文摘
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. Keywords Curvature-dimension inequality Sub-Riemannian geometry Hypoelliptic operator Spectral gap Riemannian foliations