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

Paatashvili, Vakhtang

Mem. Differ. Equ. Math. Phys. 69, 77-91 (2016).

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

Keywords:

generalized analytic functions; variable exponent; Smirnov classes of generalized analytic functions; generalized Cauchy and Cauchy type integrals

Citations:

Zbl 1346.30022

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