The Fractional Dunkl Laplacian: Extension Problem and Fundamental Solution

Consider the Dunkl Laplacian ${\mathrm{\Delta }}_k$ associated with a root system $\mathrm{\Phi }$ in ${\mathbb{R}}^d$ and a nonnegative multiplicity function $k$ on $\mathrm{\Phi }$. 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 ${\mathrm{\Delta }}_k$-Riesz kernel and prove a fractional Nash-type inequality.