Annali di Matematica Pura ed Applicata (1923 -)

, Volume 195, Issue 2, pp 323–341

# Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts

Article

## Abstract

It is known that radial symmetry is preserved by the Riesz potential operators and also by the hypersingular Riesz fractional derivatives typically used for inversion. In this paper, we collect properties, asymptotics, and estimates for the radial and spherical parts of Riesz potentials and for solutions to the Riesz potential equation of order one. Sharp estimates for spherical functions are provided in terms of seminorms, and a careful analysis of the radial part of a Riesz potential is carried out in elementary terms. As an application, we provide a two weight estimate for the inverse of the Riesz potential operator of order one acting on spherical functions.

## Keywords

Riesz potentials Singular integrals Weighted spaces Radial functions Spherical symmetry

## Mathematics Subject Classification

47G40 45E99 26A33

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