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Let (Z, d, μ) be a compact, connected, Ahlfors Q-regular metric space with Q &gt; 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.

Weak Capacity and Modulus Comparability in Ahlfors Regular Metric Spaces