Dunkl convolution and elliptic regularity for Dunkl operators

We discuss in which cases the Dunkl convolution u∗kv$u*_kv$ of distributions u,v$u,v$, 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 system An−1$A_{n-1}$ we consider the Riesz distributions (Rα)α∈C$(R_\alpha)_{\alpha \in \mathbb {C}}$ and prove that their Dunkl convolution exists and that Rα∗kRβ=Rα+β$R_\alpha *_kR_\beta = R_{\alpha +\beta }$ holds.