Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

Abstract

Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x,E)−α, where E is a closed set in X and $\alpha \in \mathbb {R}$ . We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt’s Ap classes of weights, 1 ≤ p < ∞. With the help of general Ap-weighted embedding results, we then prove (global) Hardy–Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.