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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. 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Arendt","submitted_at":"2019-03-15T04:56:35Z","abstract_excerpt":"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. 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