## Lecture 24. Sharp Sobolev inequalities

In this Lecture, we are interested in sharp Sobolev inequalities in positive curvature. Let $(\mathbb{M},g)$ be a complete and $n$-dimensional Riemannian manifold such that $\mathbf{Ricci} \ge \rho$ where $\rho > 0$. We assume $n > 2$. As we already know from Lecture 15 , we have $\mu (\mathbb{M}) < +\infty$, but as we already stressed we do not want to use Bonnet-Myers theorem, since one of our goals will be to recover it by using heat kernel techniques. Without loss of generality, and to simplify the constants, we assume that $\mu(\mathbb{M}) =1$. Our goal is to prove the following sharp result:

Theorem: For every $1 \le p \le \frac{2n}{n-2}$ and $f \in C^\infty_0(\mathbb{M})$,
$\frac{n \rho}{(n-1)(p-2)} \left( \left( \int_{\mathbb{M}} | f |^p d\mu\right)^{2/p}-\int_\mathbb{M} f^2 d\mu \right) \le \int_\mathbb{M} \Gamma(f) d\mu.$

Our proof follows an argument due to Bakry. We observe that for $p=1$, the inequality becomes
$\frac{n \rho}{(n-1)} \left( \int_\mathbb{M} f^2 d\mu - \left( \int_{\mathbb{M}} | f | d\mu\right)^{2} \right) \le \int_\mathbb{M} \Gamma(f) d\mu.$
which is the Poincare inequality with optimal Lichnerowicz constant. For $p=2$, we get the log-Sobolev inequality
$\frac{n \rho}{2(n-1)} \left( \int_\mathbb{M} f^2 \ln f^2 d\mu -\int_\mathbb{M} f^2 d\mu \ln\int_\mathbb{M} f^2 d\mu \right) \le \int_\mathbb{M} \Gamma(f) d\mu.$
We prove our Sobolev inequality in several steps.

Lemma: For every $1 \le p \le \frac{2n}{n-2}$, there exists a constant $C_p > 0$ such that for every $f \in C^\infty_0(\mathbb{M})$,
$C_p \left( \left( \int_{\mathbb{M}} | f |^p d\mu\right)^{2/p}-\int_\mathbb{M} f^2 d\mu \right) \le \int_\mathbb{M} \Gamma(f) d\mu.$

Proof: Using Jensen’s inequality, it is enough to prove the result for $p= \frac{2n}{n-2}$. We already proved the following Li-Yau inequality: For $f \in C^\infty_0(\mathbb{M})$, $f \neq 0$, $t > 0$, and $x \in \mathbb{M}$,
$\| \nabla \ln P_t f (x) \|^2 \le e^{-\frac{2\rho t}{3}} \frac{ L P_t f (x)}{P_t f (x)} +\frac{n\rho}{3} \frac{e^{-\frac{4\rho t}{3}}}{ 1-e^{-\frac{2\rho t}{3}}}.$
As a consequence we have
$\frac{ L P_t f (x)}{P_t f (x)} \ge- \frac{n\rho}{3} \frac{e^{-\frac{2\rho t}{3}}}{ 1-e^{-\frac{2\rho t}{3}}},$
which yields,
$\int_t^{+\infty} \partial_t \ln P_sf (x) ds \ge- \frac{n\rho}{3} \int_t^{+\infty} \frac{e^{-\frac{2\rho s}{3}}}{ 1-e^{-\frac{2\rho s}{3}}} ds.$
We obtain then
$P_tf(x) \le \left( \frac{1}{1-e^{-\frac{2\rho t}{3}} }\right)^{n/2} \int_\mathbb{M} f d\mu.$
This of course implies the following upper bound on the heat kernel,
$p(x,y,t) \le \left( \frac{1}{1-e^{-\frac{2\rho t}{3}} }\right)^{n/2}.$
Using Varopoulos theorem, we deduce therefore that there is constant $C_p'$ such that for every $f \in C^\infty_0(\mathbb{M})$,
$\| f \|^2_{p} \le C'_p \left( \| \sqrt{\Gamma(f)} \|^2_2 + \| f \|^2_2 \right).$
We now use the following inequality which is easy to see:
$\left( \int_{\mathbb{M}} | f |^p d\mu \right)^{2/p} \le \left( \int_{\mathbb{M}} f d\mu \right)^{2}+\left( \int_{\mathbb{M}} \left| f -\int_{\mathbb{M}} f d\mu \right|^p d\mu \right)^{2/p}$
This yields
$\left( \int_{\mathbb{M}} | f |^p d\mu \right)^{2/p} \le \left( \int_{\mathbb{M}} f d\mu \right)^{2} +(p-1)C'_p \left( \| \sqrt{\Gamma(f)} \|^2_2 + \left\| f-\int_{\mathbb{M}} f d\mu \right\|^2_2 \right)$.
We can now bound
$\left\| f-\int_{\mathbb{M}} f d\mu \right\|^2_2 \le \frac{1}{\lambda_1} \int_\mathbb{M} \Gamma(f) d\mu,$
using the Poincare inequality $\square$

