# Operators with continuous kernels

## TL;DR

This paper studies when operators on L2 spaces over open sets have continuous kernels, focusing on conditions like mapping into continuous functions and analyzing the regularity of powers of such operators.

## Contribution

It establishes conditions under which operators have continuous kernels, especially relating to their powers and self-adjointness, and explores Mercer's theorem in this context.

## Key findings

- Operators with $T L_2(\Omega) o C(ar{\Omega})$ have continuous kernels for their third power if self-adjoint and $\Omega$ is bounded.
- The condition $T L_2(\Omega) o C(ar{\Omega})$ is central to kernel regularity.
- The paper analyzes the applicability of Mercer's theorem for these operators.

## Abstract

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