Skip to main content
SpringerLink
Log in

On mappings of finite distortion that are quasiconformal in the unit disk

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We study quasiconformal mappings of the unit disk that have homeomorphic planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.

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

Access this article

Log in via an institution

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. K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, NJ, 2009.

    MATH  Google Scholar 

  2. K. Astala, T. Iwaniec, I. Prause and E. Saksman, Bilipschitz and quasiconformal rotation, stretching and multifractal spectra, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 113–154.

    Article  MathSciNet  MATH  Google Scholar 

  3. K. Astala and V. Nesi, Composites and quasiconformal mappings: new optimal bounds in two dimensions, Calc. Var. Partial Differential Equations 18 (2003), 335–355.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Clop and D. Herron, Mappings with subexponentially integrable distortion: Modulus of continuity, and distortion of Hausdorff measure and Minkowski content, Illinois J. Math. 57 (2013), 965–1008.

    Article  MathSciNet  MATH  Google Scholar 

  5. F. W. Gehring, Characteristic Properties of Quasidisks, Presses de l’Université de Montréal, Montreal, Que., 1982.

    MATH  Google Scholar 

  6. C. Guo, Generalized, quasidisks and conformality II, Proc. Amer. Math. Soc. 143 (2015), 3505–3517.

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Guo, P. Koskela and J. Takkinen, Generalized, quasidisks and conformality, Publ. Mat. 58 (2014), 193–212.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Guo and H. Xu, Generalized, quasidisks and conformality: progress and challenges, Complex Anal. Synerg. 7 (2021), article no. 2.

  9. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1934.

    MATH  Google Scholar 

  10. D. Herron and P. Koskela, Mappings of finite distortion: gauge dimension of generalized, quasi-circles, Illinois J. Math. 47 (2003), 1243–1259.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Hitruhin, Pointwise rotation for mappings with exponentially integrable distortion, Proc. Amer. Math. Soc. 144 (2016), 5183–5195.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Hitruhin, Rotational properties of homeomorphisms with integrable distortion, Conform. Geom. Dyn. 22 (2018), 78–98.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Hitruhin, Joint rotational and stretching multifractal spectra of mappings with integrable distortion, Rev. Mat. Iberoam. 35 (2019), 1649–1675.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Kauhanen, P. Koskela, J. Malý, J. Onninen and X. Zhong, Mappings of finite distortion: sharp Orlicz-conditions, Rev. Mat. Iberoam. 19 (2003), 857–872.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. T. Kemper, A boundary Harnack principle for Lipschitz domains and the principle of positive singularities, Comm. Pure Appl. Math. 25 (1972), 247–255.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Koskela and J. Onninen, Mappings of finite distortion: capacity and modulus inequalities, J. Reine Angew. Math. 599 (2006), 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Koskela and J. Takkinen, Mappings of finite distortion: formation of cusps, Publ. Mat. 51 (2007), 223–242.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Onninen and X. Zhong, A note on mappings of finite distortion: the sharp modulus of continuity, Michigan Math. J. 53 (2005), 329–335.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin-Heidelberg, 1992.

    Book  MATH  Google Scholar 

  20. I. Prause, Quasidisks and twisting of the Riemann map, Int. Math. Res. Not. IMRN 2019 (2019), 1936–1954.

    Article  MathSciNet  MATH  Google Scholar 

  21. I. Prause and S. Smirnov, Quasisymmetric distortion spectrum, Bull. Lond. Math. Soc. 43 (2011), 267–277.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Springer, Berlin-New York, 1988.

    Book  MATH  Google Scholar 

  23. A. Zapadinskaya, Modulus of continuity for quasiregular mappings with finite distortion extension, Ann. Acad. Sci. Fenn. Math. 33 (2008), 373–385.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to the anonymous referee for careful reading of the paper and his or her thoughtful comments. The work was supported by the Finnish AcademyCoe ‘Analysis and Dynamics’, ERCgrant 834728 Quamap and the Finnish Academy projects 1309940 and 13316965.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 University of Helsinki, Helsinki, Finland

    Olli Hirviniemi, Lauri Hitruhin & Eero Saksman

  2. Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101, Joensuu, Finland

    István Prause

Authors
  1. Olli Hirviniemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lauri Hitruhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. István Prause
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eero Saksman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to István Prause.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hirviniemi, O., Hitruhin, L., Prause, I. et al. On mappings of finite distortion that are quasiconformal in the unit disk. JAMA 149, 369–400 (2023). https://doi.org/10.1007/s11854-022-0248-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-022-0248-x

Search

Navigation