Global Calderón–Zygmund inequalities on complete Riemannian manifolds

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 Δu=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 ≤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 rôle of a gradient estimate of the form ∥∇u∥ L p ≤C 1 ∥u∥ L p +C 2 ∥f∥ L p , and its connections with the L p -Hessian estimate, are also discussed.