Revista Matemática Iberoamericana


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Volume 26, Issue 2, 2010, pp. 481–528
DOI: 10.4171/RMI/607

Published online: 2010-08-31

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

Xiang-Dong Li[1]

(1) Chinese Academy of Sciences, Beijing, China

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.

Keywords: Hodge decomposition, martingale transforms, Riesz transforms, Weitzenböck curvature

Li Xiang-Dong: Riesz transforms on forms and $L^p$-Hodge decomposition on complete Riemannian manifolds. Rev. Mat. Iberoamericana 26 (2010), 481-528. doi: 10.4171/RMI/607