Vakhtang Paatashvili

The Riemann Problem and Linear Singular Integral Equations with Measurable Coefficients in Lebesgue Type Spaces with a Variable Exponent

In the present work the Riemann problem for analysis functions $\phi(z)$ is considered in a class of Cauchy type integrals with density from $L^{p(t)}$ and a singular integral equation
$$ a(t)\varphi(t)+\frac{b(t)}{\pi i}\int\limits_\Gamma \frac{\varphi(\tau)}{\tau-t}\,d\tau=f(t) $$
in the space $\mathcal{L}^{p(t)}$ whose norm defined by the Lebesgue summation with a variable exponent. In both takes an integration curve is taken from a set containing non-smooth curves. The functions $G$ and $(a-b)(a+b)^{-1}$ are take from a set of measurable functions $A(p(t),\Gamma)$ which is generalization of the class $A(p)$ of I. B. Simonenko. For the Riemann problem the necessary condition of solvability and the sufficient condition are pointed out, and solutions (if any) are constructed. For the singular integral equation the necessary Noetherity condition and one sufficient Noetherity condition are established; the index is calculated and solutions are constructed.

Mathematics Subject Classification: 47B35, 30E20, 45P95, 47B38, 30E25

Key words and phrases: Riemann's boundary value problem, measurable coefficient, factorization of functions, Lebesgue space with a variable exponent, Cauchy type integrals, Noetherian operator, Smirnov class of analytic functions with variable exponents, Cauchy singular integral equations