I have a clarification question. If we have a Riemannian compact manifold $M$, then there exists constants c and $r_{0}$ such that for any radius r < $r_{0}$ we have
$$ \bigg(\frac{1}{|B_{r}(x)|} \int_{B_{r}(x)} \left|f(y) - \frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}f\right|^{\frac{np}{n-p}}dy \bigg)^{\frac{n-p}{np}}\leq c r \bigg(\frac{1}{|B_{r}(x)|} \int_{B_{r}(x)} |Df(y)|^{p} dy \bigg)^{\frac{1}{p}}$$ where $f \in C^{1}(B_{r}(x))$ and $x \in M$.
My question is, is it true that since M is compact, then the constants c and $r_{0}$ are uniform, independent of the point $x$ taken? And can you please refer me to a reference where is proven? Thank you.
See Aubin's book Some Nonlinear Problems in Riemannian Geometry.