The compactness of a weakly singular integral operator on weighted Sobolev spaces

Abstract

It is shown that the weakly singular integral operator~$\int_{-1}^{1}\big(\phi(\tau)/|\tau -t|^{\gamma}\big)\,d\tau$, where $0\lt \gamma\lt 1$, maps the weighted Sobolev space~$W_{p;\alpha,\beta}^{(n)}(\Omega)$ compactly into itself for $1\lt p\lt \infty$, $0\lt \alpha+1/q, \beta+1/q\lt 1$~and $n\in \mathbb{N}_0$.