Vakhtang Paatashvili

Certain Properties of Generalized Analytic Functions from Smirnov Class with a Variable Exponent

abstract:
Let $D$ be a simply connected domain bounded by a simple, closed, rectifiable curve $\Gamma$, $p=p(t)$ be the given on $\Gamma$ positive measurable function, and $z=z(\zeta)$, $\zeta=re^{i\vartheta}$ be conformal mapping of the circle $U=\{\zeta: |\zeta|<1\}$ onto the domain $D$.

The function $W(z)$, generalized-analytical in I. Vekua's sense, belongs to the Smirnov class $E^{p(t)}(A;B;D)$, if
(1) $W\in U^{s,2}(A;B;D)$;
(2) $\sup\limits_{0<r<1} \int\limits_0^{2\pi}|W(z(re^{i\vartheta}))|^{p(z(e^{i\vartheta}))}|z'(re^{i\vartheta})|\,d\vartheta<\infty$
(see [V. Paatashvili, Variable exponent Smirnov classes of generalized analytic functions. Proc. A. Razmadze Math. Inst. 163 (2013), 93–110]).

When $p(t)$ is Log-Hölder function continuous in $\Gamma$ and $\min p(t)=\underline{p}>1$, we considers the problems of representability of functions from $E^{p(t)}(A;B;D)$ by the generalized Cauchy integral, show the connection between the generalized Cauchy type integral and the generalized singular integral; of special interest is the question of extendability of functions from those classes, and the symmetry principle is proved.

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

Key words and phrases: Generalized analytic functions, variable exponent, Smirnov classes of generalized analytic functions, generalized Cauchy and Cauchy type integrals