Quantitative C^1 - estimates on manifolds

We prove a ${\mathsf{C}}^1$-elliptic estimate of the form $ \sup_{B(x,r/2)} |$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}$ (\psi) | \leq C \left{ \sup_{B(x,r)} |\Delta \psi| + \sup_{B(x,r)} |\psi| \right}, $validonanycompleteRiemannianmanifold$M$andforanysmoothfunction$\psi$whichisdefinedinanighbourhoodof$B(x,r)$,withanexplicitquantitativecontrolontheconstant$C=C(B(x,r))$intermsofthecurvatureofthegeodesicball$B(x,r)\subset M$.