Covariant Riesz transform on differential forms for $1<p\leq2$

TL;DR

This paper investigates the boundedness of the covariant Riesz transform on differential forms in weighted Riemannian manifolds for 1<p≤2, establishing local $L^p$ bounds without curvature derivative assumptions.

Contribution

It introduces new local $L^p$-boundedness results for the covariant Riesz transform under curvature-dimension and Weitzenböck curvature conditions, without requiring derivatives of curvature.

Findings

Established $L^p$-boundedness for the covariant Riesz transform on differential forms.

Derived Calderón-Zygmund inequalities under curvature-dimension conditions.

Provided local boundedness results without assumptions on curvature derivatives.

Abstract

In this paper, we study $L^p$-boundedness ($1<p\leq 2$) of the covariant Riesz transform on differential forms for a class of non-compact weighted Riemannian manifolds without assuming conditions on derivatives of curvature. We present in particular a local version of $L^p$-boundedness of Riesz transforms under two natural conditions, namely the curvature-dimension condition, and a lower bound on the Weitzenb\"{o}ck curvature endomorphism. As an application, the Calder\'on-Zygmund inequality for $1< p\leq 2$ on weighted manifolds is derived under the curvature-dimension condition as hypothesis.