Nonlinear Calder\'on-Zygmund inequalities for maps

TL;DR

This paper establishes a nonlinear Calderón-Zygmund inequality for maps between complete Riemannian manifolds, motivated by deriving $L^p$ bounds on geometric quantities of isometric immersions.

Contribution

It introduces a novel nonlinear Calderón-Zygmund inequality applicable to maps between complete Riemannian manifolds, extending classical linear results.

Findings

Proves a nonlinear Calderón-Zygmund inequality for manifold maps.

Provides a framework to estimate second fundamental form bounds from mean curvature.

Extends inequalities to possibly noncompact Riemannian manifolds.

Abstract

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calder\'on-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.