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).