Global Calder\'on-Zygmund inequalities on complete Riemannian manifolds

TL;DR

This survey reviews recent advances and limitations in establishing global $W^{2,p}$ regularity for solutions to the Poisson equation on complete Riemannian manifolds, highlighting geometric conditions affecting these inequalities.

Contribution

It compiles and discusses various methods for obtaining $L^p$-Hessian estimates and presents counterexamples showing the failure of these inequalities under certain geometric conditions.

Findings

$W^{2,p}$ regularity can fail even with lower curvature bounds.

Gradient estimates are closely linked to Hessian estimates.

Counterexamples demonstrate limitations of integral inequalities.

Abstract

This paper is a survey of some recent results on the validity and the failure of global $W^{2,p}$ regularity properties of smooth solutions of the Poisson equation $\Delta u = f$ on a complete Riemannian manifold $(M,g)$. We review different methods developed to obtain a-priori $L^p$-Hessian estimates of the form $\| \Hess(u) \|_{L^p} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$ under various geometric conditions on $M$ both in the case of real valued functions and for manifold valued maps. We also present explicit and somewhat implicit counterexamples showing that, in general, this integral inequality may fail to hold even in the presence of a lower sectional curvature bound. The r\^ole of a gradient estimate of the form $\| \nabla u \|_{L^{p}} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$, and its connections with the $L^{p}$-Hessian estimate, are also discussed.