Sobolev, Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality

Abstract

By adapting some ideas of M. Ledoux ([12], [13] and [14]) to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in [3]. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.