Propriété de Liouville et équation de Poisson pour le laplacien généralisé de Dunkl

Let Δ_k be the Dunkl generalized Laplacian associated to a root system R of $\text{R}{}^{\mathrm{N}}$ and a non-negative function k defined on R and invariant by the Weyl group. In this Note, we show that this differential-difference operator on $\text{R}{}^{\mathrm{N}}$ satisfies the Liouville property, then we solve the Poisson equation Δ_ku=?f by using a generalized Fourier analysis method. To cite this article: L. Gallardo, L. Godefroy, C. R. Acad. Sci. Paris, Ser. I 337 (2003).