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

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

383 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

Access to Document

10.1016/j.aim.2008.10.014

Cite this

@article{f0ed344091c8448aa38be043084a8717,

title = "New maximal functions and multiple weights for the multilinear Calder{\'o}n-Zygmund theory",

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.",

keywords = "Calder{\'o}n-Zygmund theory, Commutators, Maximal operators, Multilinear singular integrals, Weighted norm inequalities",

author = "Lerner, {Andrei K.} and Sheldy Ombrosi and Carlos P{\'e}rez and Torres, {Rodolfo H.} and Rodrigo Trujillo-Gonz{\'a}lez",

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.",

year = "2009",

month = mar,

day = "1",

doi = "10.1016/j.aim.2008.10.014",

language = "אנגלית",

volume = "220",

pages = "1222--1264",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Academic Press Inc.",

number = "4",

}

TY - JOUR

T1 - New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

AU - Lerner, Andrei K.

AU - Ombrosi, Sheldy

AU - Pérez, Carlos

AU - Torres, Rodolfo H.

AU - Trujillo-González, Rodrigo

N1 - 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.

PY - 2009/3/1

Y1 - 2009/3/1

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.

AB - 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.

KW - Calderón-Zygmund theory

KW - Commutators

KW - Maximal operators

KW - Multilinear singular integrals

KW - Weighted norm inequalities

UR - http://www.scopus.com/inward/record.url?scp=58149488770&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2008.10.014

DO - 10.1016/j.aim.2008.10.014

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

SN - 0001-8708

VL - 220

SP - 1222

EP - 1264

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 4

ER -

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

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this