Brascamp--Lieb inequalities on locally compact abelian groups

TL;DR

This paper extends the understanding of Brascamp--Lieb inequalities by establishing a structure theorem applicable to all locally compact abelian groups, unifying previous results and emphasizing Fourier invariance.

Contribution

It provides a general structure theorem for Brascamp--Lieb constants on locally compact abelian groups, unifying previous Euclidean and finitely generated group results.

Findings

Unified characterization of Brascamp--Lieb constants on locally compact abelian groups

Extension of finiteness conditions beyond Euclidean spaces

Emphasis on Fourier invariance in the general setting

Abstract

We establish a structure theorem for the Brascamp--Lieb constant formulated in the general setting of locally compact abelian groups. This extends and unifies the finiteness characterisations previously known for euclidean spaces and for finitely generated groups and their duals. We place particular emphasis on Fourier invariance throughout, reflecting the fundamental Fourier invariance of Brascamp--Lieb multilinear forms in this context.