Abstract
We give examples of the absence of representations of the “second kind” for a Beltrami equation. Some sufficient conditions are proposed for the existence of these representations. We specify the regularity “up to the boundary” of a quasiconformal homeomorphism of the unit circle onto itself in the case of a regular complex characteristic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Vekua I. N., Generalized Analytic Functions [in Russian], Fizmatgiz, Moscow (1959).
Kalyanichenko S. I. and Klimentov S. B., “The Hardy classes of solutions to the Beltrami equations,” in: Proceedings of the International School-Seminar on Geometry and Analysis Dedicated to the Memory of N. B. Efimov, Izdat. TsBBR, Rostov-on-Don, 2006, pp. 127–128.
Kalyanichenko S. I. and Klimentov S. B., “The Hardy classes of solutions to the Beltrami equations,” Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauki, No. 1, 7–10 (2008).
Goluzin G. M., Geometric Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1966).
Soldatov A. P., “The Hardy space of solutions to first-order elliptic systems,” Dokl. Math., 76, No. 2, 660–664 (2007).
Baratchart L., Leblond J., Rigat S., and Russ E., “Hardy spaces of the conjugate Beltrami equation,” J. Funct. Anal., 259, No. 2, 384–427 (2010).
Klimentov S. B., “Hardy classes of generalized analytic functions,” Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauki, No. 3, 6–10 (2003).
Klimentov S. B., “Riemann-Hilbert boundary value problem for generalized analytic functions in Smirnov classes,” Global and Stochastic Analysis, 1, No. 2, 217–240 (2011).
Klimentov S. B., “On a method of constructing the solutions of boundary-value problems of the theory of bendings of surfaces of positive curvature,” J. Math. Sci., 51, No. 2, 2230–2248 (1990).
Rudin W., “The radial variation of analytic functions,” Duke Math. J., 22, No. 2, 235–242 (1955).
Bojarsky B. V., “Weak solutions to a system of first-order differential equations of elliptic type with discontinuous coefficients,” Mat. Sb., 43, No. 4, 451–503 (1957).
Belinskiĭ P. P., General Properties of Quasiconformal Mappings [in Russian], Nauka, Novosibirsk (1974).
Ahlfors L. V., Lectures on Quasiconformal Mappings [Russian translation], Mir, Moscow (1969).
Vinogradov V. S., “On the solvability of one singular integral equation,” Dokl. Akad. Nauk SSSR, 241, No. 2, 272–274 (1978).
Mandzhavidze G. F., “The application of the theory of generalized analytic functions to the study of the displacement boundary value problems,” in: Differential and Integral Equations, Boundary Value Problems, Izdat. Tbilisisk. Univ., Tbilisi, 1979, pp. 165–186.
Klimentov S. B., “On the summability exponent of the derivatives of solution to the first-order elliptic system,” in: Abstracts of the All-Union Conference on Geometry and Analysis, Novosibirsk, 1989, p. 43.
Iwaniec T. and Martin G. J., “The Beltrami equation,” Institut Mittag-Leffler, Report No. 13 (2001/2002).
Danilov V. A., “Estimates of distortion of quasiconformal mapping in space C mα ,” Siberian Math. J., 14, No. 3, 362–369 (1973).
Sternberg S., Lectures on Differential Geometry [Russian translation], Mir, Moscow (1970).
Danilyuk I. I., Irregular Boundary Value Problems on the Plane [in Russian], Nauka, Moscow (1975).
Gakhov F. D., Boundary Value Problems [in Russian], Nauka, Moscow (1977).
Gokhberg I. Ts. and Krupnik N. Ya., An Introduction to the Theory of One-Dimensional Singular Integral Operators [in Russian], Shtiintsa, Kishinëv (1973).
Stein E. M., Singular Integrals and Differentiability Properties of Functions [Russian translation], Mir, Moscow (1973).
Garnett J., Bounded Analytic Functions [Russian translation], Mir, Moscow (1984).
Klimentov S. B., “The Riemann-Hilbert boundary value problem in the Hardy classes of generalized analytic functions,” Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauki, No. 4, 3–5 (2004).
Klimentov S. B., “A stochastic Riemann-Hilbert boundary value problem in conformal martingales of classes Hp and a bounded mean oscillator,” Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauki, No. 3, 6–12 (2004).
Kreĭn S. G., Linear Equations in Banach Spaces [in Russian], Nauka, Moscow (1971).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2014 Klimentov S.B.
The author was financially supported by the Ministry of Education and Science of the Russian Federation (Agreement 14.A18.21.0356 “Theory of Function Spaces, Operators, and Equations in Them”) and the Interior Grant 213.01-24/2013-66 of Southern Federal University.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 2, pp. 324–340, March–April, 2014.
Rights and permissions
About this article
Cite this article
Klimentov, S.B. Representations of the “second kind” for the hardy classes of solutions to the Beltrami equation. Sib Math J 55, 262–275 (2014). https://doi.org/10.1134/S0037446614020098
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446614020098