For a root system in R d furnished with its Coxeter-Weyl group W and a multiplicity nonnegative function k, we consider the associated commuting system of Dunkl operators D 1 , . . . , D d and the Dunkl-Laplacian Δ k = D 2 1 +. . .+D 2 d . This paper studies the properties of the functions u defined on an open W -invariant set Ω ⊂ R d and satisfying Δ k u = 0 on Ω (D-harmonicity).In particular, we introduce and give a complete study of a new mean value operator which characterizes D-harmonicity. As applications we prove a strong maximum principle, a Harnack's type theorem and a Bôcher's theorem for D-harmonic functions.
The purpose of this paper is to present a new theory of subharmonic functions for the Dunkl-Laplace operator ∆ k in R d associated to a root system and a multiplicity function k ≥ 0. In particular, we introduce and study a Dunkl-Newton kernel and the corresponding potential of Radon measures. As applications we give a strong maximum principle, a solution of the Poisson equation and a Riesz decomposition theorem for ∆ k-subharmonic functions.
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