Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball.
Q Can we find a constant $C=C(\kappa,r,m)$(independent on the point $p$), such that $$(\int_{B_p(r)}\phi^{2^*}/Vol(B_p(r)))^{\frac1{2^*}}\leq C (\int_{B_p(r)}|\nabla\phi|^2/Vol(B_p(r))),$$$$ \left(\frac{1}{Vol(B_p(r))}\int_{B_p(r)}\phi^{2^*}\right)^{\frac1{2^*}}\leq C \left(\frac{1}{Vol(B_p(r))}\int_{B_p(r)}|\nabla\phi|^2\right)^{1/2}, $$ for all compactly supported function $\phi$ on $B_p(R)$.
PS: I think it is proved in someone's paper or book, could anyone give me a reference?
