Riesz transforms on forms and $L^p$-Hodge decomposition on complete Riemannian manifolds

Abstract

In this paper we prove the Strong L^p -stability of the heat semigroup generated by the Hodge Laplacian on complete Riemannian manifolds with non-negative Weitzenböck curvature. Based on a probabilistic representation formula, we obtain an explicit upper bound of the L^p -norm of the Riesz transforms on forms on complete Riemannian manifolds with suitable curvature conditions. Moreover, we establish the Weak L^p -Hodge decomposition theorem on complete Riemannian manifolds with non-negative Weitzenböck curvature.