Gradient estimates under integral Ricci bounds

In this paper we study $W^{1,p}$ global regularity estimates for solutions of $Δu = f$ on Riemannian manifolds. Under integral (lower) bounds on the Ricci tensor we prove the validity of $L^p$-gradient estimates of the form $|| \nabla u ||_{L^p} \le C (|| u ||_{L^p} + || Δu||_{L^p})$. 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.

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