$L^p$-Boundedness of the Covariant Riesz Transform on Differential Forms for $p>2$

TL;DR

This paper proves the $L^p$-boundedness of the covariant Riesz transform on differential forms for $p>2$ on certain weighted Riemannian manifolds, confirming a conjecture and extending Calderón-Zygmund inequalities.

Contribution

It establishes the $L^p$-boundedness for $p>2$ of the covariant Riesz transform on differential forms on weighted manifolds, settling a conjecture and extending inequalities.

Findings

Proves $L^p$-boundedness for $p>2$ of the covariant Riesz transform.

Confirms a conjecture by Baumgarth, Devyver, and G"uneysu.

Extends Calderón-Zygmund inequalities to weighted manifolds.

Abstract

The $L^p$-boundedness for $p>2$ of the covariant Riesz transform on differential forms is proved for a class of non-compact weighted Riemannian manifolds under certain curvature and volume growth conditions, which in particular settles a conjecture of Baumgarth, Devyver and G\"uneysu~\cite{BDG-23}. As an application, the Calder\'on-Zygmund inequality for $p> 2$ is derived on weighted manifolds, which extends the recent work \cite{CCT} on manifolds without weight.