Skip to main content
SpringerLink
Log in

Extremal Extensions from the Circle to the Disk

  • Chapter
Quasiconformal Mappings and Analysis

Abstract

Let\( \Delta = \{ |z|{\rm{ }} < {\rm{1\} , }}\Gamma {\rm{ = \{ |}}z{\rm{| = 1\} , }}\partial {\rm{ = (}}\partial _x - i\partial _y )/2, \bar \partial = (\partial _x + i\partial _y )/2 \) In this exposition we shall be concerned with two kinds of extension problems from \( \Gamma to\Delta \cup \Gamma \), namely, firstly, the least non-analytic (abbreviated LNA) extensions of continuous complex-valued boundary values, and, secondly, the extremal quasiconformal (abbreviated EQC) extensions of homeomorphic boundary values. We formulate the first problem as follows. Let \( g(z),z \in \Gamma \), be a continuous complex-valued function. It will often be convenient to normalize g by the conditions

$$ Re[\bar zg(z)] = 0, z \in \Gamma , $$
((0.1))
$$ g(1) = g(i) = g( - 1) = 0. $$
((0.2))

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Lars V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Math. Studies N° 10, Princeton, 1966.

    MATH  Google Scholar 

  2. Lipman Bers, A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113–134.

    Article  MathSciNet  MATH  Google Scholar 

  3. Lipman Bers, A new proof of a fundamental inequality for quasiconformal mappings, J. d’Analyse Math. 36 (1979), 15–30.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Beurling and L. Ahlfors, The boundary correspondence under quasi-conformal mappings, Acta Math. 96 (1956), 125–142.

    Article  MathSciNet  MATH  Google Scholar 

  5. F.F. Bonsall and D. Walsh, Symbols for trace class Hankel operators with good estimates for norms, Glasgow Math. J. 28 (1986), 47–54.

    MathSciNet  MATH  Google Scholar 

  6. Clifford J. Earle, Conformally natural extensions of vector fields from S’i to B“, Procs. Am. Math. Soc. 102 (1988), 145–149.

    MathSciNet  MATH  Google Scholar 

  7. Clifford J. Earle and Li Zhong, Extremal quasiconformal mappings in plane domains,[this volume].

    Google Scholar 

  8. R. Fehhnann, Ober extremale quasikonforme Abbildungen, Comm. Math. Heiv. 56 (1981), 558–580.

    Google Scholar 

  9. Frank Forelli and Walter Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974), 593–602.

    Article  MathSciNet  MATH  Google Scholar 

  10. F.W. Gehring and E. Reich, Area distortion under quasiconformal map-pings, Ann. Acad. Sci. Fenn. 388 (1966), 1–15.

    MathSciNet  Google Scholar 

  11. Richard S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans Am. Math. Soc. 138 (1969), 399–406.

    Article  MathSciNet  MATH  Google Scholar 

  12. S.L. Krushkal’, Extremal quasiconformal mappings, Siberian Math. J. 10 (1969), 411–418.

    Article  Google Scholar 

  13. Edgar Reich, Uniqueness of Hahn-Banach extensions from certain spaces of analytic functions, Math. Zeitschr. 167 (1979), 81–89.

    Article  MATH  Google Scholar 

  14. Edgar Reich, On criteria for unique extremality of Teichmüller mappings, Ann. Acad. Sci. Fenn. AI Math. 6 (1981), 289–301.

    MATH  Google Scholar 

  15. Edgar Reich, A quasiconformal extension using the parametric representation, J. d’Analyse Math. 54 (1990), 246–258.

    Article  MATH  Google Scholar 

  16. Edgar Reich, On some related extremal problems, Rev. Roum. Math. Pures Appl. 39 (1994), 613–626.

    MATH  Google Scholar 

  17. Edgar Reich, A theorem of Fehlmann-type for extensions with bounded 8-derivatives, Complex Variables 26 (1995), 343–351.

    Article  MATH  Google Scholar 

  18. Edgar Reich and Chen Jixiu, Extensions with bounded 8-derivative, Ann. Acad. Sci. Fenn. AI Math 16 (1991), 377–389.

    MATH  Google Scholar 

  19. Edgar Reich and Kurt Strebel, On quasiconformal mappings which keep the boundary points fixed, Trans. Am. Math. Soc. 138 (1969), 211–222.

    Article  MathSciNet  MATH  Google Scholar 

  20. Edgar Reich and Kurt Strebel, Extremal quasiconformal mappings with given boundary values, Contributions to Analysis, A Collection of Papers dedicated to Lipman Bers, Academic Press, New York, 1974, pp. 375–391.

    Google Scholar 

  21. Kurt Strebel, Zur Frage der Eindeutgkeit extremaler quasikonformer Abbildungen des Einheitskreises (I); II, Comm. Math. Heiv. 36 (1961/1962), 306–323; 39 (1964), 77–89.

    Article  MathSciNet  MATH  Google Scholar 

  22. Kurt Strebel, On the existence of extremal Teichmüller mappings, J. d’Analyse Math. 30 (1976), 464–480.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Strebel, Extremal quasiconformal mappings, MSRI Report N° 0941986 (August 1986) = Resultate der Mathematik 10 (1986), 168–210.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street S.E., Minneapolis, MN, 55455, USA

    Edgar Reich

Authors
  1. Edgar Reich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, 4074 East Hall, Ann Arbor, MI, 48109-1109, USA

    Peter Duren

  2. Department of Mathematics, University of Michigan, 525 East University Street-East Hall, Ann Arbor, MI, 48109-1109, USA

    Juha Heinonen

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305-2125, USA

    Brad Osgood

  4. Department of Mathematics, University of Texas, Austin, TX, 78712-1082, USA

    Bruce Palka

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media New York

About this chapter

Cite this chapter

Reich, E. (1998). Extremal Extensions from the Circle to the Disk. In: Duren, P., Heinonen, J., Osgood, B., Palka, B. (eds) Quasiconformal Mappings and Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0605-7_18

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0605-7_18

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6836-9

  • Online ISBN: 978-1-4612-0605-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation