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 $\mathrm{\parallel }Rm\mathrm{\parallel }$. Let ${\mathrm{\Delta }}_j$ be the Hodge Laplace operator on $j$-forms. For any $p\in (1,\mathrm{\infty })$ and $\kappa >{\kappa }_0$, we show that the operator $\mathrm{\nabla }({\mathrm{\Delta }}_j+\kappa {)}^{-1\mathrm{/}2}$ is bounded on $L^p(M)$. Consequently, we obtain Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.