Operators with Continuous Kernels

Abstract

Let $\Omega \subset \mathbb {R}^d$ be open. We investigate conditions under which an operator T on $L_2(\Omega )$ has a continuous kernel $K \in C({{\overline{\Omega }}} \times \overline{\Omega })$. In the centre of our interest is the condition $T L_2(\Omega ) \subset C({{\overline{\Omega }}})$, which one knows for many semigroups generated by elliptic operators. This condition implies that $T^3$ has a kernel in $C({{\overline{\Omega }}} \times \overline{\Omega })$ if T is self-adjoint and $\Omega$ is bounded, and the power 3 is best possible. We also analyse Mercer’s theorem in our context.