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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630645X&_mathId=si1.gif&_user=111111111&_pii=S0022247X1630645X&_rdoc=1&_issn=0022247X&md5=661a4969dfefcf759a22b208b6b87265" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$ with Coxeter–Weyl group W, multiplicity function k   and weight function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630645X&_mathId=si2.gif&_user=111111111&_pii=S0022247X1630645X&_rdoc=1&_issn=0022247X&md5=f9dc261644c569e9e23daff48a113d43" title="Click to view the MathML source">ω_kclass="mathContainer hidden">class="mathCode">${\omega }_k$. In particular for Ω a W  -invariant open subset of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630645X&_mathId=si1.gif&_user=111111111&_pii=S0022247X1630645X&_rdoc=1&_issn=0022247X&md5=661a4969dfefcf759a22b208b6b87265" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$, for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630645X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1630645X&_rdoc=1&_issn=0022247X&md5=19c9d5b52fe2e535cef3aa4d914d6777" title="Click to view the MathML source">ϕ∈D(R^d)class="mathContainer hidden">class="mathCode">$\varphi \in \mathcal{D}({\mathbb{R}}^d)$ a radial function and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630645X&_mathId=si4.gif&_user=111111111&_pii=S0022247X1630645X&_rdoc=1&_issn=0022247X&md5=2b2af7b9671318f4433dd71331d828ed">class="imgLazyJSB inlineImage" height="18" width="146" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X1630645X-si4.gif">class="mathContainer hidden">class="mathCode">$u\in L_{\mathit{loc}}^1(\mathrm{\Omega },{\omega }_k(x)dx)$, we study the Dunkl-convolution product class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630645X&_mathId=si166.gif&_user=111111111&_pii=S0022247X1630645X&_rdoc=1&_issn=0022247X&md5=a6c22ab6f42644bf98bc45f4f82330c6" title="Click to view the MathML source">u⁎_kϕclass="mathContainer hidden">class="mathCode">$u{⁎}_k\varphi$ 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.