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Potential theory associated with the Dunkl Laplacian

Authors
  • Hassine, Kods
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
May 19, 2017
Volume
8
Issue
4
Pages
241–263
Identifiers
DOI: 10.1515/apam-2015-0056
Source
De Gruyter
Keywords
License
Yellow

Abstract

The main goal of this paper is to develop a potential theoretical approach to study the Dunkl Laplacian Δ k {\Delta_{k}} , which is a standard example of differential-difference operators. Introducing the Green kernel relative to Δ k {\Delta_{k}} , we prove that the Dunkl Laplacian generates a Balayage space and we investigate the associated family of harmonic measures. Therefore, by means of harmonic kernels, we give a characterization of all Δ k {\Delta_{k}} -harmonic functions on a large class of open subsets U of ℝ d {\mathbb{R}^{d}} . We also establish existence and uniqueness results of a solution of the corresponding Dirichlet problem.

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