The Brascamp--Lieb inequality on compact Lie groups and its extinction on homogeneous Lie groups

TL;DR

This paper investigates the Brascamp--Lieb inequalities on various Lie groups, establishing when the constants are finite or infinite, and relating group inequalities to their Lie algebra counterparts, with specific results for homogeneous and compact Lie groups.

Contribution

It extends Brascamp--Lieb inequalities to nonabelian Lie groups, characterizes the constants on homogeneous and compact groups, and relates group inequalities to algebraic structures.

Findings

On homogeneous Lie groups, the Brascamp--Lieb constant equals that of the Lie algebra.

For Heisenberg-like groups, only multilinear H"older inequalities occur.

Necessary and sufficient conditions for finiteness of the constant on compact Lie groups.

Abstract

We study the Brascamp--Lieb inequalities on locally compact nonabelian groups and the Brascamp--Lieb constants $\mathbf{BL}(G, \boldsymbol{\sigma}, \boldsymbol{p})$ associated to a Brascamp--Lieb datum: locally compact groups $G$ and $G_j$, a family of homomorphisms $\sigma_j: G \to G_j$ and Lebesgue indices $p_j$. We focus on homogeneous Lie groups and compact Lie groups. For homogeneous Lie groups $G$, we show that the constant $\mathbf{BL}(G, \boldsymbol{\sigma}, \boldsymbol{p})$ is equal to the constant $\mathbf{BL}(\mathfrak{g}, \boldsymbol{\mathrm{d}\sigma}, \boldsymbol{p})$, where $\mathfrak{g}$ is the Lie algebra of $G$ and $\mathrm{d}\sigma_j$ is the differential of $\sigma_j$. For Heisenberg-like groups $G$, we show that the only inequalities that can occur are multilinear H\"older inequalities. For compact Lie groups, we find necessary and sufficient conditions for finiteness of the constant $\mathbf{BL}(G, \boldsymbol{\sigma}, \boldsymbol{p})$ in terms of $\boldsymbol{\sigma}$ and $\boldsymbol{p}$ and find an explicit expression for the constant, similar to those found by Bennett and Jeong in the abelian case.