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

TL;DR

This work resolves the $L^p$-boundedness problem for the covariant Riesz transform on differential forms in the regime $p>2$ on weighted Riemannian manifolds, confirming the BDG-23 conjecture under curvature and volume-growth hypotheses. The authors develop a general $L^p$-boundedness criterion based on heat-kernel and gradient estimates and verify it via curvature-dimension-type conditions, then deduce a Hessian Calderón–Zygmund inequality for $p>2$ on weighted manifolds. The approach combines off-diagonal heat-kernel bounds, Davies-Gaffney-type estimates, and a localized $L^p$-boundedness framework (PTTS-2004) to transfer analytic control to global $L^p$ bounds. As a consequence, Calderón–Zygmund theory extends to weighted manifolds and supports $L^p$-regularity of Hessians and related PDEs on differential forms, with potential applications to Hodge theory and geometric analysis on non-compact spaces.

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 Güneysu~\cite{BDG-23}. As an application, the Calderón-Zygmund inequality for $p> 2$ is derived on weighted manifolds, which extends the recent work \cite{CCT} on manifolds without weight.

Theorems & Definitions (26)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1: TX-99
• Lemma 3.2
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Theorem 3.5