Notion of convexity in Carnot groups.

*(English)*Zbl 1077.22007The aim of this interesting paper is to study appropriate notions of convexity in the setting of Carnot groups \(G\). First, the notion of strong \(H\)-convexity is examined. Some arguments showing that the concept is to restrictive are presented. Then the notion of weakly \(H\)-convex functions is defined. A function \(u:G\rightarrow {\mathbb R}\) is weakly \(H\)-convex if for any \(g\in G\) and every \(\lambda\in [0,1]\) we have
\[
u(g\delta_\lambda(g^{-1}g'))\leq (1-\lambda_u(g) + \lambda u(g')),
\]
where \(\delta_\lambda\) is a group dilation and \(g'\) is an element of the horizontal plane \(H_g\) passing through \(g\). It is proved that a twice differentiable function is weakly \(H\)-convex iff its symmetrized horizontal Hessian is positive semi-definite at any \(g\in G\). This is the subelliptic counterpart of the classical characterization of convex functions. The intrinsic gauge in any group of Heisenberg type is weakly \(H\)-convex. Moreover, a weakly \(H\)-convex function is Lipschitz continuous with respect to the sub-Riemannian metric of \(G\). The main result of the paper says that the supremum of the absolute value of a weakly \(H\)-convex continuous function over any ball can be estimated from above by the mean value of the absolute value. The local boundedness, the continuity on effective domains of weakly \(H\)-convex functions as well as their relations to fully nonlinear differential operators in the sub-Riemannian setting are studied.

Reviewer: Leszek Skrzypczak (Poznań)