The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension

J. M. Aldaz

doi:http://doi.org/10.4007/annals.2011.173.2.10

Abstract

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d \to \infty$ as $d\to \infty$, thus answering, for the case of cubes, a longstanding open question of E. M. Stein and J. O. Strömberg.

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@article {A, MRKEY = {1606325}, AUTHOR = {Aldaz, J. M.}, TITLE = {Remarks on the {H}ardy-{L}ittlewood maximal function}, JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A}, FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics}, VOLUME = {128}, YEAR = {1998}, NUMBER = {1}, PAGES = {1--9}, ISSN = {0308-2105}, CODEN = {PEAMDU}, MRCLASS = {42B25}, MRNUMBER = {1606325}, MRREVIEWER = {Javier Soria}, ZBLNUMBER = {0892.42010}, }
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@misc{ACP, author={Aldaz, J. M. and Colzani, L. and Pérez Lázaro, J.}, TITLE={Optimal bounds on the modulus of continuity of the uncentered {H}ardy-{L}ittlewood maximal function}, NOTE={{\it {J}. {G}eom. {A}nal.}, to appear}, ARXIV={1009.1359v1}, }
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The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension

Abstract

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