Parabolic equations in sub-riemannian spaces: Boundary behavior of non-negative solutions, curvature-dimension inequalities, Li-Yau inequalities

The first part of this thesis is devoted to study the boundary behavior of non-negative solutions of diffusion equations associated with Hormander vector fields on cylinders whose bases are NTA domains. Our main results are: 1) a backward Harnack inequality for non-negative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder continuity up to the boundary of the quotient of two non-negative solutions which vanish continuously on a portion of the lateral boundary ; 3) the doubling property for the parabolic measure associated with the operator L. These results generalize to the sub-elliptic setting those theorems established in Lipschitz cylinders by Garofalo in [9], Fabes, Safonov and Yuan in [10], and Safonov and Yuan in [11]. The second part of the thesis is concern with the study of volume growth in sub-Riemannian manifolds with symmetries without the use of Jacobi fields. For instance, we were able to establish global volume comparison theorems using heat semigroup techniques recently discovered by Baudoin and Garofalo in [7] and Baudoin, Bonnefont and Garofalo in [5].