The Fractional Dunkl Laplacian: Extension Problem and Fundamental Solution

Abstract

Consider the Dunkl Laplacian ${\Delta }_k$ associated with a root system $\Phi$ in ${\mathbb{R}}^d$ and a nonnegative multiplicity function $k$ on $\Phi$. In this paper, we establish a Caffarelli-Silvestre characterization for the fractional Dunkl Laplacian through an extension problem. We also express the corresponding fundamental solution in terms of the ${\Delta }_k$-Riesz kernel and prove a fractional Nash-type inequality.