The Hajłasz capacity density condition is self-improving

Abstract

We prove a self-improvement property of a capacity density condition for a nonlocal Hajlasz gradient in complete geodesic spaces. The proof relates the capacity density condition with boundary Poincaré inequalities, adapts Keith-Zhong techniques for establishing local Hardy inequalities and applies Koskela-Zhong arguments for proving self-improvement properties of local Hardy inequalities. This leads to a characterization of the Hajlasz capacity density condition in terms of a strict upper bound on the upper Assouad codimension of the underlying set, which shows the self-improvement property of the Hajlasz capacity density condition.

Minor correction: connectivity of X is needed in some of the results for metric spaces X. This assumption was omitted in the first version