Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

Abstract

We introduce new potential type operators \(J^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}\), (α > 0, β > 0), and bi-parametric scale of function spaces \(H^{\alpha}_{\beta , p}({\mathbb{R}}^n)\) associated with Jαβ. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces \(H^{\alpha}_{\beta, p}({\mathbb{R}}^n)\) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).