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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