Potential theory associated with the Dunkl Laplacian

Abstract

The main goal of this paper is to give potential theoretical approach to study the Dunkl Laplacian ${\Delta }_k$ which is a standard example of differential-difference operators. By introducing the Green kernel relative to ${\Delta }_k$, we prove that the Dunkl Laplacian generates a balayage space and we investigate the associated family of harmonic measures. Therefore, by mean of harmonic kernels, we give a characterization of all ${\Delta }_k$-harmonic functions on large class of open subsets $U$ of ${\mathbb{R}}^d$. We also establish existence and uniqueness result of a solution of the corresponding Dirichlet problem.

21 pages