New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

Andrei K. Lerner; Sheldy Ombrosi; Carlos Pérez; Rodolfo H. Torres; Rodrigo Trujillo-González

doi:10.1016/j.aim.2008.10.014

New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

Andrei K. Lerner
, Sheldy Ombrosi
, Carlos Pérez
, Rodolfo H. Torres
, Rodrigo Trujillo-González

University of Seville
Universidad Nacional del Sur
University of Kansas
University of La Laguna

Research output: Contribution to journal › Article › peer-review

420 Scopus citations

Abstract

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.

Original language	English
Pages (from-to)	1222-1264
Number of pages	43
Journal	Advances in Mathematics
Volume	220
Issue number	4
DOIs	https://doi.org/10.1016/j.aim.2008.10.014
State	Published - 1 Mar 2009
Externally published	Yes

Bibliographical note

Funding Information:
✩ A.K. Lerner’s research supported by the Spanish Ministry of Education under the program “Programa Ramón y Cajal, 2006.” S. Ombrosi’s research supported by a fellowship from the same institution. These two authors and C. Pérez are also supported by the same institution under research grant MTM2006-05622. R. Trujillo-González is also supported by the same institution under grant MTM2005-07347. R.H. Torres’ research supported in part by the National Science Foundation under grants OISE 0126272 and DMS 0400423. * Corresponding author.

Funding

✩ A.K. Lerner’s research supported by the Spanish Ministry of Education under the program “Programa Ramón y Cajal, 2006.” S. Ombrosi’s research supported by a fellowship from the same institution. These two authors and C. Pérez are also supported by the same institution under research grant MTM2006-05622. R. Trujillo-González is also supported by the same institution under grant MTM2005-07347. R.H. Torres’ research supported in part by the National Science Foundation under grants OISE 0126272 and DMS 0400423. * Corresponding author.

Funders	Funder number
National Science Foundation	DMS 0400423, OISE 0126272
Ministerio de Educación, Cultura y Deporte	MTM2006-05622, MTM2005-07347

Keywords

Calderón-Zygmund theory
Commutators
Maximal operators
Multilinear singular integrals
Weighted norm inequalities

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10.1016/j.aim.2008.10.014

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abstract = "A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calder{\'o}n-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calder{\'o}n-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.",

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note = "Funding Information: ✩ A.K. Lerner{\textquoteright}s research supported by the Spanish Ministry of Education under the program “Programa Ram{\'o}n y Cajal, 2006.” S. Ombrosi{\textquoteright}s research supported by a fellowship from the same institution. These two authors and C. P{\'e}rez are also supported by the same institution under research grant MTM2006-05622. R. Trujillo-Gonz{\'a}lez is also supported by the same institution under grant MTM2005-07347. R.H. Torres{\textquoteright} research supported in part by the National Science Foundation under grants OISE 0126272 and DMS 0400423. * Corresponding author.",

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N2 - A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.

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New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

Abstract

Bibliographical note

Funding

Keywords

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