Radial mollifiers, mean value operators and harmonic functions in Dunkl theory

Abstract

In this paper we show how to use mollifiers to regularize functions relative to a set of Dunkl operators in Rd with Coxeter–Weyl group W, multiplicity function k and weight function ωk. In particular for Ω a W-invariant open subset of Rd, for ϕ∈D(Rd) a radial function and u∈Lloc1(Ω,ωk(x)dx), we study the Dunkl-convolution product u⁎kϕ 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.