above, we have used the fact that see (25). In conclusion, The desired result now follows from the fact that is an entire function of

The following lemma is needed for later use.

Lemma 7. (1)For any radial function on with and for any -spherical harmonic of degree we have(2)Under the change of variable the differential equationtransforms to the Bessel equation with

Proof. (1)In the polar coordinates the Dunkl Laplacian operator is expressed aswhere is the analogue of the Laplace-Beltrami operator on the sphere which, in particular, has -spherical harmonics as eigenfunctions, We refer the reader to [37] for more details on (2)ObviousNext we will list the sufficient conditions for Theorem 9.

Proposition 8. For and let be a function satisfying the following conditions: (1) is smooth on (2) is an eigenfunction of the Dunkl Laplacian with eigenvalue (3)The mapping extends to an even entire function on (4)For every and for every multi-index there exists a positive continuous increasing function on such that(5)For every and every -spherical harmonic of degree the mappingis entire on
Then, there exists such that .

Proof. Define the following function The estimate (60) (with ) shows that the integral converge absolutely, and therefore, by assumption (1) and the estimate (60) again, is smooth on
Below, we will prove that for all Let be an orthonormal basis of By (62), we have where and are the -spherical harmonic coefficients of and respectively. Here, stands for the inner product in To show that for all , it is enough to prove that for all By Lemma 7 (1), the fact that are eigenfunctions of with eigenvalue implies Further, by Lemma 7 (2), the solution of equation (64) is in the following form: where is a function which depends only on , and Because of the condition (5) on the mapping extends to an entire function of
On the other hand, it is known that the normalized Bessel function has infinitely many positive zeros Let and define For such that the identity (65) and the assumption (4) on (with ) imply, for for all In particular, if , we have for all (recall that is an increasing function). For the compact domain the estimate (68) holds true with a different constant. Moreover, by (65) and the evenness of , the map is even. Applying Lemma 5, there exists a radial function such that with where and is the Hankel transform. Here, we have used the fact that the Dunkl transform of radial functions at is a Hankel transform at (see (15)). Now, letting show in
Using again (65) and the fact that the integral (63) becomes That is, which implies that for as desired.
It will follow that with provided we prove that if a function satisfies assumptions (1)–(5) of Proposition 8 with , then for all and To prove so, it suffices to show that implies for all and However, this follows by mimicking the proof given above together with the injectivity of the Hankel transform on

We can now state the main result of this paper by putting the above propositions together.

Theorem 9. Let and There exists a smooth compactly supported function with support contained in such that if and only if satisfies the conditions listed in Proposition 8.

The following statement illustrates an interesting application of Theorem 9. We may think of it as a spectral-reformulation of the Paley-Wiener theorem for the Dunkl transform (see Fact 4).

Theorem 10. Let be a smooth function on Then, for some if and only if the following two conditions hold: (1)For each the map has an entire extension with the property that for all , there exists a constant such that(2)For an arbitrary -spherical harmonic of degree the maphas an even entire extension.

Proof. Recall from (10) the Dunkl kernel Following [38], let be the transpose of the Dunkl intertwining operator which satisfies on By [38], if and only if
We now proceed towards the proof of the direct part. Assume that for some Then, where Since it follows that This together with the Paley-Wiener theorem for the Euclidean Fourier transform on completes the proof of the property (1).
Next, we turn our attention to the property (2). For and , we have Using the Funk-Hecke formula (28) to deduce that where, by Fact 3, This shows that That is, Since and are even, it follows that the map is even. The analycity of the latter map is immediate from Preposition 6.
For the converse part, define By the assumptions (1) and (2) on and (11), the function satisfies the hypothesis of Theorem 9. Thus, there exists such that That is, which implies This finishes the proof of the converse part.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

S. Ben Said would like to thankfully acknowledge the financial support awarded by UAEU through the Start-up grant. No. G00002950.

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Copyright © 2020 Salem Ben Said and Hatem Mejjaoli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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