The Covariant Riesz Transforms on Riemannian Manifolds

TL;DR

This paper proves the boundedness of covariant Riesz transforms on differential forms over Riemannian manifolds with bounded curvature, extending harmonic analysis tools to geometric settings.

Contribution

It establishes $L^p$-boundedness of covariant Riesz transforms on manifolds with bounded Riemannian curvature, providing new Calderón-Zygmund estimates.

Findings

Boundedness of the covariant Riesz transform on $L^p$ spaces.

Calderón-Zygmund estimates for manifolds with bounded curvature.

Extension of harmonic analysis techniques to Riemannian geometry.

Abstract

We establish the $L^p$-boundedness of the local covariant Riesz transform for differential forms on manifold $M$ with bounded $\|Rm\|$. Let $\Delta_j$ be the Hodge Laplace operator on $j$-forms. For any $p \in (1, \infty)$ and $\kappa>\kappa_0$, we show that the operator $\nabla (\Delta_j + \kappa)^{-1/2}$ is bounded on $L^p(M)$. Consequently, we obtain Calder\'{o}n-Zygmund estimates for manifolds with bounded Riemannian curvature.