Abstract
Suppose \(p\ge 1\), \(w=P[F]\) is a harmonic mapping of the unit disk \({\mathbb D}\) satisfying F is absolutely continuous and \({\dot{F}}\in L^p(0, 2\pi )\), where \({\dot{F}}(e^{it})=\frac{\mathrm {d}}{\mathrm {d}t}F(e^{it})\). In this paper, we obtain Bergman norm estimates of the partial derivatives for w, i.e., \(\Vert w_z\Vert _{L^p}\) and \(\Vert \overline{w_{{\bar{z}}}}\Vert _{L^p}\), where \(1\le p<2\). Furthermore, if w is a harmonic quasiregular mapping of \({\mathbb {D}}\), then we show that \(w_z\) and \(\overline{w_{{\bar{z}}}}\) are in the Hardy space \(H^p\), where \(1\le p\le \infty \). The corresponding Hardy norm estimates, \(\Vert w_z\Vert _{p}\) and \(\Vert \overline{w_{{\bar{z}}}}\Vert _{p}\), are also obtained.
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Acknowledgements
The author of this paper would like to thank the anonymous referee for his/her helpful comments that have significant impact on this paper, and would like to thank Professor Ken-ichi Sakan for his help on the discussions of Theorem 1.1.
Funding
The research of the author was supported by NSFs of China (No. 11501220), NSFs of Fujian Province (No. 2016J01020), and the Promotion Program for Young and Middle-aged Teachers in Science and Technology Research of Huaqiao University (ZQN-PY402).
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Zhu, JF. Norm Estimates of the Partial Derivatives for Harmonic Mappings and Harmonic Quasiregular Mappings. J Geom Anal 31, 5505–5525 (2021). https://doi.org/10.1007/s12220-020-00488-x
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DOI: https://doi.org/10.1007/s12220-020-00488-x