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Using the notion of h-subdifferential, we characterize both first and second order differentiability of h-convex functions in stratified groups. Besides some new results involving the h-subdifferential of h-convex functions, we show that at all h-differentiability points of an h-convex function, the existence of a second order expansion coincides with a suitable differentiability of its horizontal gradient.

We prove some first order regularity estimates for a class of convex functions in Carnot–Caratheodory spaces, generated by Hormander vector fields. Our approach relies on both the structure of metric balls induced by Hormander vector fields and local upper estimates for the corresponding subharmonic functions.

In the class of Carnot groups, we study fine properties of sets of finite perimeter. Improving a recent result by Ambrosio–Kleiner–Le Donne, we show that the perimeter measure is local, i.e., that given any pair of sets of finite perimeter their perimeter measures coincide on the intersection of their essential boundaries. This solves a question left open in Ambrosio et al. (Calculus of variations: topics from mathematical heritage of Ennio De Giorgi. Quad Mat). As a consequence, we prove a general chain rule for BV functions in this setting.

We extend Dudley’s characterization of convex functions to the framework of h-convex distributions on stratified groups. Precisely, we prove that every distribution with nonnegative horizontal Hessian is defined by an h-convex function.