Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function

Pu Zhang

doi:10.1016/j.crma.2017.01.022

Harmonic analysis

Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function

Presented by: the Editorial Board

Pu Zhang ¹

¹ Department of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, PR China

Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 336-344.

Abstract
Résumé

Let M be the Hardy–Littlewood maximal function and b be a locally integrable function. Denote by $M_{b}$ and $[b, M]$ the maximal commutator and the (nonlinear) commutator of M with b. In this paper, the author considers the boundedness of $M_{b}$ and $[b, M]$ on Lebesgue spaces and Morrey spaces when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.

Soit M l'opérateur maximal de Hardy–Littlewood et b une fonction localement intégrable. Notons $M_{b}$ et $[b, M]$ le commutateur maximal et le commutateur (non linéaire) de M et b. Dans cette Note, l'auteur étudie la finitude de $M_{b}$ et $[b, M]$ sur les espaces de Lebesgue et les espaces de Morrey lorsque b appartient à l'espace de Lipschitz. Cela conduit à de nouvelles caractérisations de l'espace de Lipschitz.

Received: 2016-11-06
Accepted: 2017-02-01
Published online: 2017-02-20

DOI: 10.1016/j.crma.2017.01.022

Author's affiliations:

Pu Zhang ¹

¹ Department of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, PR China

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     author = {Pu Zhang},
     title = {Characterization of {Lipschitz} spaces via commutators of the {Hardy{\textendash}Littlewood} maximal function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {336--344},
     publisher = {Elsevier},
     volume = {355},
     number = {3},
     year = {2017},
     doi = {10.1016/j.crma.2017.01.022},
     language = {en},
}

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Pu Zhang. Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 336-344. doi : 10.1016/j.crma.2017.01.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.022/

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^☆ Supported by the National Natural Science Foundation of China (Grant Nos. 11571160 and 11471176).

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