Abstract
Necessary and sufficient conditions of Fredholmness, as well as index formulas, are obtained for differential boundary value problems with several unknown functions analytic on multiply connected domains. The complex conjugation and (forward or backward) shifts are involved in boundary conditions of above mentioned problems, and solutions are sought in weighted spaces with weights satisfying only the Hunt-Muckenhoupt-Wheeden condition. The results are of an explicit nature in the case of piecewise continuous coefficients.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
F. D. Gahov, Boundary value problems, Nauka, Moscow, 1977. (Russian)
I. Gohberg and N. Krupnik, On one-dimensional singular integral operators with shift, Isv. Acad. Nauk Arm. SSR. Matematica. 8 (1973), no. 1, 3–12. (Russian)
I. Gohberg and N. Krupnik, Systems of singular integral equations in weighted spaces Lp, Soviet. Math. Dokl. 10 (1969), 688–691.
R. Hunt, B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251.
R. S. Isahanov, Some boundary value problems of analytic functions, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, (Collection of articles on the equations of mathematical physics, 6), vol. 52, 1976, pp. 62–80. Russian
R. S. Isahanov, A general problem for holomorphic functions, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, (Collection of articles on the boundary problems of analytic functions, singular integral equations and some questions of harmonic analysis), vol. 65, 1980, pp. 99–109. Russian
G. S. Litvinchuk, Boundary value problems and singular integral equations with a shift, Nauka, Moscow, 1977. (Russian)
G. S. Litvinchuk, On operator approach to boundary value problems with shift and conjugation, Memoirs of the anniversary seminar on boundary value problems (F. D. Gahov anniversary Vol.), Belorussian University, Minsk, 1985, pp. 69–76. (Russian)
N. I. Lisovets, A four-element boundary value problem with piecewise continuous coefficients with shift on a composite contour, Ukrain. Mat. Zh. 35 (1983), 764–767. (Russian)
N. I. Lisovets, A boundary value problem for functions analytic in several multiply connected domains, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (1984), 75–77. (Russian)
N. I. Lisovets, Differential boundary value problem for functions analytic in several multiply connected domains, Manuscript no. 546-Uk-D84, UkrNIINTI, Kiev, 1984. (Russian)
N. I. Lisovets, A method for investigating boundary value problems for functions that are analytic in a domain, Soobshch. Akad. Nauk Gruzin. SSR 122 (1986), no. 3, 477-480. (Russian)
N. I. Lisovets, A method of investigating differential boundary value problems for analytic functions, Differentsial’nye Uravneniya 24 (1988), no. 6, 1049–1053. (Russian)
A. V. Protsenko, Differential boundary value problems for analytic functions, Ph.D. thesis, Odessa Univ., USSR, 1981. (Russian)
S. F. Skorohod, Nöether theory of multielement boundary value problems with a shift for functions analytic in a domain, Ph.D. thesis, Odessa Univ., USSR, 1984. (Russian)
S. L. Sobolev, Applications of functional analysis to mathematicap physics, Novosibirsk, 1962. (Russian)
I. Spitkovsky, Singular integral operators with PC symbols on weighted Hardy spaces, Journal of Functional Analysis 104 (1992).
N. E. Tovmasjan, Integral representations of holotnorphic functions by holomorphic densities and their applications, Lecture Notes in Math., Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), vol. 798, Springer-Verlag, Berlin and New York, 1980, pp. 439–445.
N. P. Vekua, Systems of singular integral equations and boundary value problems, Nauka, Moscow, 1970. (Russian)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media New York
About this paper
Cite this paper
Kurtz, D., Latushkin, Y., Spitkovsky, I. (1994). Boundary Value Problems for Functions Analytic on Multiply Connected Domains on Spaces with a General Weight. In: Curto, R.E., Jørgensen, P.E.T. (eds) Algebraic Methods in Operator Theory. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0255-4_33
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0255-4_33
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6683-9
Online ISBN: 978-1-4612-0255-4
eBook Packages: Springer Book Archive