Geometric second derivative estimates in Carnot groups and convexity

We prove some new a priori estimates for H_2-convex functions which are zero on the boundary of a bounded smooth domain \Omega in a Carnot group G. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature \mathcal H of the boundary of \Omega. As a consequence of our bounds we show that if G has step two, then for any smooth $H_2$-convex function in \Omega \subset G vanishing on the boundary of \Omega one has \sum_{i,j=1}^m \int_\Omega ([X_i,X_j]u)^2 dg \leq {4/3} \int_{\partial \Omega} \mathcal H |\nabla_H u|^2 d\sigma_H .