Overview
This research focuses on establishing the $L^p$-boundedness, specifically for $p>2$, of the covariant Riesz transform $\nabla(\Delta_\mu^{(k)}+\sigma)^{-1/2}$ on differential forms. The study is conducted over a class of complete weighted Riemannian manifolds.
Research Context
The investigation into the $L^p$-boundedness of the covariant Riesz transform relates to areas of harmonic analysis and differential geometry. A conjecture proposed by Baumgarth, Devyver, and G"uneysu in 2023 ($\cite{BDG-23}$) concerning the unweighted case of this transform is addressed by the current findings. Furthermore, the work builds upon previous research on Calderón–Zygmund inequalities on manifolds without weight ($\cite{CCT}$), extending these inequalities to weighted manifolds under certain conditions.
Approach
The proof strategy for establishing $L^p$-boundedness for $p>2$ relies on a heat-kernel criterion. This criterion integrates several conditions and properties:
- Local volume doubling.
- Heat kernel upper estimates.
- Kato-type curvature control.
- Gradient bounds for the heat semigroup on forms.
The application of this criterion is observed when curvature-dimension assumptions and Kato-type curvature bounds are met. Under these conditions, the criterion applies, leading to boundedness for sufficiently large values of $\sigma$.
Findings
The principal finding is the establishment of the $L^p$-boundedness, for $p>2$, of the covariant Riesz transform $\nabla(\Delta_\mu^{(k)}+\sigma)^{-1/2}$ on differential forms over the specified class of complete weighted Riemannian manifolds. This result is contingent on satisfaction of the heat-kernel criterion involving local volume doubling, heat kernel upper estimates, Kato-type curvature control, and gradient bounds for the heat semigroup on forms.
Specifically:
- The criterion applies and yields boundedness for all sufficiently large $\sigma$ when curvature-dimension assumptions and Kato-type curvature bounds are in place.
- In the unweighted setting, the result confirms a conjecture previously put forth by Baumgarth, Devyver, and G"uneysu $\cite{BDG-23}$.
- As a direct application of these findings, Calderón–Zygmund inequalities for $p>2$ are obtained on weighted manifolds. This extends the scope of earlier work $\cite{CCT}$, which addressed these inequalities on manifolds without an associated weight.