Boundedness of the Hardy–Littlewood Maximal Operator Along the Orbits of Contractive Similitudes

Abstract

In this note we obtain results regarding the preservation of homogeneity properties along the whole orbit of a given iterated function system (IFS). We have essentially two types of results. The first class of them contains negative results: it is possible for a classical IFS to have a complete non-homogeneous sequence of spaces along the orbit, starting from very classical homogeneous spaces such as those defined by Muckenhoupt weights. The second class contains positive results which can be summarized here by saying that the sequence of spaces defined by the orbit of contractive similitudes starting at a normal space in the sense of Ahlfors, Macias, and Segovia, preserves doubling. As a consequence of these results we conclude boundedness properties of the Hardy–Littlewood maximal operator along the orbits.