2026/07/26 by Diego Maldonado
#math.CA
Based on the theory of Muckenhoupt weights, short and conceptually simpler proofs are provided for the following two implications: (1) if (X, d, μ) is a space of homogeneous type where d-balls are open sets and the Lebesgue differentiation theorem holds true and if E ⊂ X is a weakly porous set whose maximal E-free hole function ρd, E is doubling, then \dist⋅, E-α ∈ A1(X, d, μ) for some α> 0; and (2) now without the assumption on the validity of Lebesgue's differentiation theorem, if \dist⋅, E-α ∈ A1(X, d, μ) for some α> 0, then ρd, E is doubling.