Non local Poincar\'e inequalities on Lie groups with polynomial volume growth

TL;DR

This paper establishes conditions under which non-local Poincaré inequalities hold on Lie groups with polynomial volume growth, extending classical inequalities to non-local settings with weighted measures.

Contribution

It provides new sufficient conditions for non-local Poincaré inequalities on Lie groups with polynomial volume growth, incorporating weighted measures.

Findings

Derived a sufficient condition for local Poincaré inequalities on Lie groups.

Established a non-local Poincaré inequality with weighted measure.

Extended classical inequalities to non-local and weighted contexts.

Abstract

Let $G$ be a real connected Lie group with polynomial volume growth, endowed with its Haar measure $dx$. Given a $C^2$ positive function $M$ on $G$, we give a sufficient condition for an $L^2$ Poincar\'e inequality with respect to the measure $M(x)dx$ to hold on $G$. We then establish a non-local Poincar\'e inequality on $G$ with respect to $M(x)dx$.