On the weak-type (1,1) of the uncentered Hardy–Littlewood maximal operator associated with certain measures on the plane

Anna K. Savvopoulou Department of Mathematical Sciences, Indiana University South Bend Christopher M. Wedrychowicz Department of Mathematics and Computer Science, Saint Mary’s College

Classical Analysis and ODEs mathscidoc: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.
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@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.
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