Abstract
In this article various types of weighted variable exponent Hardy and Smirnov classes of analytic functions in simple and doubly connected domains are introduced and studied. In particular, a wide class of those domains are revealed in which the functions from the above-mentioned classes are representable by the Cauchy-type integrals with densities of weighted variable exponent Lebesgue spaces. On the basis of these results, a solution of the Dirichlet problem in explicit form in a ring for harmonic functions, real parts of the functions of variable exponent Smirnov classes is given.
Acknowledgements
This research was supported by the grant GNSF/ST09_23_3-100. The authors express their thanks to the referee for helpful remarks.