Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ⊂ ℝ n is a bounded domain. The boundary δ G should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators \documentclass{article}\pagestyle{empty}\begin{document}$\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$\end{document} maps W p k ( G ) into W p k+1 ( G ) and W p k−1 ( G ) into W p k/p ( G ), respectively, and are bounded. Here θ ∈ S ⊂ ℝ n , where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.