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Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations

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Abstract

Let \(p\in (1, \infty )\), ? ? (0,1) and \(w\in A_{p}({\mathbb C}).\) In this article, the authors obtain a boundedness (resp., compactness) characterization of the Beurling–Ahlfors commutator \([\mathcal B, b]\) on the weighted Morrey space \(L_{w}^{p, \kappa }(\mathbb C)\) via \(\text {BMO}({\mathbb C})\) [resp., \(\text {CMO}({\mathbb C})\)], where \(\mathcal B\) denotes the Beurling–Ahlfors transform and \(b\in \text {BMO}({\mathbb C})\) [resp., \(\text {CMO}({\mathbb C})\)]. Moreover, an application to the Beltrami equation is also given.

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Acknowledgements

The authors would like to thank the referees for their carefully reading and several motivating and useful comments which indeed improve the quality of this article.

Author information

Correspondence to Dongyong Yang.

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Dachun Yang is supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 11761131002 and 11671185). Dongyong Yang is supported by the National Natural Science Foundation of China (Grant Nos. 11971402 and 11871254).

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Tao, J., Yang, D. & Yang, D. Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations. Potential Anal (2020). https://doi.org/10.1007/s11118-019-09814-7

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Keywords

  • Beurling–Ahlfors transform
  • Compactness
  • Commutator
  • Weighted Morrey space
  • Beltrami equation

Mathematics Subject Classification (2010)

  • Primary 42B20
  • Secondary 46E35