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
[Caractérisation des espaces de Lipschitz via les commutateurs de l'opérateur maximal de Hardy–Littlewood]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2016-11-06
Accepté le : 2017-02-01
Publié le : 2017-02-20

DOI : 10.1016/j.crma.2017.01.022

Affiliations des auteurs :

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},
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     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/

Bibliographie
Cité par

[1] D.R. Adams A note on Riesz potentials, Duke Math. J., Volume 42 (1975), pp. 765-778

[2] D.R. Adams Morrey Spaces, Lect. Notes Appl. Numer. Harmon. Anal., Springer International Publishing, Switzerland, 2015

[3] M. Agcayazi; A. Gogatishvili; K. Koca; R. Mustafayev A note on maximal commutators and commutators of maximal functions, J. Math. Soc. Jpn., Volume 67 (2015) no. 2, pp. 581-593

[4] J. Bastero; M. Milman; F.J. Ruiz Commutators for the maximal and sharp functions, Proc. Amer. Math. Soc., Volume 128 (2000) no. 11, pp. 3329-3334

[5] A. Bonami; T. Iwaniec; P. Jones; M. Zinsmeister On the product of functions in BMO and $H^{1}$ , Ann. Inst. Fourier, Volume 57 (2007) no. 5, pp. 1405-1439

[6] R.R. Coifman; R. Rochberg; G. Weiss Factorization theorems for Hardy spaces in several variables, Ann. Math., Volume 103 (1976) no. 2, pp. 611-635

[7] R.A. DeVore; R.C. Sharpley Maximal functions measuring smoothness, Mem. Amer. Math. Soc., Volume 47 (1984) no. 293, pp. 1-115

[8] J. Duoandikoetxea Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., Providence, RI, USA, 2001

[9] J. García-Cuerva; E. Harboure; C. Segovia; J.L. Torrea Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J., Volume 40 (1991), pp. 1397-1420

[10] L. Grafakos Classical Fourier Analysis, Grad. Texts Math., vol. 249, Springer, New York, 2014

[11] G. Hu; H. Lin; D. Yang Commutators of the Hardy–Littlewood maximal operator with BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal. (2008) (21 pages)

[12] G. Hu; D. Yang Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type, J. Math. Anal. Appl., Volume 354 (2009), pp. 249-262

[13] S. Janson Mean oscillation and commutators of singular integral operators, Ark. Mat., Volume 16 (1978), pp. 263-270

[14] S. Janson; M. Taibleson; G. Weiss Elementary characterization of the Morrey–Campanato spaces, Lect. Notes Math., Volume 992 (1983), pp. 101-114

[15] Y. Komori; T. Mizuhara Notes on commutators and Morrey spaces, Hokkaido Math. J., Volume 32 (2003), pp. 345-353

[16] M. Milman; T. Schonbek Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc., Volume 110 (1990) no. 4, pp. 961-969

[17] C.B. Morrey On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., Volume 43 (1938), pp. 126-166

[18] M. Paluszyński Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., Volume 44 (1995) no. 1, pp. 1-17

[19] J. Peetre On the theory of $L_{p, λ}$ spaces, J. Funct. Anal., Volume 4 (1969), pp. 71-87

[20] C. Segovia; J.L. Torrea Vector-valued commutators and applications, Indiana Univ. Math. J., Volume 38 (1989) no. 4, pp. 959-971

[21] C. Segovia; J.L. Torrea Higher order commutators for vector-valued Calderón–Zygmund operators, Trans. Amer. Math. Soc., Volume 336 (1993) no. 2, pp. 537-556

[22] S. Shirai Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces, Hokkaido Math. J., Volume 35 (2006), pp. 683-696

[23] P. Zhang Multiple weighted estimates for commutators of multilinear maximal function, Acta Math. Sin. Engl. Ser., Volume 31 (2015) no. 6, pp. 973-994

[24] P. Zhang; J.L. Wu Commutators of the fractional maximal functions, Acta Math. Sin., Volume 52 (2009) no. 6, pp. 1235-1238

[25] P. Zhang; J.L. Wu Commutators of the fractional maximal function on variable exponent Lebesgue spaces, Czechoslov. Math. J., Volume 64 (2014), pp. 183-197

[26] P. Zhang; J.L. Wu Commutators for the maximal functions on Lebesgue spaces with variable exponent, Math. Inequal. Appl., Volume 17 (2014) no. 4, pp. 1375-1386

Cité par Sources :

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

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