Centered Hardy–Littlewood maximal operator on the real line: Lower bounds

Paata Ivanisvili; Samuel Zbarsky

doi:10.1016/j.crma.2019.03.003

Mathematical analysis/Harmonic analysis

Centered Hardy–Littlewood maximal operator on the real line: Lower bounds

Presented by: the Editorial Board

Paata Ivanisvili ¹; Samuel Zbarsky ¹

¹ Princeton University, Princeton, NJ, USA

Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 339-344.

Abstract
Résumé

For $1 < p < \infty$ and M the centered Hardy–Littlewood maximal operator on $R$ , we consider whether there is some $ε = ε (p) > 0$ such that $| | M f | |_{p} \geq (1 + ε) | | f | |_{p}$ . We prove this for $1 < p < 2$ . For $2 \leq p < \infty$ , we prove the inequality for indicator functions and for unimodal functions.

Soient $1 < p < \infty$ et M la fonction maximale de Hardy–Littlewood sur $R$ . Nous étudions l'existence d'un $ε = ε (p) > 0$ tel que $| | M f | |_{p} \geq (1 + ε) | | f | |_{p}$ . Nous l'établissons pour $1 < p < 2$ . Pour $2 \leq p < \infty$ , nous prouvons l'inégalité pour les fonctions indicatrices et les fonctions unimodales.

Received: 2018-10-09
Accepted: 2019-03-11
Published online: 2019-03-26

DOI: 10.1016/j.crma.2019.03.003

Author's affiliations:

Paata Ivanisvili ¹; Samuel Zbarsky ¹

¹ Princeton University, Princeton, NJ, USA

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     author = {Paata Ivanisvili and Samuel Zbarsky},
     title = {Centered {Hardy{\textendash}Littlewood} maximal operator on the real line: {Lower} bounds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {339--344},
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     language = {en},
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Paata Ivanisvili; Samuel Zbarsky. Centered Hardy–Littlewood maximal operator on the real line: Lower bounds. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 339-344. doi : 10.1016/j.crma.2019.03.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.03.003/

References
Cited by

[1] L. Grafakos; S. Montgomery-Smith Best constants for uncentred maximal functions, Bull. Lond. Math. Soc., Volume 29 (1997) no. 1, pp. 60-64

[2] P. Ivanisvili; B. Jaye; F. Nazarov Lower bounds for uncentered maximal functions in any dimension, Int. Math. Res. Not., Volume 2017 (2016) no. 8, pp. 2464-2479

[3] S. Korry Fixed points of the Hardy–Littlewood maximal operator, Collect. Math., Volume 52 (2001) no. 3, pp. 289-294

[4] A.K. Lerner Some remarks on the Fefferman–Stein inequality, J. Anal. Math., Volume 112 (2010), pp. 329-349

[5] A.D. Melas; E.N. Nikolidakis Local lower norm estimates for dyadic maximal operators and related Bellman functions, J. Geom. Anal., Volume 27 (2017) no. 3, pp. 1940-1950

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