Skip to main content
SpringerLink
Log in

A note on Väisälä’s problem concerning free quasiconformal mappings

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we provide partial solutions to a problem raised by Väisälä on local properties of free quasiconformal mappings. In particular, we show that a locally free quasiconformal mapping is a globally free quasiconformal mapping under the condition of locally relative quasisymmetry.

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

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Gehring, F.W., Osgood, B.G.: Uniform domains and the quasi-hyperbolic metric. J. Anal. Math. 36, 50–74 (1979)

    Article  Google Scholar 

  2. Gehring, F.W., Palka, B.P.: Quasiconformally homogeneous domains. J. Anal. Math. 30, 172–199 (1976)

    Article  MathSciNet  Google Scholar 

  3. Hästö, P., Klén, R., Sahoo, S.K., Vuorinen, M.: Geometric properties of \(\varphi \)-uniform domains. J. Anal. 24, 57–66 (2016)

    Article  MathSciNet  Google Scholar 

  4. Huang, X., Liu, J.: Quasihyperbolic metric and quasisymmetric mappings in metric spaces. Trans. Amer. Math. Soc. 367, 6225–6246 (2015)

    Article  MathSciNet  Google Scholar 

  5. Huang, X., Liu, H., Liu, J.: Local properties of quasihyperbolic mappings in metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 41, 23–40 (2016)

    Article  MathSciNet  Google Scholar 

  6. Huang, M., Ponnusamy, S., Wang, X., Sahoo, S.K.: The Apollonian inner metric and uniform domains. Math. Nachr. 283, 1277–1290 (2010)

    Article  MathSciNet  Google Scholar 

  7. Huang, M., Li, Y., Vuorinen, M., Wang, X.: On quasimöbius maps in real Banach spaces. Israel J. Math. 198, 467–486 (2013)

    Article  MathSciNet  Google Scholar 

  8. Huang, M., Rasila, A., Wang, X., Zhou, Q.: Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces. Stud. Math. 242, 267–301 (2018)

    Article  MathSciNet  Google Scholar 

  9. Huang, M., Wang, X., Ponnusamy, S., Sahoo, S.K.: Uniform domains, John domains and quasi-isotropic domains. J. Math. Anal. Appl. 343, 110–126 (2008)

    Article  MathSciNet  Google Scholar 

  10. Li, Y., Ponnusamy, S., Vuorinen, M.: Freely quasiconformal maps and distance ratio metric. J. Aust. Math. Soc. 97, 383–390 (2014)

    Article  MathSciNet  Google Scholar 

  11. Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 383–401 (1978)

    Article  MathSciNet  Google Scholar 

  12. Rasila, A., Talponen, J.: On quasihyperbolic geodesics in banach spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 39, 163–173 (2014)

    Article  MathSciNet  Google Scholar 

  13. Tukia, P., Väisälä, J.: Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6, 303–342 (1981)

    Article  MathSciNet  Google Scholar 

  14. Väisälä, J.: Free quasiconformality in Banach spaces. I. Ann. Acad. Sci. Fenn. Ser. A I Math. 15, 355–379 (1990)

    Article  MathSciNet  Google Scholar 

  15. Väisälä, J.: Free quasiconformality in Banach spaces. II. Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 255–310 (1991)

    Article  MathSciNet  Google Scholar 

  16. Väisälä, J.: Free quasiconformality in Banach spaces . III. Ann. Acad. Sci. Fenn. Ser. A I Math. 17, 393–408 (1992)

    Article  MathSciNet  Google Scholar 

  17. Väisälä, J.: Free quasiconformality in Banach spaces. IV, analysis and topology, pp. 697–717. World Sci. Publ, River Edge, N. J. (1998)

    MATH  Google Scholar 

  18. Väisälä, J.: The free quasiworld. Freely quasiconformal and related maps in Banach spaces. Quasiconformal geometry and dynamics (Lublin 1996). Banach Cent. Publ. 48, 55–118 (1999)

    Article  Google Scholar 

  19. Vuorinen, M.: Conformal invariants and quasiregular mappings. J. Anal. Math. 45, 69–115 (1985)

    Article  MathSciNet  Google Scholar 

  20. Wang, X., Zhou, Q., Guan, T., Huang, M., Rasila, A.: A note on the relationship between quasi-symmetric mappings and \(\varphi \)-uniform domains. J. Math. Anal. Appl. 445, 1114–1119 (2017)

    Article  MathSciNet  Google Scholar 

  21. Zhou, Q., Rasila, A., Li, Y.: Free distance ratio mappings in Banach spaces. Monatsh Math. 445, 843–856 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The first author was supported by NNSF of China (No. 11901090), Department of Education of Guangdong Province, China (No. 2018KQNCX285), the Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515012289), and Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology (No. 2020B1212030010). The second author was supported by NNSF of China (No. 11601529, 11971124) and by Scientific Research Fund of Hunan Provincial Education Department (No. 20B118). The third author was supported by NNSF of China (No. 11971124) and NSF of Guangdong Province (No. 2021A1515010326).

Author information

Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, Guangdong, People’s Republic of China

    Qingshan Zhou

  2. Department of Mathematics, Hunan First Normal University, Changsha, 410205, Hunan, People’s Republic of China

    Yaxiang Li

  3. Mathematics with Computer Science Program, Guangdong Technion, 241 Daxue Road, Shantou, 515063, Guangdong, People’s Republic of China

    Antti Rasila

  4. Department of Mathematics, Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Antti Rasila

Authors
  1. Qingshan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaxiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antti Rasila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antti Rasila.

Additional information

Communicated by Adrian Constantin.

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

Zhou, Q., Li, Y. & Rasila, A. A note on Väisälä’s problem concerning free quasiconformal mappings. Monatsh Math 196, 607–616 (2021). https://doi.org/10.1007/s00605-021-01554-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-021-01554-4

Keywords

Mathematics Subject Classification

Search

Navigation