Covariant Riesz transform on differential forms for 1<p≤2

In this paper, we study L p -boundedness ( 1 < p ≤ 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 ≤ 2 on weighted manifolds is derived under the curvature-dimension condition as hypothesis.