Operators with continuous kernels

Abstract

Let $\Omega \subset {\mathbf{R}}^d$ 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.

References in corpus (1)

• The Dirichlet-to-Neumann operator on $C(\partial \Omega )$