Weighted -Inequalities for the Dunkl Transform

Abstract

We give, for , weighted -inequalities for the Dunkl transform, using, respectively, the modulus of continuity of radial functions and the Dunkl convolution in the general case. As application, we obtain, in particular, the integrability of this transform on Besov-Lipschitz spaces.

1. Introduction

Dunkl theory is a far reaching generalization of Euclidean Fourier analysis. It started twenty years ago with Dunkl’s seminal work [1] and was further developed by several mathematicians (see [2–6]) and later was applied and generalized in different ways by many authors (see [7–11]). The Dunkl operators are commuting differential-difference operators , . These operators, attached to a finite root system and a reflection group acting on , can be considered as perturbations of the usual partial derivatives by reflection parts. These reflection parts are coupled by parameters, which are given in terms of a nonnegative multiplicity function . The Dunkl kernel has been introduced by Dunkl in [12]. For a family of weight functions invariant under a reflection group , we use the Dunkl kernel and the weighted Lebesgue measure to define the Dunkl transform , which enjoys properties similar to those of the classical Fourier transform. If the parameter then , so that becomes the classical Fourier transform and the , , reduce to the corresponding partial derivatives , (see next section, Remark 1). The classical Fourier transform behaves well with the translation operator , which leaves the Lebesgue measure on invariant. However, the measure is no longer invariant under the usual translation. Trimèche has introduced in [6] the Dunkl translation operators , , on the space of infinitely differentiable functions on . At the moment an explicit formula for the Dunkl translation of a function is unknown in general. However, such formula is known when the function is radial (see next section). In particular, the boundedness of is established in this case. As a result one obtains a formula for the Dunkl convolution . An important motivation to study Dunkl operators originates from their relevance for the analysis of quantum many body systems of Calogero-Moser-Sutherland type. These describe algebraically integrable systems in one dimension and have gained considerable interest in mathematical physics (see [13]).

Let be a function in , , where denote the space with the weight function associated with the Dunkl operators given by with a fixed positive root system (see next section).

The modulus of continuity of first order of a radial function in is defined by where is the unit sphere on with the normalized surface measure . We use as a shorthand for .

For , we set where is a radial function in satisfying being the dilation of given by with , , and .

Obviously, and are nondecreasing in .

The goal of this paper is to extend to the context of Dunkl theory the results obtained by Móricz in [14] for the Fourier transform on the real line. More precisely, we give, for , sufficient conditions for weighted -inequalities of the Dunkl transform of a radial function using the modulus of continuity . In the general case, the difficulty arises in the application of the modulus of continuity for nonradial function in since whether the Dunkl translation can be defined on is still an open problem. To avoid this problem, we use the Dunkl convolution and . As application, we obtain, in particular, the integrability of this transform on Besov-Lipschitz spaces.

The contents of this paper are as follows.

In Section 2, we collect some basic definitions and results about harmonic analysis associated with Dunkl operators.

In Section 3, we give sufficient conditions for weighted -inequalities of the Dunkl transform of function which we apply in particular on Besov-Lipschitz spaces.

Along this paper we denote by the standard Euclidean scalar product on . We write, for , and we use to denote a suitable positive constant which is not necessarily the same in each occurrence. Furthermore, we denote by(i) the space of infinitely differentiable functions on ,(ii) the Schwartz space of functions in which are rapidly decreasing as well as their derivatives,(iii) the subspace of of compactly supported functions.

2. Preliminaries

In this section, we recall some notations and results in Dunkl theory and we refer to more details to the surveys [15].

Let be a finite reflection group on , associated with a root system . For , we denote by the hyperplane orthogonal to . Given , we fix a positive subsystem . We denote by a nonnegative multiplicity function defined on with the property that is -invariant. We associate with the index and the weight function given by is -invariant and homogeneous of degree . Further, we introduce the Mehta-type constant by

For every , we denote by the space and we use as a shorthand for .

By using the homogeneity of degree of , it is shown in [4] that, for a radial function in , there exists a function on such that , for all . The function is integrable with respect to the measure on and we have where is the unit sphere on with the normalized surface measure and

The Dunkl operators , , on associated with the reflection group and the multiplicity function , are the first order differential-difference operators given by where is the reflection on the hyperplane and , being the canonical basis of .

Remark 1. In case , the weighted function and the measure associated with the Dunkl operators coincide with the Lebesgue measure. The reduce to the corresponding partial derivatives. Therefore Dunkl analysis can be viewed as a generalization of classical Fourier analysis.

