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

Abstract

In this paper we exploit a global lifting method for homogeneous Hormander vector fields in order to extend the Gibbons conjecture to any second-order differential operator $$\mathcal {L}_X = \sum _{j = 1}^mX_j^2$$, where the $$X_j$$’s are linearly independent smooth vector fields on $$\mathbb {R}^n$$ satisfying Hormander’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 $$\Delta _\lambda $$-Laplacians studied by Franchi, Lanconelli and Kogoj. We also establish a comparison result for the solutions of the semi-linear equation $$\mathcal {L}_Xu+f(u) = 0$$ under suitable assumptions on the function f.