Fourier duality in the Brascamp-Lieb inequality

TL;DR

This paper establishes a Fourier duality for the Brascamp-Lieb inequality in discrete abelian groups, extending the known Euclidean duality to a broader, more general setting involving locally compact abelian groups.

Contribution

It introduces a discrete analogue of the Fourier duality for Brascamp-Lieb inequalities and identifies constants on discrete groups with those on their duals.

Findings

Discrete Brascamp-Lieb constants are identified with dual group constants.

The duality principle extends to locally compact abelian groups.

Foundations for Brascamp-Lieb constants in this general setting are developed.

Abstract

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp-Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp-Lieb constants on (finitely-generated) discrete abelian groups with Brascamp-Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp-Lieb constants formulated in this generality.