Skip to main content
Log in

Norm Estimates of the Partial Derivatives for Harmonic Mappings and Harmonic Quasiregular Mappings

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Astala, K., Koskela, P.: \(H^p\)-theory for quasiconformal mappings. Pure Appl. Math. Q. 7, 19–50 (2011)

    Article  MathSciNet  Google Scholar 

  2. Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series, vol. 48, p. xvii+677. Princeton, NJ, Princeton University Press (2009)

    MATH  Google Scholar 

  3. Benedict, S., Koskela, P., Li, X.: Weighted Hardy spaces of quasiconformal mappings. arXiv:1904.00519 (2019)

  4. Duren, P.: Theory of \(H^p\) Spaces. Academic Press, New York (1970)

    MATH  Google Scholar 

  5. Duren, P.: Harmonic Mappings in the Plane. Cambridge University Press, New York (2004)

    Book  Google Scholar 

  6. Fink, A.M., Jodeit, M.: Jensen inequalities for functions with higher monotonicities. Aequationes Math. 40, 26–43 (1990)

    Article  MathSciNet  Google Scholar 

  7. Gehring, F.: The \(L^p\)-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)

    Article  MathSciNet  Google Scholar 

  8. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, New York (2000)

    Book  Google Scholar 

  9. Kalaj, D.: Quasiconformal and harmonic mappings between Jordan domains. Math. Zeit. 260, 237–252 (2008)

    Article  MathSciNet  Google Scholar 

  10. Kalaj, D., Pavlović, M.: On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation. Trans. Am. Math. Soc. 363, 4043–4061 (2011)

    Article  MathSciNet  Google Scholar 

  11. Kalaj, D.: On Riesz type inequalities for harmonic mappings on the unit disk. arXiv:1701.04785v2

  12. Mitrinović, D., Pečarić, J., Fink, A.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, London (1993)

    Book  Google Scholar 

  13. Partyka, D., Sakan, K.: A note on non-quasiconformal harmonic extensions. Bull. Soc. Sci. Lett. Lodz 47, 51–63 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Partyka, D., Sakan, K.: On bi-Lipschitz type inequalities for quasiconformal harmonic mappings. Ann. Acad. Sci. Fenn. Ser. A1 32, 579–594 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Pavlović, M.: Boundary correspondence under harmonic quasiconformal homeomorphisma of the unit disk. Ann. Acad. Sci. Fenn. Ser. A1 27, 365–372 (2002)

    MATH  Google Scholar 

  16. Pavlović, M.: Introduction to Function Spaces on the Disk. Beograd Publisher, Beograd (2004)

    MATH  Google Scholar 

  17. Rudin, W.: Real and Complex Analysis. McGraw-Hill Education, New York (1986)

    MATH  Google Scholar 

  18. Väisälä, J.: Lectures on n-Dimensional Quasiconformal Mappings. Springer, Berlin Heidelberg (1971)

    Book  Google Scholar 

  19. Vuorinen, M.: Comformal Geometry and Quasiregular Mappings. Springer, Berlin Heidelberg (1988)

    Book  Google Scholar 

  20. Yamashita, S.: Hardy norm, Bergman norm, and univalency. Ann. Polonici Math. 43, 23–33 (1983)

    Article  MathSciNet  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jian-Feng Zhu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-020-00488-x

Keywords

Mathematics Subject Classification

Navigation