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Estimates of commutators on Herz-type spaces with variable exponent and applications

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Abstract

Let [b, T] be the commutator generated by b and T, where \(b\in \mathrm {BMO}({\mathbb {R}}^{n})\) and T is a Calderón–Zygmund singular integral operator. In this paper, the authors establish some strong type and weak type boundedness estimates including the \(L\log L\) type inequality for [b, T] on the Herz-type spaces with variable exponent. Meanwhile, the similar results for the commutators \([b,I_l]\) of fractional integral operator are also obtained. As applications, we consider the regularity in the Herz-type spaces with variable exponent of strong solutions to nondivergence elliptic equations with \(\mathrm {VMO}\) coefficients.

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Acknowledgements

The authors are very grateful to the referees for their valuable comments. This work was supported by the National Natural Science Foundation of China (Grant nos. 11926343, 11926342, 11761026, and 11671185), the Shandong Provincial Natural Science Foundation (Grant no. ZR2017MA041), and the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J18KA225).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong, 255000, China

    Hongbin Wang

  2. ICT School, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, Korea

    Zunwei Fu

  3. School of Mathematics and Statistics, Qufu Normal University, Qufu, Shandong, 273100, China

    Zunwei Fu

Authors
  1. Hongbin Wang
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  2. Zunwei Fu
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Correspondence to Zunwei Fu.

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Communicated by Jan van Neerven.

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Wang, H., Fu, Z. Estimates of commutators on Herz-type spaces with variable exponent and applications. Banach J. Math. Anal. 15, 36 (2021). https://doi.org/10.1007/s43037-021-00120-2

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  • DOI: https://doi.org/10.1007/s43037-021-00120-2

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