Hessian measures on the Heisenberg group

In this paper, we study the properties of k-convex functions on the Heisenberg group Hn, for 1⩽k⩽2n and prove the weak continuity of k-Hessian measures with respect to local uniform convergence in the Heisenberg setting. Our approach through monotonicity formulae makes use of previous research in the corresponding Euclidean case of Trudinger and Wang. The case k=2n provides an analogue of the Monge–Ampère measure of Aleksandrov for Hn. We also answer a conjecture of Garofalo and Tournier on monotonicity for the cases n>2.