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Riesz potentials of Radon measures associated to reflection groups

Authors
  • Gallardo, Léonard
  • Rejeb, Chaabane
  • Sifi, Mohamed
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
Sep 09, 2017
Volume
9
Issue
2
Pages
109–130
Identifiers
DOI: 10.1515/apam-2017-0057
Source
De Gruyter
Keywords
License
Yellow

Abstract

For a root system R on ℝ d {\mathbb{R}^{d}} and a nonnegative multiplicity function k on R, we consider the heat kernel p k ⁢ ( t , x , y ) {p_{k}(t,x,y)} associated to the Dunkl Laplacian operator Δ k {\Delta_{k}} . For β ∈ ] 0 , d + 2 γ [ {\beta\in{]0,d+2\gamma[}} , where γ = 1 2 ⁢ ∑ α ∈ R k ⁢ ( α ) {\gamma=\frac{1}{2}\sum_{\alpha\in R}k(\alpha)} , we study the Δ k {\Delta_{k}} -Riesz kernel of index β, defined by R k , β ⁢ ( x , y ) = 1 Γ ⁢ ( β / 2 ) ⁢ ∫ 0 + ∞ t β / 2 - 1 ⁢ p k ⁢ ( t , x , y ) ⁢ 𝑑 t {R_{k,\beta}(x,y)=\frac{1}{\Gamma(\beta/2)}\int_{0}^{+\infty}t^{{\beta/2}-1}p_% {k}(t,x,y)\,dt} , and the corresponding Δ k {\Delta_{k}} -Riesz potential I k , β ⁢ [ μ ] {I_{k,\beta}[\mu]} of a Radon measure μ on ℝ d {\mathbb{R}^{d}} . According to the values of β, we study the Δ k {\Delta_{k}} -superharmonicity of these functions, and we give some applications like the Δ k {\Delta_{k}} -Riesz measure of I k , β ⁢ [ μ ] {I_{k,\beta}[\mu]} , the uniqueness principle and a pointwise Hedberg inequality.

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