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

Authors: Xuan Thinh Duong, Loukas Grafakos and Lixin Yan
Journal: Trans. Amer. Math. Soc. 362 (2010), 2089-2113
MSC (2000): Primary 42B20, 42B25; Secondary 46B70, 47G30
Published electronically: October 20, 2009
MathSciNet review: 2574888
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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 $ L^1 \times \dots \times L^1 $ to weak $ L^{1/m} $ estimates for the $ m$th-order commutator of Calderón. Our results reproduce known estimates for $ m = 1, 2$ but are new for $ m \ge 3$. We also explore connections between the $ 2$nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.

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Additional Information

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, NSW, 2109, Australia

Loukas Grafakos
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Lixin Yan
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China

Keywords: Multilinear operators, approximation to the identity, generalized Calder\'on-Zygmund kernel, Calder\'on-Zygmund decomposition, commutators
Received by editor(s): January 28, 2008
Received by editor(s) in revised form: May 9, 2008
Published electronically: October 20, 2009
Additional Notes: The first author was supported by a grant from the Australia Research Council.
The second author was supported by grant DMS $0400387$ of the US National Science Foundation and by the University of Missouri Research Council
The third author was supported by NCET of Ministry of Education of China and NNSF of China (Grant No. 10771221).
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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