Weak porosity in spaces of homogeneous type

Abstract

Based on the theory of Muckenhoupt weights, short and conceptually simpler proofs are provided for the following two implications: (1) if $(X, d, \mu)$ is a space of homogeneous type where $d$-balls are open sets and the Lebesgue differentiation theorem holds true and if $E \subset X$ is a weakly porous set whose maximal $E$-free hole function $\rho_{d, E}$ is doubling, then $\dist{\cdot, E}^{-\alpha} \in A_1(X, d, \mu)$ for some $\alpha > 0$; and (2) now without the assumption on the validity of Lebesgue's differentiation theorem, if $\dist{\cdot, E}^{-\alpha} \in A_1(X, d, \mu)$ for some $\alpha > 0$, then $\rho_{d, E}$ is doubling.