Weak capacity and modulus comparability in Ahlfors regular metric spaces

TL;DR

This paper introduces new notions of capacity in Ahlfors regular metric spaces using hyperbolic fillings, demonstrating their comparability, invariance under quasisymmetry, and applications to geometric quasiconformality.

Contribution

It defines and analyzes $p$-capacity and weak covering $p$-capacity in Ahlfors regular spaces, establishing their properties and invariance, and applies these to quasiconformal mapping theory.

Findings

Establishes comparability of $p$-capacity and weak covering $p$-capacity.

Proves quasisymmetric invariance of these capacities.

Deduces a result on geometric quasiconformality of quasisymmetric maps.

Abstract

Let $(Z,d,\mu)$ be a compact, connected, Ahlfors $Q$-regular metric space with $Q>1$. Using a hyperbolic filling of $Z$, we define the notions of the $p$-capacity between certain subsets of $Z$ and of the weak covering $p$-capacity of path families $\Gamma$ in $Z$. We show comparability results and quasisymmetric invariance. As an application of our methods we deduce a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected Ahlfors $Q$-regular metric spaces.