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

TL;DR

This paper introduces Calderón-Zygmund inequalities on Riemannian manifolds, explores their dependence on geometry, and establishes criteria for their validity, impacting Sobolev space theory on noncompact manifolds.

Contribution

It defines Calderón-Zygmund inequalities on Riemannian manifolds, analyzes their geometric conditions, and derives new results for Sobolev spaces and gradient estimates.

Findings

Calderón-Zygmund inequalities can hold or fail depending on the manifold's geometry.

Established geometric criteria ensuring the validity of these inequalities.

Derived new density results for second order Sobolev spaces on noncompact 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.