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 Ω in a Carnot group $${\mathbb{G}}$$ . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature $${\mathcal{H}}$$ of ∂Ω. As a consequence of our bounds we show that if $${\mathbb{G}}$$ has step two, then for any smooth H 2-convex function in $$\Omega \subset {\mathbb{G}}$$ vanishing on ∂Ω one has $$\sum \limits _{i,j=1} ^m \int \limits_\Omega ([X_i,X_j]u)^2 \, dg \, \leq \, \frac{4}{3} \int \limits_{\partial \Omega} \mathcal H\ |\nabla_H u|^2\, d\sigma_H$$ .