an:06203927
Zbl 1272.31009
Bonfiglioli, Andrea; Lanconelli, Ermanno
A new characterization of convexity in free Carnot groups
EN
Proc. Am. Math. Soc. 140, No. 9, 3263-3273 (2012).
00323264
2012
j
31C05 26B25 43A80 35J70
Summary: 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 \(\operatorname{Lie}(\mathbb G)\).