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Notion of convexity in Carnot groups. (English) Zbl 1077.22007
The 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.

22E25 Nilpotent and solvable Lie groups
35A30 Geometric theory, characteristics, transformations in context of PDEs
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