For , the system admits a unique analytic solution on , denoted by and called the Dunkl kernel. This kernel has a unique holomorphic extension to . We have, for all and , , and, for , .

The Dunkl transform is defined for by We list some known properties of this transform. (i)The Dunkl transform of a function has the following basic property: (ii)The Dunkl transform is an automorphism on the Schwartz space .(iii)When both and are in , we have the inversion formula: (iv)(Plancherel’s theorem) the Dunkl transform on extends uniquely to an isometric automorphism on .Since the Dunkl transform is of strong-type and , then, by interpolation, we get, for with and such that , the Hausdorff-Young inequality The Dunkl transform of a radial function in is also radial and could be expressed via the Hankel transform. More precisely, according to ([4], proposition 2.4), we have the following results: where is the function defined on by , is the Hankel transform of order , and is the normalized Bessel function of the first kind and order .

The integral representation of , , is given by We note that

Trimèche has introduced in [6] the Dunkl translation operators , , on . For and , we have Notice that, for all , and for fixed As an operator on , is bounded. A priori it is not at all clear whether the translation operator can be defined for -functions with different from 2. However, according to ([11], Theorem 3.7), the operator can be extended to the space of radial functions in , , and we have, for a radial function in ,

The Dunkl convolution product of two functions and in is given by The Dunkl convolution product is commutative and, for , we have It was shown in ([11], Theorem 4.1) that when is a bounded radial function in , then initially defined on the intersection of and extends to , , as a bounded operator. In particular,

For , and using the modulus of continuity of first order of function introduced in Section 1, we define the weighted Besov-Lipschitz spaces (see [16] for the classical case) denoted by as the subspace of radial functions in satisfying

3. Weighted -Estimates for the Dunkl Transform with Sufficient Conditions

In this section, we give sufficient conditions for weighted -estimates of the Dunkl transform of function which we apply on Besov-Lipschitz spaces.

Throughout this section, we denote by the conjugate of for . According to (22) and (26), we recall the following.(i)The modulus of continuity of first order of a radial function in is defined by (2)We set, for , where is a radial function in satisfying being the dilation of given by , for all and .

For , we introduce a class of nonnegative -locally integrable radial functions on , which we denote by . We said that a function belongs to the class if there exists a constant such that for If we set then using (8) and the change of variables , we can write (31) in the form

Example 2. If , then , where we choose

Theorem 3. Let and let be a radial function in . Then for and , one has where is a constant depending only on and .

Since the proof of Theorem 3 is too long, we introduce the following technical lemma.

Lemma 4. Let be a radial function of , where , and let be a function defined on by for all . Then where and is the constant defined by

Proof. Let be a radial function in for ; then, by (20), we have , , and a.e. .
We can assert by Hausdorff-Young’s inequality that On the other hand, from (8) and (17), we get where and is the function defined on by for all .
By (9), (16), and Hölder’s inequality, we have According to (39), it follows that Integrating the two members over , this yields From (18) and (19), we get then, we obtain Let , . For and , we have which gives that Hence, from (43), we find that which proves the result.

Proof of Theorem 3. Take and put , for . Applying Hölder’s inequality and using Lemma 4, it follows from (8) and (33) that Then, we deduce Using the monotonicity of in and the fact that , , we obtain By (8), (17), and (51), we conclude that where is a constant depending only on and . This completes the proof of the theorem.

In particular, for the case and , we obtain, as application of Theorem 3, the following result which is of special interest.

Corollary 5. Let , , and ; then consider the following. (1)For , one has (2)For , one has .

Proof. Let , , and . (i)Suppose that . Using Theorem 3 with , it yields  Since , where is the ball of radius centered at , we deduce that is in .(ii)Assume now . Using Theorem 3 with and , we get  Since , we deduce that is in . This completes the proof.

For , , if we replace by in Theorem 3, we obtain analogous results in the following.

Theorem 6. Let and let be a function in . Then for and , one has where is a constant depending only on and .

Proof. Let be a function in for . Then by (24), we have, for , , a.e. . Since , we get . Using Hausdorff-Young’s inequality, we obtain
Let and . For , we have
this gives, using (57) and the property of the function , Now, take . Applying Hölder’s inequality, it follows from (33) and (59) that Then, we deduce As in the proof of Theorem 3, using the monotonicity of in , we conclude that where is a constant depending only on and . This completes the proof.

Remark 7. As consequence of Theorem 6, we deduce, in the particular case when and satisfying similar results to those obtained in Corollary 5.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was completed with the support of the DGRST research project LR11ES11 and the program CMCU 10G/1503.