Skip to main content
SpringerLink
Log in

Homeomorphisms of the Heisenberg group preserving BMO

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We provide a new geometric proof of Reimann’s theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.

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. Astala K.: A remark on quasi-conformal mappings and BMO-functions. Michigan Math. J. 30, 209–212 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich, 2011

  3. F.W. Gehring and J.C. Kelly, Quasi-conformal mappings and Lebesgue density, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 171–179. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974.

  4. Gotoh Y.: On composition operators which preserve BMO. Pacific J. Math. 201, 289–307 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Heinonen, Lectures on Analysis on Metric spaces, Springer, 2001

  6. Heinonen J., Koskela P.: Definitions of quasiconformality. Invent. Math. 120, 61–79 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Heinonen J., Koskela P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Heinonen J., Koskela P., Shanmugalingam N., Tyson J.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jones P. W.: Homeomorphisms of the line which preserve BMO. Ark. Mat. 21, 229–231 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Kinnunen, R. Korte, N. Marola, and N. Shanmugalingam, A characterization of BMO self-maps of a metric measure space. Collect. Math. 66 (2015), 405–421.

  11. Korányi A., Reimann H.M.: Foundations for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111, 1–87 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Korte R., Marola N., Shanmugalingam N.: Quasiconformality, home-omorphisms between metric measure spaces preserving quasiminimizers, and uniform density property. Ark. Math. 50, 111–134 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Magnani V.: Towards differential calculus in stratified groups. J. Aust. Math. Soc. 95, 76–128 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Reimann H.M.: Functions of bounded mean oscillation and quasiconformal mappings. Comment. Math. Helv. 49, 260–276 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  15. Staples S.: Maximal functions, A ∞-measures and quasiconformal maps. Proc. Amer. Math. Soc. 113, 689–700 (1991)

    MATH  MathSciNet  Google Scholar 

  16. Staples S.: Doubling measures and quasiconformal maps. Comment. Math. Helv. 67, 119–128 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Uchiyama A.: Weight functions of the class (A ∞) and quasi-conformal mappings. Proc. Japan Acad. 51, 811–814 (1975)

    Article  MathSciNet  Google Scholar 

  18. Vodop’yanov S.K., Greshnov A.V.: Quasiconformal mappings and BMO-spaces on metric structures. Siberian Advances in Mathematics. 8, 132–150 (1998)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076, Aalto, Finland

    Riikka Korte & Olli Saari

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Niko Marola

Authors
  1. Riikka Korte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Niko Marola
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Olli Saari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Niko Marola.

Additional information

The research is supported by the Academy of Finland, grant no. 250403, and the Väisälä Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Korte, R., Marola, N. & Saari, O. Homeomorphisms of the Heisenberg group preserving BMO. Arch. Math. 106, 175–182 (2016). https://doi.org/10.1007/s00013-015-0849-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-015-0849-z

Mathematics Subject Classification

Keywords

Search

Navigation