Article
Ufa Mathematical Journal
Volume 3, Number 1, pp. 83-91
The construction of functions with determined behavior $T_G(b)(z)$ in a singular point.
Timofeev A.Yu.
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I.N. Vekua constructed the theory of generalized analytic functions, i.e., solutions of the equation \begin{gather} \label{annotation_en_1} \partial_{\bar{z}} w + A(z)w + B(z) \overline{w} = 0, \end{gather} where $z \in G$ (G, for example, is the unit disk in the complex plane) and the coefficients $A(z),$ $B(z)$ belong to $L_p(G),$ $p > 2.$ The Vekua theory for the solutions of \eqref{annotation_en_1} is closely related to the theory of holomorphic functions due to the so-called similarity principle. In this case, the $T_G$-operator plays an important role. The $T_G$-operator is right-inverse to $\frac{\partial}{\partial \bar{z}},$ where $\frac{\partial}{\partial \bar{z}}$ is understood in Sobolev’s sense. In the research the author offers a scheme of constructing in the unit disk $G$ the function $b(z)$ with determined behavior $T_G(b)(z)$ in a singular point $z = 0,$ where $T_G$ is an integral Vekua operator. The paper formulates the conditions of $b(z)$ under which the function $T_G(b)(z)$ is continous.