Methods and Applications of Analysis

Volume 15 (2008)

Number 4

Bessel and Flett Potentials associated with Dunkl Operators on $\Bbb R^d$

Pages: 477 – 494

DOI: http://dx.doi.org/10.4310/MAA.2008.v15.n4.a5

Authors

Néjib Ben Salem

Anis El Garna

Samir Kallel

Abstract

Analogous of Bessel and Flett potentials are defined and studied for the Dunkl transform associated with a family of weighted functions that are invariant under a reflection group. We show that the Dunkl-Bessel potentials, of positive order, can be represented by an integral involving the k-heat transform and we give some applications of this result. Also, we obtain an explicit inversion formula for the Dunkl-Flett potentials, which are interpreted as negative fractional powers of a certain operator expressed in terms of the Dunkl-Laplacian.

Keywords

Dunkl operator; Poisson transform; heat transform; Bessel potential; Flett potential

2010 Mathematics Subject Classification

32A55

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