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.