A new characterization of convexity in free Carnot groups

A characterization of convex functions in $\mathbb{R}^N$ 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 $\mathbb{G}$ when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps $x\mapsto Ax$ of the Euclidean case must be replaced by suitable group isomorphisms $x\mapsto T_A(x)$, whose differential preserves the first layer of the stratification of $\operatorname {Lie}(\mathbb{G})$.