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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.
Received: 24.01.2011
Citation:
A. Y. Timofeev, “Construction of functions with determined behavior $T_G(b)(z)$ at a singular point”, Ufimsk. Mat. Zh., 3:1 (2011), 85–93; Ufa Math. J., 3:1 (2011), 83–91
Linking options:
https://www.mathnet.ru/eng/ufa84 https://www.mathnet.ru/eng/ufa/v3/i1/p85
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Abstract page: | 393 | Russian version PDF: | 111 | English version PDF: | 15 | References: | 67 | First page: | 2 |
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