Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part II

Using the curvature-dimension inequality proved in Part I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup P t corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequality. These results require that the gradient of P t f remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold.