The Calderón–Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds

We introduce the concept of Calderón–Zygmund inequalities on Riemannian manifolds. For 10C2>0. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón–Zygmund inequality (like new denseness results for second order LpLp-Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.