27 research outputs found
Radial mollifiers, mean value operators and harmonic functions in Dunkl theory
In this paper we show how to use mollifiers to regularise functions relative to a set of Dunkl operators in R d with Coxeter-Weyl group W , multiplicity function k and weight function ω k. In particular for Ω a W-invariant open subset of R d , for ϕ ∈ D(R d) a radial function and u ∈ L 1 loc (Ω, ω k (x)dx), we study the Dunkl-convolution product u * k ϕ and the action of the Dunkl-Laplacian and the volume mean operators on these functions. The results are then applied to obtain an analog of the Weyl lemma for Dunkl-harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators
Support properties of the intertwining and the mean value operators in Dunkl's analysis
In this paper we show that the Dunkl intertwining operator has a compact support which is invariant by the associated Coxeter-Weyl group. This property enables us to determine explicitely the support of the volume mean value operator, a fundamental tool for the study of harmonic functions relative to the Dunkl-Laplacian operator
Newtonian potentials and subharmonic functions associated to root systems
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