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This article is cited in 12 scientific papers (total in 12 papers)
On the boundedness of a singular integral operator in the space $C^\alpha(\overline G)$
D. S. Anikonov
Abstract:
The article considers an operator of the form
$$
[Au](x)=\int_G\frac{f(x,s)}{|x-y|^m}u(y)\,dy,
$$
where $G$ is a bounded domain in $\mathbf R^m$ with a smooth boundary, $x\in G$, $S\in\Omega$, $\Omega=\{s: s\in\mathbf R^m,|s|=1\}$, $u(y)\in C^\alpha(\overline G)$, $0<\alpha<1$. It is proved that if the function $f(x,s)$ satisfies a Hölder condition with exponent $\lambda$, $\alpha<\lambda<1$, and the condition
\begin{equation}
\int_{\Omega_1}f(x,s)\,ds=0\qquad x\in G
\end{equation}
(where $\Omega_1$ is any polysphere), then the operator is bounded from $C^\alpha(\overline G)$ to $C^\alpha(\overline G)$. Moreover, if $f(x,s)=g(s)$, then in order that the operator $A$ should be defined and bounded from $C^\alpha(\overline G)$ to $C^\alpha(\overline G)$ the condition (1) is necessary.
Bibliography: 6 titles.
Received: 25.10.1976
Citation:
D. S. Anikonov, “On the boundedness of a singular integral operator in the space $C^\alpha(\overline G)$”, Math. USSR-Sb., 33:4 (1977), 447–464
Linking options:
https://www.mathnet.ru/eng/sm2960https://doi.org/10.1070/SM1977v033n04ABEH002432 https://www.mathnet.ru/eng/sm/v146/i4/p515
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