The Fractional Dunkl Laplacian: Extension Problem and Fundamental Solution

Consider the Dunkl Laplacian $\Delta_k$ associated with a root system $\Phi$ in $\R^d$ 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.