A unified point of view on boundedness of Riesz type potentials

Abstract

We introduce a natural extension of the Riesz potentials to quasimetricmeasure spaces with an upper doubling measure. In particular, theseoperators are defined when the underlying space has components of differingdimensions. We study the behavior of the potential on classical and variableexponent Lebesgue spaces, obtaining necessary and sufficient conditions forits boundedness. The technique we use relies on a geometric property ofthe measure of the balls which holds both in the doubling and non-doublingsituations, and allows us to present our results in a unified way.