On~the boundedness of a~singular integral operator in the space $C^\alpha(\overline G)$

D. S. Anikonov

Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 447-464

Résumé

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, $x\in G$, $S\in\Omega$, $\Omega=\{s: s\in\mathbf R^m,|s|=1\}$, $u(y)\in C^\alpha(\overline G)$, $0\alpha1$. It is proved that if the function $f(x,s)$ satisfies a Hölder condition with exponent $\lambda$, $\alpha\lambda1$, 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.