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

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• 作者：Lé ; onard Gallardo^a ; ^{Leonard.Gallardo@lmpt.univ-tours.fr" class="auth_mail" title="E-mail the corresponding author} ; Chaabane Rejeb^b ; ^a ; ^{chaabane.rejeb@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Dunkl-Laplacian operator ; Dunkl convolution product ; Generalized volume mean value operator ; Dunkl harmonic functions ; Weyl's lemma
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：1142-1162
• 全文大小：464 K

文摘

In this paper we show how to use mollifiers to regularize functions relative to a set of Dunkl operators in R^d${\mathbb{R}}^d$ with Coxeter–Weyl group W, multiplicity function k   and weight function ω_k${\omega }_k$. In particular for Ω a W  -invariant open subset of R^d${\mathbb{R}}^d$, for ϕ∈D(R^d)$\varphi \in \mathcal{D}({\mathbb{R}}^d)$ a radial function and $u\in L_{\mathit{loc}}^1(\mathrm{\Omega },{\omega }_k(x)dx)$, we study the Dunkl-convolution product u⁎_kϕ$u{⁎}_k\varphi$ 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.