The Covariant Riesz Transforms on Riemannian Manifolds

We establish the $L^p$-boundedness of the local covariant Riesz transform for differential forms on manifold $M$ with bounded $\|Rm\|$. Let $\Delta_j$ be the Hodge Laplace operator on $j$-forms. For any $p \in (1, \infty)$ and $\kappa>\kappa_0$, we show that the operator $\nabla (\Delta_j + \kappa)^{-1/2}$ is bounded on $L^p(M)$. Consequently, we obtain Calder\'{o}n-Zygmund estimates for manifolds with bounded Riemannian curvature.