Dunkl convolution and elliptic regularity for Dunkl operators

TL;DR

This paper investigates the properties of Dunkl convolution for distributions, establishes elliptic regularity results for Dunkl operators, and explicitly computes convolutions for type A root systems, advancing understanding of Dunkl analysis.

Contribution

It introduces conditions for Dunkl convolution of distributions, proves elliptic regularity theorems for elliptic Dunkl operators, and computes convolutions for Dunkl-type Riesz distributions in type A systems.

Findings

Dunkl convolution can be defined under specific conditions for distributions.

Elliptic regularity holds for a class of Dunkl operators, ensuring smoothness of solutions.

Explicit convolution formulas are derived for Dunkl-type Riesz distributions in type A systems.

Abstract

We discuss in which cases the Dunkl convolution of distributions, possibly both with non-compact support, can be defined and study its analytic properties. We prove results on the (singular-)support of Dunkl convolutions. Based on this, we are able to prove a theorem on elliptic regularity for a certain class of Dunkl operators, called elliptic Dunkl operators. Finally, for the root systems of type A we consider the Dunkl-type Riesz distributions, prove that their Dunkl convolution exists and compute their convolution.