## Lecture 22. Sobolev inequality and volume growth

In this Lecture, we show how Sobolev inequalities on a Riemannian manifold are related to the volume growth of metric balls. The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I give below I learnt it from my colleague Bañuelos. It bypasses the Marcinkiewicz interpolation theorem by using the Stein’s maximal ergodic lemma.

Let $(\mathbb{M},g)$ be a complete Riemannian manifold and let $L$ be the Laplace-Beltrami operator of $\mathbb{M}$. As usual, we denote by $P_t$ the semigroup generated by $P_t$ and we assume $P_t 1=1$.

We have the following so-called maximal ergodic lemma, which was first proved by Stein. We give here the probabilistic proof since it comes with a nice constant but you can find the original (non probabilistic) proof here.

Lemma:(Stein’s maximal ergodic theorem) Let $p > 1$. For $f \in L^p_\mu(\mathbb{M})$, denote $f^*(x)=\sup_{t \ge 0} |P_t f(x)|$. We have
$\| f^* \|_{L^p_\mu(\mathbb{M})} \le \frac{p}{p-1} \| f \|_{L^p_\mu(\mathbb{M})}.$

Proof: For $x \in \mathbb{M}$, we denote by $(X_t^x)_{t \ge 0}$ the Markov process with generator $L$ and started at $x$. We fix $T > 0$. By construction, for $t \le T$, we have,
$P_{T-t}f (X_T^x) =\mathbb{E} \left( f (X_{2T-t}^x) | X_T^x \right),$
and thus
$P_{2(T-t)}f (X_T^x) =\mathbb{E} \left( (P_{T-t} f) (X_{2T-t}^x) | X_T^x \right).$
As a consequence, we obtain
$\sup_{0 \le t \le T} | P_{2(T-t)}f (X_T^x) | \le \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) | \mid X_T^x\right) .$
Jensen’s inequality yields then
$\sup_{0 \le t \le T} | P_{2(T-t)}f (X_T^x) |^p \le \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) |^p \mid X_T^x\right).$
We deduce
$\mathbb{E} \left( \sup_{0 \le t \le T} | P_{2(T-t)}f (X_T^x) |^p \right) \le \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) |^p \right).$
Integrating the inequality with respect to the Riemannian measure $\mu$, we obtain
$\left\| \sup_{0 \le t \le T} | P_{2(T-t)}f | \right\|_p \le \left( \int_\mathbb{M} \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) |^p \right)d\mu(x)\right)^{1/p}.$
By reversibility, we get then
$\left\| \sup_{0 \le t \le T} | P_{2(T-t)}f | \right\|_p \le \left( \int_\mathbb{M} \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_t^x) |^p \right)d\mu(x)\right)^{1/p}.$
We now observe that the process $(P_{T-t} f) (X_t^x)$ is martingale and thus Doob’s maximal inequality gives
$\mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_t^x) |^p \right)^{1/p} \le \frac{p}{p-1} \mathbb{E} \left( | f(X_T^x)|^p \right)^{1/p}.$
The proof is complete. $\square$

We now turn to the theorem by Varopoulos.

Theorem: Let $n > 0$, $0 < \alpha < n$, and $1 < p< \frac{n}{\alpha}$. If there exists $C > 0$ such that for every $t > 0$, $x,y \in \mathbb{M}$,
$p(x,y,t) \le \frac{C}{t^{n/2}},$
then for every $f \in L^p_\mu(\mathbb{M})$,
$\| (-L)^{-\alpha/2} f \|_{\frac{np}{n-p\alpha}} \le \left( \frac{p}{p-1} \right)^{1-\alpha/n} \frac{ 2n C^{\alpha / n}}{ \alpha (n-p\alpha) \Gamma(\alpha /2)} \|f \|_p$

