Skip to main content
SpringerLink
Log in

Extendability of Classes of Maps and New Properties of Upper Sets

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We continue to study upper sets \({\widetilde{A}=\{(x,r)\in A\times R_+ :\exists y\in A\setminus\{x\}, r=|x-y|\}}\) equipped by hyperbolic metric. We define analogous of quasiconvexity, simply connectedness and nearlipschitz functions. We give a new definition of quasisymmetry as nearlipschitz characteristic on \({\widetilde{A}}\). In the final part in terms of upper sets we give the following extension property of \({A\subset R^2}\). For \({0\le\varepsilon\le \delta}\), each \({(1+\varepsilon)}\)-bilipschitz map f : A → R 2 has an extension to a \({(1+C\varepsilon)}\)-bilipschitz map F : R 2 → R 2.

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. Alestalo P., Trotsenko D.A., Väisälä J.: Isometric approximation. Isr. J. Math. 125, 61–82 (2001)

    Article  MATH  Google Scholar 

  2. Alestalo P., Trotsenko D.A., Väisälä J.: Linear bilipschitz extension property. Sib. Math. J. 44(6), 259–268 (2003)

    Article  Google Scholar 

  3. Alestalo P., Trotsenko D.A.: Plane sets allowing bilipschitz extensions. Math. Scand. 105(1), 134–146 (2009)

    MathSciNet  MATH  Google Scholar 

  4. David G.: Hausdorff dimension of uniformly non flat sets with topology. Publ. Math. 48, 187b–225 (2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Pommerenke C.: Uniformly perfect sets and the Poincare metric. Arch. Math. 32, 192–199 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Trotsenko D.A.: Upper sets and uniform domains. Math. Rep. (Romanian Academy) 2(52), 553–562 (2000)

    MathSciNet  Google Scholar 

  8. Trotsenko D.A.: Extension of nearly conformal spatial quasiconformal mappings. Sib. Math. J. 28(6), 966–971 (1987) (translated)

    Article  MathSciNet  MATH  Google Scholar 

  9. Trotsenko D.A.: Continuation from the domain and approximation of spatial quasiconformal mappings with a small distortion coefficient. Dokl. Akad. Nauk SSSR 270(6), 1331–1333 (1983) (Russian)

    MathSciNet  Google Scholar 

  10. Trotsenko, D.A.: Approximation of space mappings with bounded distortion by similarities. Sib. Math. J. 27, 946–954 (1986) [translation from Sib. Mat. Zh. 27, No.6(160), 196–205 (1986)]

  11. Trotsenko D.A., Väisälä J.: Upper sets and quasisymmetric maps. Ann. Acad. Sci. Fennicae Math. 24, 465–488 (1999)

    MATH  Google Scholar 

  12. Väisälä J.: Bilipschitz and quasisymmetric extension properties. Ann. Acad. Sci. Fenn. Math. 11, 239–274 (1986)

    MATH  Google Scholar 

  13. Väisälä J., Vuorinen M., Wallin H.: Thick sets and quasisymmetric maps. Nagoya Math. J. 135, 121–148 (1994)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics SO RAN, Novosibirsk, Russia

    D. A. Trotsenko

Authors
  1. D. A. Trotsenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. A. Trotsenko.

Additional information

Communicated by Matti Vuorinen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trotsenko, D.A. Extendability of Classes of Maps and New Properties of Upper Sets. Complex Anal. Oper. Theory 5, 967–984 (2011). https://doi.org/10.1007/s11785-010-0096-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-010-0096-z

Keywords

Search

Navigation