Boundary Value Problems
Volume 2005 (2005), Issue 1, Pages 43-71
doi:10.1155/BVP.2005.43

Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in Lp(·)(Γ)

V. Kokilashvili,1 V. Paatashvili,1 and S. Samko2

1A. Razmadze Mathematical Institute, Georgian Academy of Sciences, Tbilisi 380093, Georgia
2Faculty of Science and Technology, University of Algarve, Faro 8000, Portugal

Received 9 July 2004

Copyright © 2005 V. Kokilashvili et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the Riemann boundary value problem Φ+(t)=G(t)Φ−(t)+g(t), for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces Lp(·)(Γ) with variable exponent. We consider both the case when the coefficient G is piecewise continuous and it may be of a more general nature, admitting its oscillation. The explicit formulas for solutions in the variable exponent setting are given. The related singular integral equations in the same setting are also investigated. As an application there is derived some extension of the Szegö-Helson theorem to the case of variable exponents.