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Let $H: L^2(M) \longrightarrow L^2(M)$ be a bounded operator. Here, $M$ can be a Riemanniannian manifold, or some open subset of $\mathbb{R}^n$.

Question: What can I say about the Schwartz Kernel $k$ of $H$? Can I conclude that it is in some Sobolev space $H^s(M \times M)$ for some $s<0$?

What if $H$ is an operator on $C^0(M)$?

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This suggests that the answer should be positive, but my knowledge here is limited. –  Behaviour Jun 5 at 3:11

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