Revista Matemática Iberoamericana

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

Xiang-Dong Li

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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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 481-528.

Dates
First available in Project Euclid: 4 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1275671309

Mathematical Reviews number (MathSciNet)
MR2677005

Zentralblatt MATH identifier
1197.53052

Subjects
Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
Secondary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx] 60J65: Brownian motion [See also 58J65]

Keywords
Hodge decomposition martingale transforms Riesz transforms Weitzenböck curvature

Citation

Li, Xiang-Dong. Riesz transforms on forms and $L^p$-Hodge decomposition on complete Riemannian manifolds. Rev. Mat. Iberoamericana 26 (2010), no. 2, 481--528. https://projecteuclid.org/euclid.rmi/1275671309


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