Localization and dimension free estimates for maximal functions

In the recent paper [A. Naor, T. Tao, Random martingales and localization of maximal inequalities, J. Funct. Anal. 259 (3) (2010) 731–779], Naor and Tao introduce a new class of measures with a so-called micro-doubling property and present, via martingale theory and probability methods, a localization theorem for the associated maximal functions. As a consequence they obtain a weak type estimate in a general abstract setting for these maximal functions that is reminiscent of the ‘nlogn result’ of Stein and Strömberg in Euclidean spaces. The purpose of this work is twofold. First we introduce a new localization principle that localizes not only in the time-dilation parameter but also in space. The proof uses standard covering lemmas and selection processes. Second, we show that a uniform condition for micro-doubling in the Euclidean spaces provides dimension free estimates for their maximal functions in all LpLp with p>1p>1. This is done introducing a new technique that allows to differentiate through dimensions.