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

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.