Local and nonlocal Poincar\'e inequalities on Lie groups

TL;DR

This paper establishes local and nonlocal Poincaré inequalities on noncompact Lie groups with sub-Riemannian structures, analyzing the growth of constants and extending to nonlocal settings with finite measures.

Contribution

It proves new local and nonlocal Poincaré inequalities on Lie groups, including growth estimates for constants and handling nondoubling groups.

Findings

Constant growth is at most exponential with radius

Nonlocal Poincaré inequality holds for suitable measures

Growth is exponential for nondoubling groups

Abstract

We prove a local $L^p$-Poincar\'e inequality, $1\leq p < \infty$, on noncompact 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 nondoubling, then its growth is indeed, in general, exponential. We also prove a nonlocal $L^2$-Poincar\'e inequality with respect to suitable finite measures on the group.