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.

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.