Three results in Dunkl theory

TL;DR

This paper advances Dunkl theory by proving a geometric Paley-Wiener theorem, establishing bounds for Dunkl translations in one dimension, and characterizing the support of related distributions in higher dimensions.

Contribution

It introduces new geometric and analytical results in Dunkl theory, including a Paley-Wiener theorem and bounds for Dunkl translations.

Findings

Proved a geometric Paley-Wiener theorem for Dunkl transform

Derived an optimal $L^p\to L^p$ bound for Dunkl translations in dimension 1

Characterized the support of distributions associated with Dunkl translations in higher dimensions

Abstract

In this article, we establish first a geometric Paley-Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the $L^p\to L^p$ norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.