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Potential theory associated with the Dunkl Laplacian

Potential theory associated with the Dunkl Laplacian AbstractThe main goal of this paper is to develop a potential theoretical approach to study the Dunkl Laplacian Δk{\Delta_{k}}, which is a standard example of differential-difference operators. Introducing the Green kernel relative to Δk{\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 Δk{\Delta_{k}}-harmonic functions on a large class of open subsets U of ℝd{\mathbb{R}^{d}}. We also establish existence and uniqueness results of a solution of the corresponding Dirichlet problem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

Potential theory associated with the Dunkl Laplacian

Advances in Pure and Applied Mathematics , Volume 8 (4): 23 – Oct 1, 2017

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References (30)

Publisher
de Gruyter
Copyright
© 2017 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6090
eISSN
1869-6090
DOI
10.1515/apam-2015-0056
Publisher site
See Article on Publisher Site

Abstract

AbstractThe main goal of this paper is to develop a potential theoretical approach to study the Dunkl Laplacian Δk{\Delta_{k}}, which is a standard example of differential-difference operators. Introducing the Green kernel relative to Δk{\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 Δk{\Delta_{k}}-harmonic functions on a large class of open subsets U of ℝd{\mathbb{R}^{d}}. We also establish existence and uniqueness results of a solution of the corresponding Dirichlet problem.

Journal

Advances in Pure and Applied Mathematicsde Gruyter

Published: Oct 1, 2017

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