**Vakhtang Paatashvili**

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

**abstract:**

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