An application of a global lifting method for homogeneous Hörmander vector fields to the Gibbons conjecture

In this paper we exploit a global lifting method for homogeneous Hörmander vector fields in order to extend the Gibbons conjecture to any second-order differential operator LX=∑j=1mXj2, where the Xj’s are linearly independent smooth vector fields on Rn satisfying Hörmander’s rank condition and fulfilling a suitable homogeneity property with respect to a family of non-isotropic dilations. The class of these operators comprehends the sub-Laplacians on Carnot groups, the smooth Grushin-type operators and the smooth Δ λ-Laplacians studied by Franchi, Lanconelli and Kogoj. We also establish a comparison result for the solutions of the semi-linear equation LXu+ f(u) = 0 under suitable assumptions on the function f.