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Construction of functions with a given behavior of $$T_{G}(b)(z)$$ at a singular point. (Construction of functions with determined behavior of $$T_{G}(b)(z)$$ at a singular point.) (Russian. English summary) Zbl 1240.30207
Let $$G$$ be a domain. I. N. Vekua [Обобщенные аналитические функции. Moskva: Nauka (1988; Zbl 0698.47036)] developed the theory of generalized analytic functions as solutions to the equations $\partial_{\overline{z}}w+A(z)w+B(z)\overline{w}=0, \quad z\in G,$ with $$A, B\in L_{p}(G)$$, $$p>2$$, using the so-called $$T_{G}$$-operator, which is the right inverse to the operator $$\frac{\partial}{\partial \overline{z}}$$ with differentiation in the Sobolev sense. In the reviewed article, for the unit disc $$G=\{z: | z| <1\}$$ and a given continuity modulus $$\mu$$, a function $$b=b(\zeta)$$ is constructed such that in the singular point $$z=0$$, $$T_{G}(b)(z)$$ has the behavior described by $$\mu$$. The scheme for the computation of $$T_{G}f$$ using residue theory is given. Conditions are established under which $$T_{G}(b)(z)$$ is continuous at $$z=0$$.
##### MSC:
 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, $$p$$-analytic, etc.)
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