Geometric second derivative estimates in Carnot groups and convexity

Abstract

We prove some new a priori estimates for H_2-convex functions which are zero on the boundary of a bounded smooth domain Ωin 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 Ω. As a consequence of our bounds we show that if G has step two, then for any smooth $H_2$-convex function in Ω\subset G vanishing on the boundary of Ωone has \sum_{i,j=1}^m \int_Ω([X_i,X_j]u)^2 dg \leq {4/3} \int_{\partial Ω} \mathcal H |\nabla_H u|^2 dσ_H .