Gradient estimates under integral Ricci bounds

Abstract

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.

9 pages, comments are welcome! Minor corrections. This paper was merged into arXiv:2207.08545