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Variational inequalities for the commutators of rough operators with BMO functions

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Abstract

Starting with the relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the bounded mean oscillation (BMO) functions can be obtained from their weighted variational estimates, we establish similar variational estimates for the commutators of the BMO functions with rough singular integrals, which do not admit any weighted variational estimates. The proof involves several Littlewood-Paley-type inequalities with the commutators as well as Bony decomposition and related para-product estimates.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11871096, 11471033, 11571160 and 11601396), Thousand Youth Talents Plan of China (Grant No. 429900018-101150(2016)), Funds for Talents of China (Grant No. 413100002), the Fundamental Research Funds for the Central Universities (Grant No. FRF-BR-16-011A, 2014KJJCA10) and Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130003110003). The authors express their deep gratitude to the referees for giving many helpful suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Bei**g, Bei**g, 100083, China

    Yan** Chen

  2. School of Mathematical Sciences, Bei**g Normal University, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Bei**g, 100875, China

    Yong Ding

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Guixiang Hong

  4. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454003, China

    Honghai Liu

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  1. Yan** Chen
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  2. Yong Ding
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  3. Guixiang Hong
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  4. Honghai Liu
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Correspondence to Yan** Chen.

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Chen, Y., Ding, Y., Hong, G. et al. Variational inequalities for the commutators of rough operators with BMO functions. Sci. China Math. 64, 2437–2460 (2021). https://doi.org/10.1007/s11425-019-1713-x

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  • DOI: https://doi.org/10.1007/s11425-019-1713-x

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