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A predual of a predual of Bσ and its applications to commutators

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Abstract

A predual of Bσ-spaces is investigated. A predual of a predual of Bσ-spaces is also investigated, which can be used to investigate the boundedness property of the commutators. The relation between Herz spaces and local Morrey spaces is discussed. As an application of the duality results, one obtains the boundedness of the singular integral operators, the Hardy-Littlewood maximal operators and the fractional integral operators, as well as the commutators generated by the bounded mean oscillation (BMO) and the singular integral operators. What is new in this paper is that we do not have to depend on the specific structure of the operators. The results on the boundedness of operators are formulated in terms of Ḃσ-spaces and Bσ-spaces together with the detailed comparison of the ones in Herz spaces and local Morrey spaces. Another application is the nonsmooth atomic decomposition adapted to Bσ-spaces.

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Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research (C) (Grant No. 16K05209) and the Japan Society for the Promotion of Science. The authors are thankful to Doctor Shohei Nakamura for his pointing out (4.13). The authors are also thankful to Professor Eiichi Nakai for his discussion. Finally, the authors express their deep gratitue to four anonymous reviewers for their fruitful comments to the paper.

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  1. Department of Mathematics and Information Science, Tokyo Metropolitan University, Tokyo, 192-0397, Japan

    Yoshihiro Sawano & Hiroko Yoshida

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  1. Yoshihiro Sawano
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Correspondence to Yoshihiro Sawano.

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Sawano, Y., Yoshida, H. A predual of a predual of Bσ and its applications to commutators. Sci. China Math. 61, 1437–1472 (2018). https://doi.org/10.1007/s11425-017-9187-3

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