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

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 ${\mathrm{\Gamma }}_2$ 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.