Potential theory associated with the Dunkl Laplacian

TL;DR

This paper develops a potential theory framework for the Dunkl Laplacian, introducing Green kernels and harmonic measures, and characterizes harmonic functions and solutions to the Dirichlet problem.

Contribution

It provides the first potential theoretical approach to the Dunkl Laplacian, including Green kernel construction and harmonic function characterization.

Findings

Dunkl Laplacian generates a balayage space.

Characterization of harmonic functions via harmonic kernels.

Existence and uniqueness of Dirichlet problem solutions.

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.