TITLE: The contractivity of the heat semigroup in $L^p$ spaces
QUESTION [5 upvotes]: 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).

REPLY [6 votes]: For the record, the result requested in the question is given in Theorem 1.3.3 in: E.B. Davies, Heat Kernels and Spectral Theory, DOI:10.1017/CBO9780511566158.
The assumptions are:

$L$ is a positive definite self-adjoint operator given by a quadratic form $Q$ on the Hilbert space $L^2(\Omega)$, where $\Omega$ is a $\sigma$-finite measure space;
if $u$ is in the domain of the form, then $|u|$ is in the domain of the form, too, and $Q(|u|) \leqslant Q(u)$ (this is one of the equivalent conditions of Theorem 1.3.2 in the same book);
if $u$ is in the domain of the form and $u \geqslant 0$, then $v = \min\{u,1\}$ is in the domain of the form, too, and $Q(v) \leqslant Q(u)$ (this is one of the equivalent conditions of Theorem 1.3.3 therein).

The proof proceeds by proving first that $\exp(-t L)$ is contractive on $L^\infty$, using duality to get the result for $L^1$, and applying Riesz–Thorin interpolation to conclude.