The Hajłasz Capacity Density Condition is Self-improving

Abstract

We prove a self-improvement property of a capacity density condition for a nonlocal Hajłasz gradient in complete geodesic spaces with a doubling measure. 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 Hajłasz 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 Hajłasz capacity density condition.