PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2011 | 65 | 2 |
Tytuł artykułu

On a modification of the Poisson integral operator

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Given a quasisymmetric automorphism \(\gamma\) of the unit circle \(\mathbb{T}\) we define and study a modification \(P_{\gamma}\) of the classical Poisson integral operator in the case of the unit disk \(\mathbb{D}\). The modification is done by means of the generalized Fourier coefficients of \(\gamma\). For a Lebesgue’s integrable complexvalued function \(f\) on \(\mathbb{T}\), \(P_{\gamma}[f]\) is a complex-valued harmonic function in \(\mathbb{D}\) and it coincides with the classical Poisson integral of \(f\) provided \(\gamma\) is the identity mapping on \(\mathbb{T}\). Our considerations are motivated by the problem of spectral values and eigenvalues of a Jordan curve. As an application we establish a relationship between the operator \(P_{\gamma}\), the maximal dilatation of a regular quasiconformal Teichmuller extension of \(\gamma\) to \(\mathbb{D}\) and the smallest positive eigenvalue of \(\gamma\).
Rocznik
Tom
65
Numer
2
Opis fizyczny
Daty
wydano
2011
online
2016-07-27
Twórcy
Bibliografia
  • Beurling, A., Ahlfors, L. V., The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125-142.
  • Duren, P., Theory of \(H^p\)-Spaces, Dover Publications, Inc., Mineola, New York, 2000.
  • Gaier, D., Konstruktive Methoden der konformen Abbildung, Springer-Verlag, Berlin, 1964.
  • Garnett, J. B., Bounded Analytic Functions, Academic Press, New York, 1981.
  • Kellogg, O. D., Foundations of Potential Theory, Dover Publications, Inc., New York, 1953.
  • Krushkal, S. L., On the Grunsky coefficient conditions, Siberian Math. J. 28 (1987), 104-110.
  • Krushkal, S. L., Grunsky coefficient inequalities, Carath´eodory metric and extremal quasiconformal mappings, Comment. Math. Helv. 64 (1989), 650-660.
  • Krushkal, S. L., Univalent holomorphic functions with quasiconformal extensions (variational approach), Handbook of Complex Analysis: Geometric Function Theory. Vol. 2 (ed. by R. K¨uhnau), Elsevier B.V., 2005, pp. 165-241.
  • Krushkal, S. L., Quasiconformal Extensions and Reflections, Handbook of Complex Analysis: Geometric Function Theory. Vol. 2 (ed. by R. K¨uhnau), Elsevier B.V.,
  • 2005, pp. 507-553.
  • Krzyż, J. G., Conjugate holomorphic eigenfunctions and extremal quasiconformal reflection, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 305-311.
  • Krzyż, J. G., Generalized Fredholm eigenvalues of a Jordan curve, Ann. Polon. Math. 46 (1985), 157-163.
  • Krzyż, J. G., Quasicircles and harmonic measure, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 19-24.
  • Krzyż, J. G., Quasisymmetric functions and quasihomographies, Ann. Univ. Mariae Curie-Skłodowska Sect. A 47 (1993), 90-95.
  • Krzyż, J. G., Partyka, D., Generalized Neumann-Poincar´e operator, chord-arc curves and Fredholm eigenvalues, Complex Variables Theory Appl. 21 (1993), 253-263.
  • Kuhnau, R., Zu den Grunskyschen Koeffizientenbedingungen, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 125-130.
  • Kuhnau, R., Quasikonforme Fortsetzbarkeit, Fredholmsche Eigenwerten und Grunskysche Koeffizientenbedingungen, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 383-391.
  • Kuhnau, R., Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend fur Q-quasikonforme Fortsetzbarkeit?, Comment. Math. Helv. 61 (1986), 290-307.
  • Kuhnau, R., Koeffizientenbedingungen vom Grunskyschen Typ und Fredholmsche Eigenwerte, Ann. Univ. Mariae Curie-Skłodowska Sect. A 58 (2004), 79-87.
  • Kuhnau, R., A new matrix characterization of Fredholm eigenvalues of quasicircles, J. Anal. Math. 99 (2006), 295-307.
  • Kuhnau, R., New characterizations of Fredholm eigenvalues of quasicircles, Rev. Roumaine Math. Pures Appl. 51 (2006), no. 5-6, 683-688.
  • Partyka, D., Spectral values and eigenvalues of a quasicircle, Ann. Univ. Mariae Curie-Skłodowska Sec. A 46 (1993), 81-98.
  • Partyka, D., The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle, Topics in Complex Analysis (Warsaw, 1992), Banach Center Publ.,
  • 31, Polish Acad. Sci., Warsaw, 1995, pp. 303-310.
  • Partyka, D., Some extremal problems concerning the operator \(B_{\gamma}\) , Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 163-184.
  • Partyka, D., The generalized Neumann-Poincare operator and its spectrum, Dissertationes Math. (Rozprawy Mat.) 366 (1997), 125 pp.
  • Partyka, D., Eigenvalues of quasisymmetric automorphisms determined by VMO functions, Ann. Univ. Mariae Curie-Skłodowska Sec. A 52 (1998), 121-135.
  • Partyka, D., The Grunsky type inequalities for quasisymmetric automorphisms of the unit circle, Bull. Soc. Sci. Lett. Łódź Ser. Rech. Deform. 31 (2000), 135-142.
  • Partyka, D., Sakan, K., A conformally invariant dilatation of quasisymmetry, Ann. Univ. Mariae Curie-Skłodowska Sec. A 53 (1999), 167-181.
  • Pommerenke, Ch., Univalent Functions, Vandenhoeck & Ruprecht, Gottingen, 1975.
  • Schiffer, M., Fredholm eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981), 149-164.
  • Schober, G., Numerische, insbesondere approximationstheoretische behandlung von
  • funktionalgleichungen, Estimates for Fredholm Eigenvalues Based on Quasiconformal Mapping, Lecture Notes in Math. 333, Springer-Verlag, Berlin, 1973, pp. 211-217.
  • Shen, Y., Generalized Fourier coefficients of a quasi-symmetric homeomorphism and Fredholm eigenvalue, J. Anal. Math. 112 (2010), no. 1, 33-48.
  • Warschawski, S. E., On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc. 12 (1961), 614-620.
  • Zając, J., Quasihomographies in the theory of Teichmuller spaces, Dissertationes Math. (Rozprawy Mat.) 357 (1996), 102 pp.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.ojs-doi-10_17951_a_2011_65_2_121-137
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.