A new characterization of convexity in free Carnot groups

A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x → Ax of the Euclidean case must be replaced by suitable group isomorphisms x → T_A(x), whose differential preserves the first layer of the stratification of Lie(G).