Abstract dyadic cubes, maximal operators and Hausdorff content

Let μ   be a locally finite Borel measure and DD a family of measurable sets equipped with a certain dyadic structure. For E⊂RnE⊂Rn and 0<α≤n0<α≤n, by α-dimensional Hausdorff content we meanHμα(E)=inf⁡∑jμ(Qj)α/n, where the infimum is taken over all coverings of E   by countable families of the abstract dyadic cubes {Qj}⊂D{Qj}⊂D. In this paper we study the boundedness of the Hardy–Littlewood maximal operator MDμ adapted to DD and μ  , that is, we prove the strong type (p,p)(p,p) inequality∫(MDμf)pdHμα≤22p+2min⁡(1,p)−(α/n)∫|f|pdHμα for α/nt})≤4(n/α)α/nt−α/n∫|f|α/ndHμα,t>0, where the integrals are taken in the Choquet sense.