Pointwise estimate for the Hardy–Littlewood maximal operator on the orbits of contractive mappings

Let M n denote the Hardy–Littlewood maximal operator on the n -th iteration of a given iterated function system (IFS). We give sufficient conditions on the IFS in order to obtain a pointwise estimate for M n in terms of the composition of M 0 and a discrete Hardy–Littlewood type maximal operator. As a corollary we prove the uniform preservation of Muckenhoupt condition along the Hutchinson orbits induced by such an IFS.