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Prime ends and quasiconformal mappings

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References

  1. L. V. Ahlfors,Conformal Invariants; Topics in Geometric Function Theory, McGraw-Hill, New York, 1973.

    MATH  Google Scholar 

  2. A. Beurling,Ensembles exceptionelles, Acta Math.72 (1940), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Carathéodory,Über die Begrenzung einfach zusammenhängender Gebiete, Ann. Math.73 (1913), 323–370.

    Article  Google Scholar 

  4. E. F. Collingwood,Cluster set theorems for arbitrary functions with applications to function theory, Ann. Acad. Sci. Fenn. A I336/8 (1963), 1–15.

    Google Scholar 

  5. E. F. Collingwood and A. J. Lohwater,The Theory of Cluster Sets, Cambridge Univ. Press, 1966.

  6. E. F. Collingwood and G. Piranian,The mapping theorems of Carathéodory and Lindelöf, J. Math. Pures Appl. (9)43 (1964), 187–199.

    MATH  MathSciNet  Google Scholar 

  7. T. Erkama,Group actions and extension problems for maps of balls, Ann. Acad. Sci. Fenn. A I556 (1973), 1–31.

    Google Scholar 

  8. H. Freudenthal,Enden and Primenden, Fund. Math.39 (1952), 189–210.

    MathSciNet  Google Scholar 

  9. F. W. Gehring,The Carathéodory convergence theorem for quasiconformal mappings in space, Ann. Acad. Sci. Fenn. A I336/11 (1963), 1–21.

    Google Scholar 

  10. F. W. Gehring,A remark on domains quasiconformally equivalent to a ball, Ann. Acad. Sci. Fenn. A I,2 (1976), 147–155.

    MathSciNet  Google Scholar 

  11. F. W. Gehring and J. Väisälä,The coefficients of quasiconformality of domains in space, Acta Math.114 (1965), 1–70.

    Article  MATH  MathSciNet  Google Scholar 

  12. V. M. Goldstein and S. K. Vodopjanov,Metric completion of a domain by using a conformal capacity invariant under quasi-conformal mappings, Dokl. Akad. Nauk SSSR238 (1978), 1042–1044 (Russian).

    MathSciNet  Google Scholar 

  13. T. Hall,Sur ta mesure harmonique de certains ensembles, Ark. Mat. Astr. Fys. 25A,28 (1937), 1–8.

    Google Scholar 

  14. F. Hausdorff,Erweiterung einer stetigen Abbildung, Fund. Math.30 (1938), 40–47.

    MATH  Google Scholar 

  15. B. Kaufmann,Über die Berandung ebener und räumlicher Gebiete, Math. Ann. 103 (1930), 70–144.

    Article  MathSciNet  Google Scholar 

  16. P. Koebe,Abhandlung zur Theorie der konformen Abbildung, J. Reine Angew. Math.145 (1915), 177–225.

    Google Scholar 

  17. T. Kuusalo,Generalized conformal capacity and quasiconformal metrics, Proc. of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings, Brasov, 1971, pp. 193–202.

  18. J. lelong-Ferrand,Construction de métriques pour lesquelles les transformations quasiconformes sont Lipschitziennes, Symp. Math.18 (1976), 407–420.

    MathSciNet  Google Scholar 

  19. E. Lindelöf,Sur un principe général de l'analyse et ses applications à la théorie de la représentation conforme, Acta Soc. Sci. Fenn.46, no. 4 (1915), 1–35.

    Google Scholar 

  20. S. Mazurkiewicz,Über die Definition der Primenden, Fund. Math.26 (1936), 272–279.

    MATH  Google Scholar 

  21. S. Mazurkiewicz,Recherches sur la théorie des bouts premiers, Fund. Math.33 (1945), 177–228.

    MATH  MathSciNet  Google Scholar 

  22. R. Näkki,Boundary behavior of quasiconformal mappings in n-space, Ann. Acad. Sci. Fenn. A I484 (1970), 1–50.

    Google Scholar 

  23. R. Näkki,Extension of quasiconformal mappings between wedges, Proc. of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings, Brasov, 1971, pp. 229–233.

  24. R. Näkki,Continuous boundary extension of quasiconformal mappings, Ann. Acad. Sci. Fenn. A I511 (1972), 1–10.

    Google Scholar 

  25. R. Näkki,Extension of Loewner's capacity theorem, Trans. Amer. Math. Soc.180 (1973), 229–236.

    Article  MATH  MathSciNet  Google Scholar 

  26. B. Palka,On the uniform convergence of quasiconformal mappings, Trans. Amer. Math. Soc.184 (1973), 137–152.

    Article  MathSciNet  Google Scholar 

  27. G. Piranian,The boundary of a simply connected domain, Bull. Amer. Math. Soc.64 (1958), 45–55.

    MATH  MathSciNet  Google Scholar 

  28. Ju. G. Rešetnjak,Spatial mappings with bounded distortion, Sibirsk Mat. Ž.8 (1967), 629–658 (Russian).

    MathSciNet  Google Scholar 

  29. E. C. Schlesinger,Conformal invariants and prime ends, Amer. J. Math.80 (1958), 83–102.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Tsuji,Beurling's theorem on exceptional sets, Tohoku Math. J.2 (1950), 113–125.

    Article  MATH  MathSciNet  Google Scholar 

  31. P. Urysohn,Über ein Problem von Herrn C. Carathéodory, Fund. Math.6 (1924), 229–235.

    Google Scholar 

  32. J. Väisälä,Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math.229, Springer-Verlag, Berlin and New York, 1971.

    MATH  Google Scholar 

  33. M. Vuorinen,On the existence of angular limits of n-dimensional quasiconformal mappings, to appear.

  34. V. Weniaminoff,Sur un problème de la représentation conforme de M. Carathéodory,—Rec. Math. Soc. Moscou31 (1922), 91–93.

    Google Scholar 

  35. W. P. Ziemer,Extremal length and p-capacity, Michigan Math. J.16 (1969), 43–51.

    Article  MATH  MathSciNet  Google Scholar 

  36. V. A. Zorič,Correspondence of the boundaries in Q-quasiconformal mappings of a sphere, Dokl. Akad. Nauk SSSR145 (1962), 1209–1212 (Russian).

    MathSciNet  Google Scholar 

  37. V. A. Zorič,Boundary properties of a class of mappings in space, Dokl. Akad. Nauk SSSR153 (1963), 23–26 (Russian).

    MathSciNet  Google Scholar 

  38. V. A. Zorič,Determination of boundary elements by means of sections, Dokl. Akad. Nauk SSSR164 (1965), 736–739 (Russian).

    MathSciNet  Google Scholar 

  39. V. A. Zorič,Angular boundary values of quasiconformal mappings of a ball, Dokl. Akad. Nauk SSSR179 (1967), 1492–1494 (Russian).

    Google Scholar 

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Part of this research was done while the author was visiting at the University of Michigan in 1974–75. The research was supported by the Academy of Finland.

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Näkki, R. Prime ends and quasiconformal mappings. J. Anal. Math. 35, 13–40 (1979). https://doi.org/10.1007/BF02791061

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