Comparison of Hardy--Littlewood and dyadic maximal functions on spaces of homogeneous type (Q2573437)

In this paper, the authors state a Calderón-Zygmund decomposition of the domain of a real integrable function in the setting of metric measure spaces. Their method is based in the original construction of dyadic type families given by \textit{M. Christ} in [Colloq. Math. 60/61, No.~2, 601--628 (1990; Zbl 0758.42009)]. Then, they compare the level sets of the Hardy-Littlewood maximal function and the level sets of the dyadic maximal function, built on these dyadic families. As applications, they compare the Muckenhoupt classes of weights defined through the \(d\)-balls and through this dyadic sets and prove that they are equivalent under the assumption of the doubling property. Reverse Hölder inequalities for \(A_p\) weights on spaces of homogeneous type are also proved.