Covariant Riesz transform on differential forms for $1<p\le 2$

In this paper, we study $L^p$-boundedness ($1<p\le 2$) of the covariant Riesz transform on differential forms for a class of non-compact weighted Riemannian manifolds without assuming conditions on derivatives of curvature. We present in particular a local version of $L^p$-boundedness of Riesz transforms under two natural conditions, namely the curvature-dimension condition, and a lower bound on the Weitzenböck curvature endomorphism. As an application, the Calderón-Zygmund inequality for $1<p\le 2$ on weighted manifolds is derived under the curvature-dimension condition as hypothesis.