The contractivity of the heat semigroup in $L^p$ spaces

Question

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).

Answer 1

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.

Answer 2

I will sketch the construction for a general Riemannian Manifold, thought as a triple $(M,d_g,Vol_g)$ (Smooth manifold $M$, Riemannian Metric $d_g$, volume form $Vol_g$) and at the end provide references for 'Riemannian' metric measure spaces $(X,d,m)$ (Polish space (X,d) and Borel, boundedly finite positive measure $m$), where you can cite the property you seek.

The contractivity of the Heat Flow $t \mapsto h_t(f^0)$ for $f^0 \in L^2$ follows quite directly if you regard it (actually define it) as a 'gradient flow trajectory starting from $f^0$' associated with the Dirichlet energy functional $$ L^2 \ni f \mapsto Dir(f):= \begin{cases} \int |\nabla f|^2 \, d Vol_g, &\text{if }f \in W^{1,2} \\ +\infty, &\text{otherwise} \end{cases} $$ Roughly, a gradient flow trajectory starting from $f^0$ is a locally absolutely continuous curve $t \mapsto h_t(f^0) \in L^2$ satisfying: $$ \begin{cases} \dot{h_t(f^0)} \in -\partial^- Dir(h_t(f^0)), &a.e. t>0 \\ L^2-\lim_{t\rightarrow 0}h_t(f^0) =f^0.& \end{cases} $$ where $\partial^-Dir(f)$ is the subdifferential of $Dir$ at $f$. It is well known that this construction agrees with the heat flow on $\mathbb{R}^n$ or Riemannian manifolds and that the above system is equivalent to the heat equation. Existence and uniqueness of gradient flow trajectories are granted by the fact that $L^2$ is an Hilbert space, and $Dir$ is convex and lower semicontinuous functional. Moreover, in the framework of Riemannian manifolds, the flow map $h_t :L^2 \rightarrow L^2$ is linear (thanks to the fact that $Dir$ is quadratic) and contractive, meaning in this language, that $h_t(f+g)=h_t(f)+h_t(g)$ and $$ \| h_t(f)\|_{L^2} \le \|f\|_{L^2},$$ for all $f,g \in L^2$ and $t\ge 0$. By density of $L^2\cap L^p$ in $L^p$ for any $p$, you can extend $h_t$ in any $L^p$ space (and bring the contraction estimate to all the $L^p$ spaces).

References (links to have a read)

The discussion (for you to see) is taken from this book https://www.springer.com/gp/book/9783030386122, but you should probably cite:

Gradient flows: http://www2.stat.duke.edu/~sayan/ambrosio.pdf

Heat flow on metric measure space: Theorem 4.16 of http://cvgmt.sns.it/media/doc/paper/1645/Heatsubmitted-revised4.3.pdf