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
[Fonction maximale centrée de Hardy–Littlewood : bornes inférieures]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2018-10-09
Accepté le : 2019-03-11
Publié le : 2019-03-26

DOI : 10.1016/j.crma.2019.03.003

Affiliations des auteurs :

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},
     publisher = {Elsevier},
     volume = {357},
     number = {4},
     year = {2019},
     doi = {10.1016/j.crma.2019.03.003},
     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/

Bibliographie
Cité par

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[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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