## Certain properties of generalized analytic functions from Smirnov class with a variable exponent.(English)Zbl 1369.30071

Summary: Let $$D$$ be a simply connected domain bounded by a simple, closed, rectifiable curve $$\Gamma$$, let $$p=p(t)$$ be a positive measurable function given on $$\Gamma$$, and $$z=z(\zeta)$$, $$\zeta=re^{i\vartheta}$$ be a 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)
$$\sup\limits_{0<r<1} \int\limits_0^{2\pi}|W(z(re^{i\vartheta}))|^{p(z(e^{i\vartheta}))}|z\prime(re^{i\vartheta})|\,d\vartheta<\infty$$ (see [V. Paatashvili, Proc. A. Razmadze Math. Inst. 163, 93–110 (2013; Zbl 1346.30022)]).
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.

### MSC:

 30H15 Nevanlinna spaces and Smirnov spaces

Zbl 1346.30022
Full Text:

### References:

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