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Characterization of compactness of commutators of bilinear singular integral operators


Authors: Lucas Chaffee, Peng Chen, Yanchang Han, Rodolfo H. Torres and Lesley A. Ward
Journal: Proc. Amer. Math. Soc. 146 (2018), 3943-3953
MSC (2010): Primary 42B20; Secondary 47B07, 42B35, 47G99
DOI: https://doi.org/10.1090/proc/14050
Published electronically: June 11, 2018
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Abstract: The commutators of bilinear Calderón-Zygmund operators and pointwise multiplication with a symbol in $ \textup {CMO}$ are bilinear compact operators on products of Lebesgue spaces. We show that, for certain non-degenerate Calderón-Zygmund operators, the symbol being in $ \textup {CMO}$ is not only sufficient but actually necessary for the compactness of the commutators.


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Additional Information

Lucas Chaffee
Affiliation: Department of Mathematics, Western Washington University, 516 High Street, Bellingham, Washington 98225
Email: Lucas.Chaffee@wwu.edu

Peng Chen
Affiliation: Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guang-zhou, 510275, People’s Republic of China
Email: chenpeng3@mail.sysu.edu.cn

Yanchang Han
Affiliation: School of Mathematical Sciences, South China Normal University, Guang-zhou, 510631, People’s Republic of China
Email: hanych@scnu.edu.cn

Rodolfo H. Torres
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: torres@ku.edu

Lesley A. Ward
Affiliation: School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes SA 5095, Australia
Email: lesley.ward@unisa.edu.au

DOI: https://doi.org/10.1090/proc/14050
Keywords: Calder\'on--Zygmund theory, singular integrals, commutators, bilinear operators, compact operators, bounded mean oscillation, CMO
Received by editor(s): September 6, 2017
Received by editor(s) in revised form: December 9, 2017
Published electronically: June 11, 2018
Additional Notes: The second author was supported by NNSF of China 11501583, Guangdong Natural Science Foundation 2016A030313351, the Fundamental Research Funds for the Central Universities 161gpy45, and by the Australian Research Council, Grant No. ARC-DP160100153.
The third author was supported by Guangdong Province Natural Science Foundation Grant No. 2017A030313028.
The second and fifth authors were supported by the Australian Research Council, Grant No. ARC-DP160100153.
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2018 American Mathematical Society