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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 \(z=0\).
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
Full Text: MNR