A Variation on H\"older-Brascamp-Lieb Inequalities

TL;DR

This paper introduces a generalized framework for H"older-Brascamp-Lieb inequalities using a new size notion that extends $L^p$ norms, providing necessary and sufficient conditions for these inequalities to hold and conditions for extremizers.

Contribution

It develops a novel size concept for these inequalities and characterizes when they are valid, extending previous classifications and linking to Young's convolution inequality.

Findings

Established necessary and sufficient conditions for generalized inequalities.

Identified conditions for the existence of extremizers.

Connected the framework to classical convolution inequalities.

Abstract

The H\"older-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized H\"older-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlying linear maps match those of the convolution inequality of Young.