Nikodym sets and maximal functions associated with spheres

Alan Chang; Georgios Dosidis; Jongchon Kim

doi:10.4171/RMI/1519

Nikodym sets and maximal functions associated with spheres

Alan Chang
, Georgios Dosidis
, Jongchon Kim

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

4 Citations (Scopus)

629 Downloads (CityUHK Scholars)

Abstract

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp L^p-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that L^p-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures. © 2024 Real Sociedad Matemática Española.

Original language	English
Pages (from-to)	1009-1056
Journal	Revista Matematica Iberoamericana
Volume	41
Issue number	3
Online published	13 Dec 2024
DOIs	https://doi.org/10.4171/RMI/1519
Publication status	Published - 15 Apr 2025

Funding

We thank David Beltran, Loukas Grafakos, Sanghyuk Lee, Malabika Pramanik, Andreas Seeger, Tongou Yang, Joshua Zahl for helpful discussions. We thank Joshua Zahl for pointing us to [24] and for sketching the proof of Theorem 3.5. We thank the anonymous referees for their helpful comments and suggestions. J. Kim was partially supported by a grant from the Research Grants Council of the Hong Kong Administrative Region, China (Project no. CityU 21309222). G. Dosidis was partially supported by the Primus research programme PRIMUS/21/SCI/002 of the Charles University. Part of the work was done while A. Chang and G. Dosidis were visiting the Hausdorff Research Institute for Mathematics in Bonn during the research trimester \u201CInteractions between Geometric measure theory, Singular integrals, and PDE\u201D, which was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany\u2019s Excellence Strategy, EXC-2047/1-390685813.

Research Keywords

Hausdorff dimension
maximal functions
Nikodym sets for spheres

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This full text is made available under CC-BY 4.0. https://creativecommons.org/licenses/by/4.0/

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title = "Nikodym sets and maximal functions associated with spheres",

abstract = "We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp Lp-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that Lp-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures. {\textcopyright} 2024 Real Sociedad Matem{\'a}tica Espa{\~n}ola.",

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TY - JOUR

T1 - Nikodym sets and maximal functions associated with spheres

AU - Chang, Alan

AU - Dosidis, Georgios

AU - Kim, Jongchon

PY - 2025/4/15

Y1 - 2025/4/15

N2 - We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp Lp-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that Lp-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures. © 2024 Real Sociedad Matemática Española.

AB - We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp Lp-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that Lp-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures. © 2024 Real Sociedad Matemática Española.

KW - Hausdorff dimension

KW - maximal functions

KW - Nikodym sets for spheres

UR - https://www.scopus.com/pages/publications/105003414751

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-105003414751&origin=recordpage

U2 - 10.4171/RMI/1519

DO - 10.4171/RMI/1519

M3 - RGC 21 - Publication in refereed journal

SN - 0213-2230

VL - 41

SP - 1009

EP - 1056

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

IS - 3

ER -

Nikodym sets and maximal functions associated with spheres

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