Gradient estimates under integral Ricci bounds

TL;DR

This paper establishes $L^p$-gradient estimates for solutions to the Laplace equation on Riemannian manifolds with integral Ricci bounds, extending regularity theory under weaker curvature conditions.

Contribution

It proves $L^p$-gradient estimates under integral Ricci bounds and constructs a counterexample showing the optimality of pointwise Ricci lower bounds.

Findings

Proved $L^p$-gradient estimates under integral Ricci bounds.

Constructed a counterexample for Ricci curvature bounds.

Explored the relation between $L^p$-gradient estimates and Sobolev spaces.

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.