Hausdorff content and the Hardy–Littlewood maximal operator on metric measure spaces

Abstract

Let (X,d,μ) be a complete metric measure space and μ be a non-negative Borel regular measure satisfying the doubling condition with some dimensional constant d. We prove that the Hausdorff content of codimension α∈[0,∞), denoted by Hα, and the Hardy–Littlewood maximal operator M satisfy the strong-type inequality∫X(Mu)pdHα≤C∫XupdHα,0≤u∈Lloc1(X), whenever p∈(max⁡{0,1−α/d},∞). If μ further satisfies some reverse doubling condition with some other dimensional constant κ, then for the endpoint case p=1−α/d with α∈[0,d)∩[0,κ], we also obtain the corresponding weak-type estimate for Hα and M. The fundamental point in the proofs is to introduce and develop a theory of the dyadic Hausdorff content HDα, which is a Choquet capacity comparable to Hα and has the strong subadditivity property.