On the Green function and Poisson integrals of the Dunkl Laplacian

TL;DR

This paper investigates the Green function for the Dunkl Laplacian in the unit ball, deriving fundamental formulas and sharp estimates that differ significantly from classical Laplacian potential theory.

Contribution

It establishes the existence of the Green function for the Dunkl Laplacian and derives new formulas and estimates specific to this operator.

Findings

Derived the Poisson-Jensen formula for Dunkl subharmonic functions

Obtained Hardy-Stein identities for Dunkl Poisson integrals

Provided sharp estimates of the Green, Newton, and Poisson kernels in rank one case

Abstract

We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian $\Delta_k$ in $\mathbb{R}^d$. As applications we derive the Poisson-Jensen formula for $\Delta_k$-subharmonic functions and Hardy-Stein identities for the Poisson integrals of $\Delta_k$. We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in $\mathbb{R}^d$. These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.