Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation

Fabrice Baudoin (1) , Michel Bonnefont (2)

1 Department of Mathematics
2 IMB - Institut de Mathématiques de Bordeaux

We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the Γ2 operator may not be bounded from below but the horizontal Bakry-Emery 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-Garofalo is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one.

Domaines: Probabilités [math.PR]; Géométrie différentielle [math.DG]; Analyse fonctionnelle [math.FA]

Fabrice Baudoin, Michel Bonnefont. Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation. Nonlinear Analysis: Theory, Methods and Applications, 2015, 126, pp.159-169. ⟨10.1016/j.na.2015.06.025⟩. ⟨hal-01138611⟩