Maximum and comparison principles for convex functions on the Heisenberg group

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