Weighted estimates for fractional integrals with Distances to Bounded Median Porous Sets and applications to Hardy--Sobolev Inequalities

Abstract

Weighted estimates for the fractional integral operator $I_{\alpha }$ are established and subsequently applied to derive corresponding Hardy--Sobolev inequalities. The weights are constructed from distance functions to bounded median porous sets and possess mixed homogeneity, which enables us to extend earlier results obtained for porous sets to a significantly broader class of geometries.

17 pages