Regularized Brascamp-Lieb inequalities via Optimal Transport and Study of Equality Cases

TL;DR

This paper advances the understanding of regularized Brascamp-Lieb inequalities by applying optimal transport theory, characterizing extremizers, and exploring conditions for finiteness and Gaussian optimizers.

Contribution

It introduces a novel approach using anisotropic Caffarelli's contraction and heat flow methods to analyze regularized inequalities and their extremizers.

Findings

Characterization of all optimizers for regularized Brascamp-Lieb inequalities.

Conditions for the finiteness of the Brascamp-Lieb constant.

Identification of Gaussian extremizers and their properties.

Abstract

We consider regularized Brascamp-Lieb inequalities using the theory of optimal transportation, more precisely an anisotropic version of Caffarelli's contraction theorem. Furthermore, we provide a full picture concerning the issues of finiteness of the Brascamp-Lieb constant and of the existence of Gaussian extremizers. We also find all optimizers for these regularized Brascamp-Lieb inequalities by employing heat flow methods that were already used to settle this question for the non-regularized Brascamp-Lieb inequality and introducing new ideas to deal with several difficulties, which do not appear for the non-regularized Brascamp-Lieb datum. Finally, we give some interesting applications.