Construction of functions with determined behavior $T_G(b)(z)$ at a singular point
A. Y. Timofeev Syktyvkar State University, Syktyvkar, Russia
Abstract:
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.
Keywords:
$T_G$-operator, singular point, modulus of continuity.
UDC:
517.9
Received: 24.01.2011
Fulltext:
PDF file (428 kB)
