On a modification of the Poisson integral operator

Dariusz Partyka

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011), / Harvested from The Polish Digital Mathematics Library

Résumé

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 $.

Détail

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:289763

Bibliographie

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.