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Improved Regularity of Harmonic Diffeomorphic Extensions on Quasihyperbolic Domains

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Abstract

Let \(\mathbb{X}\) be a Jordan domain satisfying certain hyperbolic growth conditions. Assume that φ is a homeomorphism from the boundary \(\partial \mathbb{X}\) of \(\mathbb{X}\) onto the unit circle. Denote by h the harmonic diffeomorphic extension of φ from \(\mathbb{X}\) onto the unit disk. We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of h. These generalize the Sobolev regularity of h in [A. Koski, J. Onninen, Sobolev homeomorphic extensions, J. Eur. Math. Soc. 23 (2021) 4065–4089, Theorem 3.1].

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Acknowledgements

The authors would like to thank Prof. Chang-Yu Guo for a careful reading of the manuscript and for many suggestions.

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Authors and Affiliations

  1. MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Zhuang Wang

  2. Frontiers Science Center for Nonlinear Expectations (Ministry of Education of China), Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, 266237, China

    Haiqing Xu

Authors
  1. Zhuang Wang
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  2. Haiqing Xu
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Correspondence to Haiqing Xu.

Additional information

The authors were partially supported by the Young Scientist Program of the Ministry of Science and Technology of China (2021YFA1002200). The first author was supported by National Natural Science Foundation of China (12101226). The second author was supported by Shandong Provincial Natural Science Foundation (ZR2021QA032), and partially supported by the National Natural Science Foundation of China (12101362).

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Wang, Z., Xu, H. Improved Regularity of Harmonic Diffeomorphic Extensions on Quasihyperbolic Domains. Acta Math Sci 43, 373–386 (2023). https://doi.org/10.1007/s10473-023-0121-8

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  • DOI: https://doi.org/10.1007/s10473-023-0121-8

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