The Haj{\l}asz capacity density condition is self-improving

TL;DR

This paper demonstrates that a capacity density condition related to nonlocal Hajlasz gradients in geodesic spaces inherently improves itself, linking it to boundary inequalities and codimension bounds.

Contribution

It establishes a self-improvement property of the Hajlasz capacity density condition by connecting it with boundary Poincaré inequalities and Assouad codimension bounds.

Findings

Capacity density condition is self-improving.

Characterization via upper Assouad codimension.

Extension of Keith-Zhong and Koskela-Zhong techniques.

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 Poincar\'e 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.