Some results on Dunkl analysis : A survey

Abstract

In this article, we present a survey of some new results obtained in [2,8]. First, we give a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension. We also investigate the Lp → Lq boundedness properties of the Riesz potentials Iακ and the related fractional maximal function Mκ,α associated to the Dunkl transform. Finally we prove the Lp-boundedness, (1 < p < ∞) of the Riesz transforms in the Dunkl setting.

Keywords

• Dunkl operator
• Paley–Wiener theorem
• Generalized translations
• Riesz potentials
• Riesz transforms