Horizontal normal map on the Heisenberg group

We consider the notion of H-subdifferential and H-normal map of a function u on the Heisenberg group, based on the sub-Riemannian structure. The aim of this paper is to show how these objects, like in the Euclidean setting, fit well in the investigation of a function. In particular, a characterization of the convexity can be given via the nonemptiness of the H-subdifferential at every point g; furthermore, a convex function can be recovered through a well-known Rockafellar integration result adapted to the structure of the Heisenberg group. Moreover, we prove a monotonicity result for the H-normal map when a suitable strictly convex radial function u is considered, and we suggest a definition of the Monge-Ampère measure of a function v via the horizontal subgradient of v