Weak Capacity and Modulus Comparability in Ahlfors Regular Metric Spaces

Abstract

Let (Z, d, μ) 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 theweak covering p-capacity of path families Γ in Z.We show comparability results and quasisymmetric invariance.As an application of our methodswe deduce a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected Ahlfors Q-regular metric spaces.

Keywords

• weak capacity
• weak covering capacity
• modulus
• quasisymmetry
• quasi-isometry
• ahlfors regular metric space
• gromov hyperbolic metric space
• ahlfors regular conformal dimension