Adjoint Brascamp-Lieb inequalities

TL;DR

This paper introduces an adjoint form of Brascamp-Lieb inequalities, connecting them to entropy inequalities, and demonstrates their applications in harmonic analysis, including properties of Gowers norms and reverse inequalities for tomographic transforms.

Contribution

It presents a novel adjoint version of Brascamp-Lieb inequalities, linking them to entropy inequalities and expanding their applications in harmonic analysis.

Findings

Reproved a log-convexity property of Gowers uniformity norms.

Established reverse $L^p$ inequalities for tomographic transforms.

Connected adjoint inequalities to entropy Brascamp-Lieb inequalities.

Abstract

The Brascamp-Lieb inequalities are a generalization of the H\"older, Loomis-Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper we introduce an "adjoint" version of these inequalities, which can be viewed as an $L^p$ version of the entropy Brascamp-Lieb inequalities of Carlen and Cordero-Erausquin. As applications, we reprove a log-convexity property of the Gowers uniformity norms, and establish some reverse $L^p$ inequalities for various tomographic transforms. We conclude with some open questions.