Curvature-dimension estimates for the Laplace–Beltrami operator of a totally geodesic foliation. (October 2015)

• Record Type: Journal Article
• Authors: Baudoin, Fabrice; Bonnefont, Michel
• Is Part Of: Nonlinear analysis. Volume 126(2015)
• Page Start: 159
• Page End: 169
• Publication Date: 2015-10
• Subjects: Laplacian -- Riemannian foliation -- Curvature dimension inequalities; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.06.025
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study Bakry-Émery type estimates for the Laplace–Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the Γ 2 operator may not be bounded from below but the horizontal Bakry-Émery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang–Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin and Garofalo (2015) is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one.