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

We establish the \(L^p\)-boundedness, for \(p>2\), of the covariant Riesz transform \(\nabla(\Delta_\mu^{(k)}+\sigma)^{-1/2} \) on differential forms over a class of complete weighted Riemannian manifolds.

The proof is based on an heat-kernel criterion involving local volume doubling, heat kernel upper estimates, Kato-type curvature control, and gradient bounds for the heat semigroup on forms.

Under curvature-dimension assumptions and Kato-type curvature bounds, this criterion applies and yields boundedness for all sufficiently large \(\sigma\).

In particular, in the unweighted case, the result confirms a conjecture of Baumgarth, Devyver and Güneysu~\cite{BDG-23}.

As an application, we obtain Calderón--Zygmund inequalities for \(p>2\) on weighted manifolds, which extends the recent work \cite{CCT} on manifolds without weight.