Localization and dimension free estimates for maximal functions

In the recent paper [J. Funct. Anal. {\bf 259} (2010)], 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 `$n\log n$ result' of Stein and Str\"omberg 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 indeed dimension free estimates for their maximal functions in all $L^p$ with $p>1$. This is done introducing a new technique that allows to differentiate through dimensions.