The Calder\'on-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds

Abstract

We introduce the concept of Calder\'on-Zygmund inequalities on Riemannian manifolds. For $1<p<\infty$, these are inequalities of the form $$ \left\Vert \mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert u\right\Vert _{L^p}+C_{2}\left\Vert \Delta u\right\Vert _{L^p}, $$ valid a priori for all smooth functions $u$ with compact support, and constants $C_1\geq 0$, $C_2>0$. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calder\'on-Zygmund inequality (like new denseness results for second order $\mathsf{L}^p$-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.