Proof: We first observe that the bound
$p(x,y,t) \le \frac{C}{t^{n/2}},$
implies that $|P_t f(x)| \le \frac{C^{1/p}}{t^{n/2p}} \| f \|_p$. Denote $I_\alpha f (x)=(-L)^{-\alpha/2} f (x)$. We have
$I_\alpha f (x)=\frac{1}{\Gamma(\alpha /2) } \int_0^{+\infty} t^{\alpha /2 -1 }P_t f (x) dt$
Pick $\delta > 0$, to be later chosen, and split the integral in two parts:
$I_\alpha f (x)=J_\alpha f(x) +K_\alpha f (x),$
where $J_\alpha f (x)=\frac{1}{\Gamma(\alpha /2) } \int_0^{\delta} t^{\alpha /2 -1 }P_t f (x) dt$ and $K_\alpha f (x)=\frac{1}{\Gamma(\alpha /2) } \int_\delta^{+\infty} t^{\alpha /2 -1 }P_t f (x) dt$. We have
$| J_\alpha f (x) | \le \frac{1}{\Gamma(\alpha /2) } \int_0^{+\infty} t^{\alpha /2 -1 }dt | f^* (x) | =\frac{2}{\alpha \Gamma(\alpha /2) } \delta^{\alpha /2} | f^* (x) |.$
On the other hand,
$| K_\alpha f(x)| \le \frac{1}{\Gamma(\alpha /2) } \int_\delta^{+\infty} t^{\alpha /2 -1 } | P_t f (x)| dt$
$\le \frac{C^{1/p}}{\Gamma(\alpha /2) } \int_\delta^{+\infty} t^{\frac{\alpha} {2}-\frac{n}{2p} -1 } dt \| f \|_p$
$\le \frac{C^{1/p}}{\Gamma(\alpha /2) } \frac{1}{-\frac{\alpha} {2}+\frac{n}{2p} }\delta^{\frac{\alpha} {2}-\frac{n}{2p} } \| f \|_p .$
We deduce
$| I_\alpha f (x) | \le \frac{2}{\alpha \Gamma(\alpha /2) } \delta^{\alpha /2} | f^* (x) |+ \frac{C^{1/p}}{\Gamma(\alpha /2) } \frac{1}{-\frac{\alpha} {2}+\frac{n}{2p} }\delta^{\frac{\alpha} {2}-\frac{n}{2p} } \| f \|_p.$
Optimizing the right hand side of the latter inequality with respect to $\delta$ yields
$| I_\alpha f (x) |\le \frac{ 2n C^{\alpha / n}}{ \alpha (n-p\alpha) \Gamma(\alpha /2)} \|f \|^{\alpha p /n}_p |f^*(x)|^{1-p\alpha/n}.$
The proof is then completed by using Stein’s maximal ergodic theorem $\square$

A special case, of particular interest, is when $\alpha =1$ and $p=2$. We get in that case the following Sobolev inequality:

Theorem: Let $n > 2$. If there exists $C > 0$ such that for every $t > 0$, $x,y \in \mathbb{M}$,
$p(x,y,t) \le \frac{C}{t^{n/2}},$
then for every $f \in C^\infty_0(\mathbb{M})$,
$\| f \|_{\frac{2n}{n-2}} \le 2^{1-1/n} \frac{ 2n C^{1 / n}}{ (n-2) \sqrt{\pi}} \| \sqrt{\Gamma(f)} \|_2$.

We mention that the constant in the above Sobolev inequality is not sharp even in the Euclidean case.

Combining the above with the Li-Yau upper bound for the heat kernel, we deduce the following theorem:

Theorem: Assume that $\mathbf{Ric} \ge 0$ and that there exists a constant $C > 0$ such that for every $x \in \mathbb{M}$ and $r \ge 0$, $\mu (B(x,r)) \ge C r^n$, then there exists a constant $C'=C'(n) > 0$ such that for every $f \in C^\infty_0(\mathbb{M})$,
$\| f \|_{\frac{2n}{n-2}} \le C' \| \sqrt{\Gamma(f)} \|_2$

In many situations, heat kernel upper bounds with a polynomial decay are only available in small times the following result is thus useful:

Theorem: Let $n > 0$, $0 < \alpha < n$, and $1 < p < \frac{n}{\alpha}$. If there exists $C > 0$ such that for every $0 < t \le 1$, $x,y \in \mathbb{M}$,
$p(x,y,t) \le \frac{C}{t^{n/2}},$
then, there is constant $C'$ such that for every $f \in L^p_\mu(\mathbb{M})$,
$\| (-L+1)^{-\alpha/2} f \|_{\frac{np}{n-p\alpha}} \le C' \|f \|_p$

Proof: We apply the Varopoulos theorem to the semigroup $Q_t=e^{-t} P_t$. Details are let to the reader $\square$

The following corollary shall be later used:

Corollary: Let $n > 2$. If there exists $C > 0$ such that for every $0 < t \le 1$, $x,y \in \mathbb{M}$,
$p(x,y,t) \le \frac{C}{t^{n/2}},$
then there is constant $C'$ such that for every $f \in C^\infty_0(\mathbb{M})$,
$\| f \|_{\frac{2n}{n-2}} \le C' \left( \| \sqrt{\Gamma(f)} \|_2 + \| f \|_2 \right)$

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