A new characterization of the Riesz potential spaces with the aid of a composite wavelet transform

We introduce a composite wavelet-like transform generated by the so-called beta-semigroup and a wavelet measure. This wavelet-like transform enables one to obtain a new explicit inversion formula for the Riesz potentials and a new characterization of the Riesz potential spaces. The usage of the concept "beta-semigroup", which is a natural generalization of the well-known Gauss-Weierstrass and Poisson semigroups, enables one to minimize the number of conditions on wavelet measure, no matter how big the order of Riesz's potentials is. (C) 2010 Elsevier Inc. All rights reserved.