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
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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