Integral Operators in Sobolev Spaces on Domains with Boundary
Jörg Witte · Mathematische Nachrichten · 1999
Abstract Properties of integral operators with weak singularities arc investigated. It is assumed that G ⊂ ℝn is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators \documentclass{article}\pagestyle{empty}\begin{document}$\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left( u \right)d\partial G_ u }$\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 Wpk(G) into Wpk+1(G) and Wpk−1(G) into Wpk/p(G), respectively, and are bounded. Here θ ∈ S ⊂ ℝn, 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.