I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact manifold with or 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 was looking at some sources but what I could find is a global inequality where only the function $f(y)$ itself is bounded by the derivative not $f(y)-f(z)$. If you can point me to a source, where I could find such an inequality, I would be grateful. Thank you.
(This is just a comment without too many details, and I'm not sure whether it will help you, but it is too long to fit into a comment, so I used the answer field)
If you want to do something like that on a manifold, then you need to say what $|.|$ is, i.e. you need a metric, so you are probably referring to Riemannian manifolds. If you have a compact Riemannian manifold, then in suitable local charts the metric can be bounded from below and from above by the Euclidean metric, i.e. $$ g(v,v) \le C \langle v, v\rangle \le C^\prime g(v, v)$$ If you choose geodesic coordinate neighbourhoods you have, in addition, radial isometries, which allows to retain the radial growth in the inequalities. So, unless you are picky about the constant $c$ appearing in your inequality's right hand side, you get the desired inequality on the manifold simply by adapting the Euclidean ones, using well known techniques from Riemannian geometry.
As a side remark, a global $L^p$ bound for $f$ by $Df$ cannot be true since (on compact $M$) you can always add an arbitrary constant to $f$ without changing the derivative and without getting something which is not integrable, but you can increase the $L^p$ norm as much as you like. You need a bound at least at some fixed point.