  In this paper we study $W^{1,p}$ global regularity estimates for solutions of
$\Delta u = f$ on Riemannian manifolds. Under integral (lower) bounds on the
Ricci tensor we prove the validity of $L^p$-gradient estimates of the form $||
\nabla u ||_{L^p} \le C (|| u ||_{L^p} + || \Delta u||_{L^p})$. We also
construct a counterexample which proves that the previously known constant
lower bounds on the Ricci curvature are optimal in the pointwise sense. The
relation between $L^p$-gradient estimates and different notions of Sobolev
spaces is also investigated.
