WeightedLp-Inequalities for the Dunkl Transform

Chokri Abdelkefi,Faten Rached

doi:10.1155/2014/702151

Abstract

We give, for1<p≤2, weightedLp-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.

Highlights

Dunkl theory is a far reaching generalization of Euclidean Fourier analysis
The Dunkl operators are commuting differential-difference operators Ti, 1 ≤ i ≤ d. These operators, attached to a finite root system R and a reflection group W acting on Rd, can be considered as perturbations of the usual partial derivatives by reflection parts
For a family of weight functions wk invariant under a reflection group W, we use the Dunkl kernel and the weighted Lebesgue measure wk(x)dx to define the Dunkl transform Fk, which enjoys properties similar to those of the classical Fourier transform

Summary

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 [2,3,4,5,6]) and later was applied and generalized in different ways by many authors (see [7,8,9,10,11]). Trimeche has introduced in [6] the Dunkl translation operators τx, x ∈ Rd, on the space of infinitely differentiable functions on Rd. At the moment an explicit formula for the Dunkl translation τx(f) of a function f is unknown in general. The difficulty arises in the application of the modulus of continuity for nonradial function f in Lp since whether the Dunkl translation τx can be defined on L1k(Rd) is still an open problem. To avoid this problem, we use the Dunkl convolution and ωp,k(f). (iii) D(Rd) the subspace of C∞(Rd) of compactly supported functions

Preliminaries

Weighted Lp-Estimates for the Dunkl Transform with Sufficient Conditions