The optimal constants in Holder-Brascamp-Lieb inequalities for discrete Abelian groups

TL;DR

This paper determines the best possible constants for Holder-Brascamp-Lieb inequalities in discrete Abelian groups, extending previous results to include finite groups through detailed analysis.

Contribution

It extends the understanding of optimal constants in Holder-Brascamp-Lieb inequalities to all discrete Abelian groups, including finite groups, building on prior work for finitely generated and torsion-free groups.

Findings

Optimal constants are identified for all discrete Abelian groups.

Analysis of finite groups is crucial for the complete characterization.

The results unify previous partial findings in the field.

Abstract

The optimal constants are found for Lebesgue norm multilinear inequalities of Holder-Brascamp-Lieb type for arbitrary discrete Abelian groups. Previously a criterion for finiteness of the constants had been established for finitely generated Abelian groups, and the optimal constant had been found in the torsion-free case. The main step here is the analysis of finite groups.