Newtonian Potentials and Subharmonic Functions Associated to Root Systems

Abstract

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.

Riesz potentials of Radon measures associated to reflection groups

“…Hence, by Proposition 3.3 in [10], it suffices to prove that for every r > 0, S x 0 ,β,r is D-superharmonic on R d . To do this, we have only to show that S x 0 ,β,r is of class C 2 on R d and ∆ k S x 0 ,β,r ≤ 0 on R d (see [10], Proposition 4.1).…”

“…Finally, we recall that • a function u of class C 2 on Ω is D-subharmonic in the sense of (1.10) if and only if ∆ k u ≥ 0 on Ω (see [10]). …”

“…the case of the Dunkl-Newton kernel) has been done in [10]. So, we will deal with the case β ̸ = 2. i) Suppose that β > 2.…”

“…It is also known (see [21]) that for all fixed x ∈ R d , the function p k (t, x, .) solves the Dunkl heat equation We note that when β = 2, we obtain the Dunkl-Newton kernel which has been introduced and studied in [10]. Let x ∈ R d , x ̸ = 0, be fixed and W.x be its W -orbit.…”

“…In particular, we will study the sub-or-superharmonicity of these functions in the sense of the Dunkl-Laplace operator and we will describe explicitly their ∆ k -Riesz measures. This notion of subharmonicity, which generalizes the classical one 1 has been introduced and studied in some details in [10]. More [8] and which properties will be recalled in the next section.…”