Local and non-local Poincaré inequalities on Lie groups

We prove a local (Formula presented.) -Poincaré inequality, (Formula presented.), on non-compact Lie groups endowed with a sub-Riemannian structure. We show that the constant involved grows at most exponentially with respect to the radius of the ball, and that if the group is non-doubling, then its growth is indeed, in general, exponential. We also prove a non-local (Formula presented.) -Poincaré inequality with respect to suitable finite measures on the group.