Boundary Value Problems
Volume 2005 (2005), Issue 1, Pages 43-71
Boundary value problems for analytic functions in the class of Cauchy-type integrals
with density in
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.
We study the Riemann boundary value problem , for analytic functions in the class of analytic functions represented by the
Cauchy-type integrals with density in the spaces with variable exponent. We consider both the case when the coefficient 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.