We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge-Ampère type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge-Ampère measures for convex functions in the Heisenberg group.

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We prove a ball-box theorem for nonsmooth Hörmander vector fields of step s ≥ 2.

We introduce a new class of curvature PDOs describing relevant properties of real hypersurfaces of C nþ1 : In our setting, the pseudoconvexity and the Levi form play the same role as the convexity and the real Hessian matrix play in the real Euclidean one. Our curvature operators are second-order fully nonlinear PDOs not elliptic at any point. However, when computed on generalized s-pseudoconvex functions, we shall show that their characteristic form is nonnegative definite with kernel of dimension one. Moreover, we shall show that the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurfaces. These properties allow us to prove a strong comparison principle, leading to symmetry theorems for domains with constant curvatures and to identification results for domains with comparable curvatures. r

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