# MathSciDoc: An Archive for Mathematician ∫

#### Classical Analysis and ODEsmathscidoc:1701.05005

Arkiv for Matematik, 52, (2), 367-382, 2012.11
Suppose$μ$is a positive measure on $\mathbb{R}^{2}$ given by$μ$=$ν$×$λ$, where$ν$and$λ$are Radon measures on $\mathcal{S}^{1}$ and $\mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}$ , respectively, which do not vanish on any open interval. We prove that if for either$ν$or$λ$there exists a set of positive measure$A$in its domain for which the upper and lower$s$-densities, 0<$s$≤1, are positive and finite for every$x$∈$A$then the uncentered Hardy–Littlewood maximal operator$M$_{$μ$}is weak-type (1,1) if and only if$ν$is doubling and$λ$is doubling away from the origin. This generalizes results of Vargas concerning rotation-invariant measures on $\mathbb{R}^{n}$ when$n$=2.
@inproceedings{anna2012on,
title={On the weak-type (1,1) of the uncentered Hardy–Littlewood maximal operator associated with certain measures on the plane},
author={Anna K. Savvopoulou, and Christopher M. Wedrychowicz},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203637877654063},
booktitle={Arkiv for Matematik},
volume={52},
number={2},
pages={367-382},
year={2012},
}

Anna K. Savvopoulou, and Christopher M. Wedrychowicz. On the weak-type (1,1) of the uncentered Hardy–Littlewood maximal operator associated with certain measures on the plane. 2012. Vol. 52. In Arkiv for Matematik. pp.367-382. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203637877654063.