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

Abstract

Let [bT] 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 [bT] 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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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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Keywords

  • Boundedness estimate
  • Commutator
  • Herz-type spaces
  • Variable exponent

Mathematics Subject Classification

  • 42B20
  • 42B35
  • 46E30