On the lack of dimension free estimates in Lp for maximal functions associated to radial measures

TL;DR

This paper demonstrates that for small p near 1, dimension-free Lp bounds for maximal functions associated with radial measures do not exist, with specific examples like Gaussian measures showing improved estimates.

Contribution

It extends previous results by showing the lack of dimension-free bounds for small p and provides concrete examples where better estimates are achieved.

Findings

No dimension-free Lp bounds near p=1 for radial measures

Existence of measures like Gaussian with improved estimates

Extension of Aldaz's results to small p near 1

Abstract

In a recent article J. Aldaz proved that the weak L1 bounds for the centered maximal operator associated to finite radial measures cannot be taken independently with respect to the dimension. We show that at least for small p near to 1 the same result holds for the Lp bounds of such measures with decreasing densities. We also give some concrete examples, that include Gaussian measure, where better estimates with respect to the general case are obtained.