Flett potentials associated with differential-difference Laplace operators

Abstract

A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|1−1/mΔk, where Δk is the Dunkl Laplace (differential and difference) operator on R. Here, k ≥ 0 and m∈N\{0}. In the (k = 0, m = 1) case, our family of potentials reduces to the classical one studied by Flett [Proc. London Math. Soc. s3-22, 385–451 (1971)]. An explicit inversion formula of the Flett potentials is obtained for functions belonging to C0(R) and weighted Lp spaces, 1 ≤ p < ∞. As a tool, we use a wavelet-like transforms generated by a Poisson type semigroup and signed Borel measures. In this context, a fundamental theorem proving an almost everywhere convergence of a convolution operator for an approximate identity was given. The k = 0 case is already new.