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

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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Published 1 January 2008