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

Let (chi, d, mu) a complete metric measure space and mu be a non-negative Borel regular measure satisfying the doubling condition with some dimensional constant d. We prove that the Hausdorif content of codimension alpha is an element of[0,infinity), denoted by H-alpha, and the Hardy Littlewood maximal operator M satisfy the strong-type inequality integral(x) (Mu)(p) dH(alpha) <= C integral(x) u(p) dH(alpha), 0 <= u is an element of L-loc(1) (chi), whenever p is an element of(max{0, 1 - alpha/d}, infinity). If mu further satisfies some reverse doubling condition with some other dimensional constant kappa, then for the endpoint case p = 1 - alpha/d with alpha is an element of[0, d) boolean AND [0, kappa], we also obtain the corresponding weak-type estimate for H-alpha and M. The fundamental point in the proofs is to introduce and develop a theory of the dyadic Hausdorff content H-D(alpha), which is a Choquet capacity comparable to H-alpha and has the strong subadditivity property. (C) 2016 Elsevier Inc. All rights reserved.