Flett potentials associated with differential-difference Laplace operators

A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|(1-1/m)& UDelta;(k), where & UDelta;(k) is the Dunkl Laplace (differential and difference) operator on R. Here, k & GE; 0 and m & ISIN;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 C-0(R) and weighted L-p spaces, 1 & LE; p < & INFIN;. 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.

Automated summary

This article investigates a large family of Flett potentials, which includes some classical cases that have been studied before. The explicit inversion formula of Flett potentials is obtained using a tool generated by a Poisson type semigroup and signed Borel measures. The article also proves the almost everywhere convergence of a convolution operator. The research in this article is of great significance for understanding and applying Flett potentials.