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Transactions of the American Mathematical Society

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End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

Authors: Kangwei Li, José María Martell, Henri Martikainen, Sheldy Ombrosi and Emil Vuorinen
Journal: Trans. Amer. Math. Soc. 374 (2021), 97-135
MSC (2010): Primary 42B20, 42B25, 42B35
Published electronically: October 20, 2020
MathSciNet review: 4188179
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Abstract: In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding “finite” case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm classes. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.

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Additional Information

Kangwei Li
Affiliation: BCAM, Basque Center for Applied Mathematics, Mazarredo 14, E-48009 Bilbao, Spain
Address at time of publication: Center for Applied Mathematics, Tianjin University, Weijin Road 92, 300072 Tianjin, People’s Republic of China
MR Author ID: 977289

José María Martell
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain
MR Author ID: 671782
ORCID: 0000-0001-6788-4769

Henri Martikainen
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, FI-00014 University of Helsinki, Finland
MR Author ID: 963282

Sheldy Ombrosi
Affiliation: Department of Mathematics, Universidad Nacional del Sur, Bahía Blanca, Argentina
MR Author ID: 713193

Emil Vuorinen
Affiliation: Centre for Mathematical Sciences, University of Lund, P.O.B. 118, 22100 Lund, Sweden
MR Author ID: 1178205
ORCID: 0000-0001-8986-0725

Keywords: Multilinear Muckenhoupt weights, Rubio de Francia extrapolation, multilinear Calderón-Zygmund operators, bilinear Hilbert transform, vector-valued inequalities, mixed-norm estimates
Received by editor(s): February 15, 2019
Received by editor(s) in revised form: March 9, 2020
Published electronically: October 20, 2020
Additional Notes: The first author was supported by Juan de la Cierva - Formación 2015 FJCI-2015-24547, by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718, and through project MTM2017-82160-C2-1-P funded by (AEI/FEDER, UE) and acronym “HAQMEC”
The second author was supported by the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554) and the European Research Council through the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT
The third author was supported by the Academy of Finland through the grants 294840 and 306901, and by the three-year research grant 75160010 of the University of Helsinki. He is a member of the Finnish Centre of Excellence in Analysis and Dynamics Research
The fourth author was supported by CONICET PIP 11220130100329CO, Argentina.
The fifth author was supported by the Jenny and Antti Wihuri Foundation.
Article copyright: © Copyright 2020 American Mathematical Society