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The theorem of Busemann-Feller-Alexandrov in Carnot groups. (English) Zbl 1071.22004
The authors prove a version of the Busemann-Feller-Alexandrov theorem for the class of weakly \(H\)-convex functions in Carnot groups. Precisely, let \(\mathbf G\) be a Carnot group of step \(r=2\) with a system \(X_1,\dots,X_m\) of bracket generating left-invariant vector fields. If \(u\in C({\mathbf G})\) is a weakly \(H\)-convex function, then the horizontal second derivatives \(X_iX_ju\) exist at a.e. point in \(\mathbf G.\) More precisely, for \(dg\)-a.e. point \(g_0\in{\mathbf G}\) there exists a polynomial of weighted degree \(\leq 2,\) \(P_u(g;g_0),\) such that \[ \lim_{g\to g_0}\frac{u(g)-P_u(g;g_0)}{d(g,g_0)^2}=0. \]

MSC:
22E20 General properties and structure of other Lie groups
22E30 Analysis on real and complex Lie groups
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