Some super-Poincar\'e inequalities for gaussian-like measures on stratified Lie groups

TL;DR

This paper extends super-Poincaré inequalities to subelliptic measures on Metivier groups, using Hardy-type inequalities, thereby broadening the understanding of functional inequalities in non-commutative geometric settings.

Contribution

It introduces new super-Poincaré inequalities for Gaussian-like measures on stratified Lie groups, extending previous probabilistic results to a broader class of subelliptic measures.

Findings

Established super-Poincaré inequalities for measures on Metivier groups

Extended previous results using Hardy-type inequalities

Provided a probabilistic interpretation of the inequalities

Abstract

We continue the $U$-bound program initiated in [J. Funct. Anal. 258, 814-851 (2010)] and prove super-Poincar\'e inequalities for a class of subelliptic probability measures defined on M\'etivier groups, the main ingredient in the proof being a Hardy-type inequality. In doing so, we recover and extend some previous results from the probabilistic viewpoint.