Riesz Type Potentials in the framework of quasi-metric spaces equipped with upper doubling measures

TL;DR

This paper extends Riesz potentials to quasi-metric spaces with upper doubling measures, establishing boundedness conditions on Lebesgue spaces and applying results to spaces with components of different dimensions.

Contribution

It introduces a unified approach to Riesz potential boundedness in quasi-metric spaces with upper doubling measures, including non-doubling cases and spaces with unequal dimensions.

Findings

Necessary and sufficient conditions for boundedness on Lebesgue spaces.

A geometric property of measures enabling unified boundedness proofs.

Application to Riesz potentials on composite spaces with different dimensions.

Abstract

The purpose of this paper is threefold. First the natural extension of Riesz potentials to the context of quasi metric measure spaces for the class of upper doubling measures are studied on Lebesgue spaces, obtaining necessary and sufficient conditions on a upper doubling measure. Second, we exhibit a geometric property of the measure of the ball which permit prove the boundedness in a unified way, both in of the doubling as non doubling situation. Third, we show that the result can be applied to a type Riesz potential operator defined over a space formed by two components which are not of necessarily equal dimensions.