Failure of the Hörmander kernel condition for multilinear Calderón–Zygmund operators

Loukas Grafakos; Danqing He; Lenka Slavíková

doi:10.1016/j.crma.2019.04.002

Harmonic analysis

Failure of the Hörmander kernel condition for multilinear Calderón–Zygmund operators
[Insuffisance de la condition de noyau de Hörmander pour les opérateurs multilinéaires de Calderón–Zygmund]

Présenté par : Yves Meyer

Loukas Grafakos ¹ ; Danqing He ² ; Lenka Slavíková ¹

¹ Department of Mathematics, University of Missouri, Columbia MO 65211, USA
² Department of Mathematics Sun Yat-sen (Zhongshan) University, Guangzhou, Guangdong, China

Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 382-388.

Abstract (VO)
Résumé

It is well known that the Hörmander smoothness condition $\sup_{y \neq 0} \int_{| x | \geq 2 | y |} | K (x - y) - K (x) | d x < \infty$ implies weak-type $(1, 1)$ estimates for associated $L^{2}$ -bounded Calderón–Zygmund operators. It has been an open question to know whether Hörmander's condition also suffices to guarantee weak-type $(1, 1, 1 / 2)$ estimates for bilinear Calderón–Zygmund operators that are bounded at one point. In this paper, we provide a negative answer to this question.

Il est bien connu que la condition de lissage de Hörmander $\sup_{y \neq 0} \int_{| x | \geq 2 | y |} | K (x - y) - K (x) | d x < \infty$ implique des estimations faibles de type $(1, 1)$ pour les opérateurs de Calderón–Zygmund $L^{2}$ -bornés. La question s'est alors posée de savoir si cette condition de Hörmander est également suffisante pour assurer des estimations faibles de type $(1, 1, 1 / 2)$ pour les opérateurs bilinéaires de Calderón–Zygmund qui sont bornés en un point. Nous donnons ici une réponse négative à cette question.

Reçu le : 2018-12-10
Accepté le : 2019-04-05
Publié le : 2019-04-01

DOI : 10.1016/j.crma.2019.04.002

Affiliations des auteurs :

Loukas Grafakos ¹ ; Danqing He ² ; Lenka Slavíková ¹

¹ Department of Mathematics, University of Missouri, Columbia MO 65211, USA
² Department of Mathematics Sun Yat-sen (Zhongshan) University, Guangzhou, Guangdong, China

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     author = {Loukas Grafakos and Danqing He and Lenka Slav{\'\i}kov\'a},
     title = {Failure of the {H\"ormander} kernel condition for multilinear {Calder\'on{\textendash}Zygmund} operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {382--388},
     publisher = {Elsevier},
     volume = {357},
     number = {4},
     year = {2019},
     doi = {10.1016/j.crma.2019.04.002},
     language = {en},
}

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Loukas Grafakos; Danqing He; Lenka Slavíková. Failure of the Hörmander kernel condition for multilinear Calderón–Zygmund operators. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 382-388. doi : 10.1016/j.crma.2019.04.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.04.002/

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Cité par Sources :

^☆ The first author was supported by the Simons Foundation (No. 315380). The second author was supported by the NNSF of China (No. 11701583), the Guangdong Natural Science Foundation (No. 2017A030310054) and the Fundamental Research Funds for the Central Universities (No. 17lgpy11).

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