We now want to prove that the optimal $C_p$ in the previous inequality satisfies $C_p \ge \frac{n \rho}{(n-1)(p-2)}$. We assume $p > 2$ and consider the functional
$\frac{\left( \int_{\mathbb{M}} | f |^p d\mu\right)^{2/p}-\int_\mathbb{M} f^2 d\mu}{ \int_\mathbb{M} \Gamma(f) d\mu}.$
Classical non linear variational principles on the functional provide then a positive non trivial solution of the equation
$C_p(f^{p-1}-f)=-Lf.$
Set $f=u^r$ where $r$ is a constant to be later chosen. By the chain rule for diffusion operators, we get
$C_p ( u^{r(p-1)}-u^r)=-ru^{r-1}Lu -r(r-1)u^{r-2} \Gamma(u).$
Multiplying by $u^{-r} \Gamma(u)$ and integrating yields
$C_p \left( \int_{\mathbb{M}} u^{r(p-2)} \Gamma(u) d\mu -\int_{\mathbb{M}} \Gamma (u) d\mu \right)=-r \int_{\mathbb{M}} \frac{Lu}{u} \Gamma(u) d\mu -r (r-1) \int_{\mathbb{M}} \frac{\Gamma(u)^2}{u^2} d\mu.$
Now, integrating by parts,
$\int_{\mathbb{M}} u^{r(p-2)} \Gamma(u) d\mu=-\frac{1}{r(p-2)+1} \int_{\mathbb{M}} u^{r(p-2)+1} Lu d\mu.$
On the other hand, multiplying
$C_p ( u^{r(p-1)}-u^r)=-ru^{r-1} -r(r-1)u^{r-2} \Gamma(u),$
by $u^{1-r} Lu$ and integrating with respect to $\mu$ yields
$C_p \left( \int_{\mathbb{M}} u^{r(p-2)+1} Lu d\mu - \int_{\mathbb{M}} uLu d\mu \right)=-r \int_{\mathbb{M}} (Lu)^2 d\mu -r(r-1) \int_{\mathbb{M}} \frac{Lu}{u} \Gamma(u) d\mu.$
Combining the previous computations gives
$C_p \left( r(p-2) +1\right) \int_{\mathbb{M}} u^{r(p-2)} \Gamma(u) d\mu = r \int_{\mathbb{M}} (Lu)^2 d\mu +r(r-1) \int_{\mathbb{M}} \frac{Lu}{u} \Gamma(u) d\mu +C_p \int_{\mathbb{M}} \Gamma(u) d\mu$
Hence, we have
$C_p (p-2) \int_{\mathbb{M}} \Gamma(u) d\mu= \int_{\mathbb{M}} (Lu)^2 d\mu + \int_{\mathbb{M}} \frac{Lu}{u} \Gamma(u) d\mu +(r-1) \left( r(p-2) +1\right) \int_{\mathbb{M}} \frac{\Gamma(u)^2}{u^2} d\mu$
We have from Bochner’s inequality,
$\Gamma_2 (u^s)\ge \frac{1}{n} ( L u^s)^2 +\rho \Gamma(u^s).$
Once again, $s$ is a parameter that will be later decided. Using the chain, to rewrite the previous inequality, leads after tedious computations to
$\Gamma_2(u) +(s-1) \frac{1}{u} \Gamma ( u , \Gamma(u)) +(s-1)^2 \frac{\Gamma(u)^2}{u^2} \ge \rho \Gamma(u) +\frac{1}{n} (Lu)^2 +\frac{2}{n} (s-1) \frac{1}{u} Lu \Gamma (u) +\frac{1}{n} (s-1)^2 \frac{1}{u^2} \Gamma(u)^2.$
After integration and integration by parts, we see that
$\rho \int_{\mathbb{M}} \Gamma(u) d\mu \le \left( 1 -\frac{1}{n} \right) \int_{\mathbb{M}} (Lu)^2 d\mu-s' \left( 1+\frac{2}{n} \right) \int_{\mathbb{M}} \frac{1}{u} Lu \Gamma (u) d\mu+s'\left( 1+s'\left( 1-\frac{1}{n} \right) \right) \int_{\mathbb{M}} \frac{\Gamma(u)^2}{u^2} d\mu,$
where $s'=s-1$. Combining the previous inequalities we can eliminate the term $\int_{\mathbb{M}} \frac{1}{u} Lu \Gamma (u) d\mu$. Chosing
$\frac{s'}{r}=(p-1) \frac{n-1}{n+2},$
we see that the coefficient in front of $\int_{\mathbb{M}} (Lu)^2 d\mu$ is zero and we are left with
$\left( C_p \frac{(p-2)(n-1)}{n} -\rho \right)\int_\mathbb{M} \Gamma(u) d\mu \ge K(s',r) \int_{\mathbb{M}} \frac{\Gamma(u)^2}{u^2} d\mu,$
for some constant $K(s',r)$ which is seen to be non-negative as soon as $2 < p \le \frac{2n}{n-2}$. We conclude
$C_p \frac{(p-2)(n-1)}{n} -\rho \ge 0.$

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