Local and non‐local Poincaré inequalities on Lie groups

We prove a local L p $L^p$ -Poincaré inequality, 1 ⩽ < ∞ $1\leqslant \infty$ , on non-compact Lie groups endowed with sub-Riemannian structure. show that the constant involved grows at most exponentially respect to radius of ball, and if group is non-doubling, then its growth indeed, in general, exponential. also non-local 2 $L^2$ inequality suitable finite measures group.