An $L^p$ version of Hardy's theorem for the Dunkl transform

J. Aust. Math. Soc. 77 (2004), 371-385

An version of Hardy's theorem for the Dunkl transform

Léonard Gallardo
Faculté des Sciences
Département de Mathématiques
Parc de Grandmont
37200 Tours
France
gallardo@univ-tours.fr

and

Khalifa Trimèche
Faculté des Sciences de Tunis
Département de Mathématiques
Campus Universitaire
1060 Tunis
Tunisie
khlifa.trimeche@fst.rnu.tn

Abstract

In this paper, we give a generalization of Hardy's theorems for the Dunkl transform ${\mathcal{F}}_D$ on ${\mathbb{R}}^d$ . More precisely for all

and $p, q \in [1, + \infty]$ , we determine the measurable functions

on ${\mathbb{R}}^d$ such that $e^{a\|x\|^2}f \in L^p_k({\mathbb{R}}^d)$ and $e^{b\|y\|^2}{\mathcal{F}}_D(f) \in L^q_k({\mathbb{R}}^d)$ , where $L^p_k({\mathbb{R}}^d)$ are the Lebesgue spaces associated with the Dunkl transform.

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