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

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 Γ2Γ2 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.