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

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).

• Is you heat semigroup symmetric (i.e. no drift)? If yes, then the proof is quite straightforward: it is a contraction on $L^\infty$ (by a direct verification) and on $L^1$ (by duality), and the result follows now by the Riesz–Thorin interpolation theorem. – Mateusz Kwaśnicki Oct 2 at 9:36
• @MateuszKwaśnicki: The heat semigroup is generated by the Friedrichs extension of the Laplace-Beltrami operator, so it has no drift. As I've said, I know how to do the proof, but if one wants to do it properly it 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. – Alex M. Oct 2 at 9:45
• Do you accept the fact that the semigroup is Markovian? If yes, you can simply refer to Theorem 1.4.1 in Davies's Heat kernels and spectral theory. – Mateusz Kwaśnicki Oct 2 at 9:57
• @MateuszKwaśnicki: No, that hypothesis is too restrictive. Also, if you take a look at the proof, it is not really done - Davies cites some result of Gross, who in turn does not prove anything but cites some other two works, none of which does what Davies (and Gross) claim to show. I know that theorem, and what Davies has done is a bit unethical. – Alex M. Oct 2 at 10:28
• Wait, I am lost. (1) The heat kernel on a Riemannian manifold is Markovian, 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? – Mateusz Kwaśnicki Oct 2 at 10:51

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.

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).

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

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

• I fail to see how this proves contractivity on $L^p$ for $p\neq 2$. For that you should use the "Markovianity" of the Dirichlet energy as indicated in Mateusz's answer. – MaoWao Oct 5 at 10:29
• One can prove that, for $u \colon \mathbb{R}\rightarrow [0,\infty]$ convex and l.s.c. with $u(0)=0$, the mapping $t\mapsto \int u(h_t(f))\, dVol_g$ is nonincreasing. Apply this to $u(\cdot)=|\cdot|^p$, for any $p \in [1,\infty)$ to get $$\|h_t(f)\|_{L^p}\le \|f\|_{L^p}, \qquad \forall f \in L^2\cap L^p, t\ge 0.$$ The case $p=\infty$ is to be treated differently. Is this what you asked? – Francesco Nobili Oct 5 at 10:50
• For $p=\infty$, there is a 'weak maximum principle' to be shown, namely $$f \le C, \quad a.e. \quad \Rightarrow \quad h_t(f) \le C, \quad a.e.,\forall t>0.$$ Again, this can be shown to be true to cover the case of $L^\infty$ – Francesco Nobili Oct 5 at 11:11
• Of course both of these facts can be proven, but the question was about how to prove this (or, more precisely, give a reference for these facts). – MaoWao Oct 5 at 11:12
• Okay, I was making first sure to identify your question. I did not report a proof in the answer, but links to read this fact. It is all in the references I gave, e.g. (a)-(b) in Theorem 4.16 of cvgmt.sns.it/media/doc/paper/1645/Heatsubmitted-revised4.3.pdf – Francesco Nobili Oct 5 at 11:15