Regularized Brascamp--Lieb inequalities

TL;DR

This paper extends the understanding of inverse Brascamp--Lieb inequalities by employing heat-flow techniques to regularized functions, generalizing previous theorems and connecting forward and reverse inequalities with Gaussian optimality.

Contribution

It introduces a heat-flow approach to regularized inverse Brascamp--Lieb inequalities, extending Barthe--Wolff's theorem and unifying forward-reverse inequalities with Gaussian saturation.

Findings

Extended Barthe--Wolff theorem to regularized functions

Connected forward and reverse Brascamp--Lieb inequalities

Reproduced Gaussian saturation property in a broader framework

Abstract

Given any (forward) Brascamp--Lieb inequality on euclidean space, a famous theorem of Lieb guarantees that gaussian near-maximizers always exist. Recently, Barthe and Wolff used mass transportation techniques to establish a counterpart to Lieb's theorem for all non-degenerate cases of the inverse Brascamp--Lieb inequality. Here we build on work of Chen--Dafnis--Paouris and employ heat-flow techniques to understand the inverse Brascamp--Lieb inequality for certain regularized input functions, in particular extending the Barthe--Wolff theorem to such a setting. Inspiration arose from work of Bennett, Carbery, Christ and Tao for the forward inequality, and we recover their generalized Lieb's theorem using a clever limiting argument of Wolff. In fact, we use Wolff's idea to deduce regularized inequalites in the broader framework of the forward-reverse Brascamp--Lieb inequality, in particular allowing us to recover the gaussian saturation property in this framework first obtained by Courtade, Cuff, Liu and Verd\'u.