Poisson integrals and kelvin transform associated to dunkl-laplacian operator

In this paper, we will first solve the Dirichlet problem for the unit ball around 0 associated to the Dunkl-Laplacian operator. Then, we will study the boundary behavior of Poisson integrals associated to these differential-difference operators. Finally, we will define the Kelvin transform in Dunkl's theory and prove that the Kelvin transform preserves h-harmonic functions. As an application, we will restate Xu's results about Maxwell representations of h-harmonics polynomials in [11] and give simpler proofs.