Dunkl convolution and elliptic regularity for Dunkl operators

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.