Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 447-464
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D. S. Anikonov. On the boundedness of a singular integral operator in the space $C^\alpha(\overline G)$. Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 447-464. http://geodesic.mathdoc.fr/item/SM_1977_33_4_a0/
@article{SM_1977_33_4_a0,
author = {D. S. Anikonov},
title = {On~the boundedness of a~singular integral operator in the space $C^\alpha(\overline G)$},
journal = {Sbornik. Mathematics},
pages = {447--464},
year = {1977},
volume = {33},
number = {4},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1977_33_4_a0/}
}
TY - JOUR
AU - D. S. Anikonov
TI - On the boundedness of a singular integral operator in the space $C^\alpha(\overline G)$
JO - Sbornik. Mathematics
PY - 1977
SP - 447
EP - 464
VL - 33
IS - 4
UR - http://geodesic.mathdoc.fr/item/SM_1977_33_4_a0/
LA - en
ID - SM_1977_33_4_a0
ER -
%0 Journal Article
%A D. S. Anikonov
%T On the boundedness of a singular integral operator in the space $C^\alpha(\overline G)$
%J Sbornik. Mathematics
%D 1977
%P 447-464
%V 33
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1977_33_4_a0/
%G en
%F SM_1977_33_4_a0
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.
