Sobolev's inequality for Riesz potentials of functions in non-doubling Morrey spaces

Abstract

The space introduced by Morrey [13] in 1938 has become a useful tool of the study for the existence and regularity of solutions of partial differential equations. In the present paper, we aim to establish Sobolev’s inequality for the Riesz potentials of functions in generalized Morrey spaces in the non-doubling setting, as extensions of Gogatishvili-Koskela [4], Orobitg-Perez [14] and Sawano-Sobukawa-Tanaka [19]. Let X be a separable metric space with a nonnegative Radon measure . For simplicity, write jx yj for the distance of x and y. We assume that (fxg) = 0 and 0 0, where B(x , r ) denotes the open ball centered at x of radius r > 0. In this paper, may or may not be doubling. Let G be an open set in X . We define the Riesz potential of order for a nonnegative measurable function f on G by