Geometric properties of infinite graphs and the Hardy–Littlewood maximal operator

We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy–Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak-type boundedness of the centered Hardy–Littlewood maximal operator. Several non-trivial examples of infinite graphs are given to illustrate the differences among these properties.The first author has been partially supported by the Spanish Government grant MTM2016-75196-P (MINECO / FEDER, UE) and the Catalan Autonomous Government grant 2017SGR358. The second author has been partially supported by the Spanish Government grants MTM2016-75196-P, MTM2016-76808-P, Grupo UCM 910346 and the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554