On the Green function and Poisson integrals of the Dunkl Laplacian

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.

25 pages