Hardy and uncertainty inequalities on stratified Lie groups

TL;DR

This paper establishes Hardy and uncertainty inequalities on stratified Lie groups, introducing bounded operators involving the sub-Laplacian and deriving related inequalities such as the Heisenberg-Pauli-Weyl inequality.

Contribution

It proves boundedness of specific operators on stratified Lie groups and derives new uncertainty inequalities, including a logarithmic version and a generalized Heisenberg-Pauli-Weyl inequality.

Findings

Operators $T_\alpha$ are bounded on $L^p(G)$.

Derived a logarithmic uncertainty inequality.

Established a generalized Heisenberg-Pauli-Weyl inequality.

Abstract

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_\alpha: f \mapsto |.|^{-\alpha} L^{-\alpha/2} f$, where $|.|$ is a homogeneous norm, $0 < \alpha < Q/p$, and $L$ is the sub-Laplacian, are bounded on the Lebesgue space $L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg-Pauli-Weyl inequality, relating the $L^p$ norm of a function $f$ to the $L^q$ norm of $|.|^\beta f$ and the $L^r$ norm of $L^{\delta/2} f$.