I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of
$$ \int_{B_{r}(x)} |f(y) – f(z)|^{p} dy \leq c r^{n+p-1} \int_{B_{r}(x)} |Df(y)|^{p} |y-z|^{1-n} dy$$
where $f \in C^{1}(B_{r}(x))$ and $z \in B_{r}(x)$.
I would be grateful for any source you can point me to. I am also interested to know if there is a version of this inequalilty if the manifold has boundary.
Thank you.
Best Answer
The same inequality holds on Riemannian manifolds, at least if you want it for small $r$. Fix a point $x\in M$ and let $r_0$ be the injectivity radius at it. If $0<r\leq\frac12r_0$, then $\exp_x:U_r\to B_r(x)$ is a diffeomorphism and bi-Lipschitz continuous with a Lipschitz constant independent of $r$. Here $B_r(x)\subset M$ is the ball in the Riemannian metric and $U_r\subset T_xM$ is the ball of radius $r$ in the tangent plane at $x$. The tangent plane is just a Euclidean space, so your original estimate holds in $U_r$, and you can apply it to $f\circ\exp_p:U_r\to\mathbb R$. Since the exponential map is a bi-Lipschitz diffeomorphism $\exp_x:U_r\to B_r(x)$, you get the same estimate (possibly with worse a constant) on $B_r(x)$. If you take $r$ very small, you can make the Lipschitz constant arbitrarily close to one and give an arbitrarily small loss of constant in the final estimate. If you have some curvature bounds (especially if the manifold is compact), the constant $c$ in the estimate can be chosen uniformly; without any such assumptions it will depend on $x$.