Construction of functions with determined behavior $T_G(b)(z)$ at a~singular point

A. Y. Timofeev

Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 83-91

Résumé

I. N. Vekua developed the theory of generalized analytic functions, i.e., solutions of the equation \begin{equation} \partial_{\overline z}w+A(z)w+B(z)\overline w=0, \tag{0.1} \end{equation} where $z\in G$ ($G$, for example, is the unit disk on a complex plane) and the coefficients $A(z)$, $B(z)$ belong to $L_p(G)$, $p>2$. The Vekua theory for the solutions of $(0.1)$ 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 $\frac\partial{\partial\overline z}$, where $\frac\partial{\partial\overline z}$ is understood in Sobolev's sense. The author suggests a scheme for constructing the function $b(z)$ in the unit disk $G$ with determined behavior $T_G(b)(z)$ at a singular point $z=0$, where $T_G$ is an integral Vekua operator. The paper states the conditions for $b(z)$ under which the function $T_G(b)(z)$ is continuous.

Détail

Keywords: $T_G$-operator, singular point, modulus of continuity.