Weighted Hardy and Smirnov classes and the Dirichlet problem for a ring within the framework of variable exponent analysis

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.

Keywords:

• Hardy and Smirnov classes
• variables exponent
• Cauchy-type integral
• harmonic functions
• Dirichlet problem

AMS Subject Classifications:

• 30E20
• 30E25
• 30D55
• 47B38
• 42B20

Acknowledgements

This research was supported by the grant GNSF/ST09_23_3-100. The authors express their thanks to the referee for helpful remarks.