Comparison of Hardy–Littlewood and dyadic maximal functions on spaces of homogeneous type ✩

We obtain a comparison of the level sets for two maximal functions on a space of homogeneous type: the Hardy–Littlewood maximal function of mean values over balls and the dyadic maximal function of mean values over the dyadic sets introduced by M. Christ in [M. Christ, A T (b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601–628]. As applications to the theory of Ap weights on this setting, we compare the standard and the dyadic Muckenhoupt classes and we give an alternative proof of reverse Hölder type inequalities.  2005 Elsevier Inc. All rights reserved.