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1995 | 31 | 1 | 303-310
Tytuł artykułu

The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle

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Abstrakty
EN
This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichmüller self-mapping φ of the unit disc Δ. In particular, the norm of the generalized harmonic conjugation operator $A_γ:ℍ → ℍ$ is determined by the maximal dilatation of φ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue of some other quasisymmetric automorphism σ is given.
Rocznik
Tom
31
Numer
1
Strony
303-310
Opis fizyczny
Daty
wydano
1995
Twórcy
  • Department of Mathematics, Maria Curie-Skłodowska University, Pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland
Bibliografia
  • [A] L. V. Ahlfors, Lectures on Quasiconformal Mappings, D. Van Nostrand, Princeton, 1966.
  • [BS] S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1951), 205-249.
  • [B1] B. Bojarski, Homeomorphic solution of Beltrami systems, Dokl. Akad. Nauk SSSR 102 (1955), 661-664 (in Russian).
  • [B2] B. Bojarski, Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43 (1957), 451-503 (in Russian).
  • [G] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
  • [K] J. G. Krzyż, Quasicircles and harmonic measure, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 19-24.
  • [KP] J. G. Krzyż and D. Partyka, Generalized Neumann-Poincaré operator, chord-arc curves and Fredholm eigenvalues, Complex Variables, Theory Appl. 21 (1993), 253-263.
  • [Kü] R. Kühnau, Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend für Q-quasikonforme Fortsetzbarkeit?, Comment. Math. Helv. 61 (1986), 290-307.
  • [L] O. Lehto, Univalent Functions and Teichmüller Spaces, Graduate Texts in Math. 109 Springer, New York, 1987.
  • [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973.
  • [Ł] K] J. Ławrynowicz and J. G. Krzyż, Quasiconformal Mappings in the Plane: Parametrical Methods, Lecture Notes in Math. 978, Springer, Berlin, 1983.
  • [P1] D. Partyka, Spectral values of a quasicircle, Complex Variables Theory Appl., to appear.
  • [P2] D. Partyka, Generalized harmonic conjugation operator, Proceedings of the Fourth Finnish-Polish Summer School in Complex Analysis at Jyväskylä, Ber. Univ. Jyväskylä Math. Inst. 55 (1993), 143-155.
  • [P3] D. Partyka, Spectral values and eigenvalues of a quasicircle, preprint.
  • [S] M. Schiffer, The Fredholm eigenvalues of plane domains, Pacific J. Math. 7 (1957), 1187-1225.
  • [St1] K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises I, Comment. Math. Helv. 36 (1962), 306-323.
  • [St2] K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises II, ibid. 39 (1964), 77-89.
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bwmeta1.element.bwnjournal-article-bcpv31z1p303bwm
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