# Diffusion semigroup generated by Laplacian

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion semigroup generated by $\Delta$, and pick a function $f \in L^2(M)$. I am looking for a proof of the following $$\Vert P_t f\Vert_{L^2} \leq Ce^{-ta}\Vert f\Vert_{L^1}, t \geq 1$$ A reference would also be appreciated.

Could we also say something like $\Vert \nabla P_t f\Vert_{L^2} \leq Ce^{-ta}\Vert f\Vert_{L^1}, t \geq 1$?

• I think you may be able to find this in E.B. Davies' book Heat Kernels and Spectral Theory. Roughly, it should be equivalent to the ultracontractivity of the semigroup $P_t$ and/or the boundedness of the heat kernel. Your second question should follow from the first, integrating by parts and using the spectral theorem which says that $\Delta P_t$ is bounded on $L^2$. I will try to look at my copy of Davies tomorrow and fill in details. – Nate Eldredge Apr 13 '15 at 0:49

First, note that if your inequality holds with $t=1$ then it holds with any $t \ge 1$. (Since the spectrum of $-\Delta$ is bounded below by $a$, we have $\|P_t f\|_2 \le e^{-ta} \|f\|_2$. Replacing $f$ by $P_1 f$ and $t$ by $t-1$ and using the semigroup property, if your inequality holds for $t=1$ we get $$\|P_t f\|_2 \le e^{-a(t-1)} \|P_1 f\|_2 \le C e^{-at} \|f\|_1.$$ So we are asking whether $P_1$ is a bounded operator from $L^1$ to $L^2$.
Since $P_1$ is symmetric on $L^2$, by taking adjoints, this is equivalent to asking whether $P_1$ is a bounded operator from $L^2$ to $L^\infty$. If we have $P_t$ bounded from $L^2$ to $L^\infty$ for all $t > 0$, we say $P_t$ is ultracontractive. In Lemma 2.1.2 of E.B. Davies, Heat Kernels and Spectral Theory, you can find the sketch of a proof that ultracontractivity implies the existence of a bounded heat kernel. But we care more about the converse, which is really trivial. Suppose $P_t$ admits an integral kernel $p(t,x,y)$ such that for each $t$, $p(t,\cdot,\cdot)$ is bounded. Then the desired inequality clearly follows with $C = e^{a} \sup_{x,y \in M} p(1,x,y)$.
A famous result of Li and Yau says that a complete Riemannian manifold with Ricci curvature bounded below has a heat kernel that satisfies Gaussian upper bounds. In particular, for each $t > 0$, $p(t,\cdot,\cdot)$ is bounded. This may be overkill but it certainly works.
Your second statement essentially follows from the first, at least for $t > 1$. The spectral theorem says that $\Delta P_t$ is bounded from $L^2$ to $L^2$ for each $t > 0$. (Essentially, this follows from the fact that $-\Delta$ is a positive operator, and $x e^{-x}$ is a bounded function on $[0, \infty)$.) Now integrating by parts gives \begin{align*} \|\nabla P_{1+\epsilon} f\|_2 &= \|\nabla P_{1+\epsilon} f\|_2 \\ &= \sqrt{-\langle P_{1+\epsilon} f, \Delta P_\epsilon P_1 f\rangle} \\ &\le \sqrt{\|P_\epsilon\|_{2 \to 2} \|\Delta P_\epsilon\|_{2 \to 2}} \|P_1 f\|_2\end{align*} and then use the previous statement.