Boundary Value Problems for Analytic Functions and Singular
Operators in the Variable Exponent Spaces with General Weights
We consider the Riemann boundary value problem for analytic
functions in the class of analytic functions represented by the
Cauchy type integral with density in the generalized Lebesgue
spaces with variable exponent. We consider both the cases when the
coefficient G is piecewise continuous or it may be of a more
general nature, admitting its oscillation. The solvability
conditions are derived and in all the cases of solvability the
explicit formulas are given. Following the approach of I.Simonenko,
we make use of the results on the explicit solution of the boundary
value problem to obtain the weight results for Cauchy singular
integral operator in Lebesgue spaces with variable exponent, among
them some extension of the well known Helson-Szego theorem.
