Gauge functions, eikonal equation and Bôcher theorem on stratified Lie groups (Italian Title: Funzioni gauge, equazione eikonale e Teorema di B^ocher sui gruppi di Lie stratificati)

Let $L =\Sum_{i=1}^m X_i^2$ be a real sub-Laplacian on a Carnot group G and denote by $\nabla_L =(X_1,...,X_m)$ the intrinsic gradient related to L. Our aim in this present note is to analyze some features of the L-gauge functions on G, i.e., the homogeneous functions d such that L(d^\gamma) = 0 in $G\setminus\{0\}$ (for some $\gamma\in R\setminus\{0\}$). We consider the relation of L-gauge functions with: the L-Eikonal equation $|\nabla_L u| = 1$ in G; the Mean Value Formulas for the L-harmonic functions; the fundamental solution for L; the B^ocher-type theorems for nonnegative L-harmonic functions in the punctured open sets $\Omega\setminus\{x_0\}$.