The Brascamp--Lieb inequality on compact Lie groups and its extinction on homogeneous Lie groups
Michael G. Cowling, Ji Li, Chong-Wei Liang
TL;DR
This work extends Brascamp–Lieb inequalities to nonabelian locally compact groups, establishing a canonical reduction framework and sharp finiteness criteria across homogeneous and compact Lie groups. For homogeneous groups, the BL constant is shown to coincide with its Lie-algebra counterpart via dilations, and noncommutativity yields an obstruction to non‑Hölder inequalities in Heisenberg‑like groups. In the compact case, finiteness is characterized by explicit codimension inequalities on Lie ideals and normal subgroups, and the constant is computable via an open-subgroup maximum, paralleling the abelian theory. The results unify nonabelian BL theory, produce computable constants, and connect to prior abelian results, broadening applications in harmonic analysis and related fields.
Abstract
We study the Brascamp--Lieb inequalities on locally compact nonabelian groups and the Brascamp--Lieb constants $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ associated to a Brascamp--Lieb datum: locally compact groups $G$ and $G_j$, a family of homomorphisms $σ_j: G \to G_j$ 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 $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ is equal to the constant $\mathbf{BL}(\mathfrak{g}, \boldsymbol{\mathrm{d}σ}, \boldsymbol{p})$, where $\mathfrak{g}$ is the Lie algebra of $G$ and $\mathrm{d}σ_j$ is the differential of $σ_j$. For Heisenberg-like groups $G$, we show that the only inequalities that can occur are multilinear Hölder inequalities. For compact Lie groups, we find necessary and sufficient conditions for finiteness of the constant $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ in terms of $\boldsymbolσ$ and $\boldsymbol{p}$ and find an explicit expression for the constant, similar to those found by Bennett and Jeong in the abelian case.