The Covariant Riesz Transforms on Riemannian Manifolds

Abstract

We establish the $L^p$-boundedness of the local covariant Riesz transform for differential forms on manifold $M$ with bounded $\Vert Rm\Vert$. 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 Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.

26pages