On the Green Function and Poisson Integrals of the Dunkl Laplacian

We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian △ _k in (mathbb {R}^{d}). As applications we derive the Poisson-Jensen formula for △ _k -subharmonic functions and Hardy-Stein identities for the Poisson integrals of △ _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.