Vakhtang Paatashvili

On Some Properties of Analytic Functions from Smirnov Class with a Variable Exponent

abstract:
Let $D$ be a simply connected domain bounded by a simple closed rectifiable curve $\Gm$ and $L^{p(t)}(D)$ denote the Lebesgue space with variable exponent.
The present work reveals different conditions regarding the functions $p(t)$ and the domain $D$ under fulfilment of which the Cauchy type integrals with density from $L^{p(t)}(\Gm)$ belong to the Smirnov class $E^{p(t)}(D)$.
When the domain $D$ is bounded by the Lavrent'yev curve, the analogue of the well-known Smirnov's theorem is stated: if $\phi\in E^{p_1(\cdot)}(D)$, $\phi^{+}(t)\in L^{p_2(t)}(\Gm)$, then $\phi\in E^{\wt{p}(t)}(D)$, where $\wt{p}(t)=\max(p_1(t),p_2(t))$.

Mathematics Subject Classification: 47B38, 42B20, 45P05

Key words and phrases: Smirnov classes of analytic functions, variable exponent, Cauchy type integral, regular curves, Lavrent'yev curves