Gradient estimates under integral Ricci bounds

Abstract

In this paper we study W 1,p global regularity estimates for solutions of ∆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 ||∇u|| L p ≤ C(||u|| L p +||∆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.