Multilinear operators with non-smooth kernels and commutators of singular integrals

Xuan Thinh Duong; Loukas Grafakos; Lixin Yan

doi:10.1090/S0002-9947-09-04867-3

Multilinear operators with non-smooth kernels and commutators of singular integrals

Xuan Thinh Duong, Loukas Grafakos, Lixin Yan

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60 Citations (Scopus)

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Abstract

We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calderón-Zygmund kernels. As a consequence, we deduce endpoint L¹ X ⋯ X L¹ to weak L^1/m estimates for the mth-order commutator of Calderón. Our results reproduce known estimates for m =1, 2 but are new for m ≥ 3. We also explore connections between the 2nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.

Original language	English
Pages (from-to)	2089-2113
Number of pages	25
Journal	Transactions of the American Mathematical Society
Volume	362
Issue number	4
DOIs	https://doi.org/10.1090/S0002-9947-09-04867-3
Publication status	Published - Apr 2010

Bibliographical note

Copyright 2010 American Mathematical Society. First published in Transactions of the American Mathematical Society, Vol.362, No. 4, published by the American Mathematical Society. The original article can be found at http://www.ams.org/journals/tran/

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title = "Multilinear operators with non-smooth kernels and commutators of singular integrals",

abstract = "We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calder{\'o}n-Zygmund kernels. As a consequence, we deduce endpoint L1 X ⋯ X L1 to weak L1/m estimates for the mth-order commutator of Calder{\'o}n. Our results reproduce known estimates for m =1, 2 but are new for m ≥ 3. We also explore connections between the 2nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.",

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AU - Grafakos, Loukas

AU - Yan, Lixin

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N2 - We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calderón-Zygmund kernels. As a consequence, we deduce endpoint L1 X ⋯ X L1 to weak L1/m estimates for the mth-order commutator of Calderón. Our results reproduce known estimates for m =1, 2 but are new for m ≥ 3. We also explore connections between the 2nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.

AB - We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calderón-Zygmund kernels. As a consequence, we deduce endpoint L1 X ⋯ X L1 to weak L1/m estimates for the mth-order commutator of Calderón. Our results reproduce known estimates for m =1, 2 but are new for m ≥ 3. We also explore connections between the 2nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.

UR - https://www.scopus.com/pages/publications/77949780794

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EP - 2113

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 4

ER -

Multilinear operators with non-smooth kernels and commutators of singular integrals

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