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Sharp \(L^{p,\infty }\rightarrow L^q\) Estimates for the Dyadic-Like Maximal Operators

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Abstract

For each \(1\le q<p<\infty \) we study the sharp versions of the \(L^{p,\infty }\rightarrow L^q\) estimates for the dyadic maximal operator on \(\mathbb {R}^n\). Actually, this is done in the more general setting of maximal operators associated with a tree-like structure. The proof rests on a novel combination of the Bellman function technique and optimization arguments.

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Acknowledgments

Research supported by the NCN grant DEC-2012/05/B/ST1/00412. The author would like to thank an anonymous referee for the careful reading of the first version of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 , Warsaw, Poland

    Adam Osȩkowski

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  1. Adam Osȩkowski
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Correspondence to Adam Osȩkowski.

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Communicated by Loukas Grafakos.

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Osȩkowski, A. Sharp \(L^{p,\infty }\rightarrow L^q\) Estimates for the Dyadic-Like Maximal Operators. J Fourier Anal Appl 20, 911–933 (2014). https://doi.org/10.1007/s00041-014-9338-1

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  • DOI: https://doi.org/10.1007/s00041-014-9338-1

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