Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p \in [1, \infty]$, but this requires a bit of work. The thing is that I need to use this contractivity property in a work of mine, and I don't want to waste journal pages by providing a (not so short) proof of a result that I believe is already known and quite a classic. I already know of theorem X.55 in volume 2 of Reed & Simon, but this only treats the case of $M$ of finite measure.

*Edit*: Crucially, I do not want to assume the existence of the heat kernel (this must be obtained as a by-product, in fact), and in general I am willing to assume only those facts about the heat semigroup that are immediate consequences of its construction through functional calculus.

Could you please point me to a citeable reference that proves the $L^p$-contractivity of the heat semigroup (or any other more general result from which this contractivity can be immediately obtained)?

Once again: I am not looking for a proof, I already have one. I need a reference to include in an article, instead of my own proof, and for arbitrary manifolds (not just of finite measure).

properlyit takes about one full A4 page, which is quite much for a result that for sure is known. A journal reviewer might reject my work for publication precisely because it wastes space in the journal with results already proven. $\endgroup$ – Alex M. Oct 2 at 9:45Heat kernels and spectral theory. $\endgroup$ – Mateusz Kwaśnicki Oct 2 at 9:57isMarkovian, is it not? (2) Where in the proof of Davies's Thm 1.4.1 does the author cite Gross? (3) What exactly is missing in the proof of that theorem? $\endgroup$ – Mateusz Kwaśnicki Oct 2 at 10:51