Regularized Brascamp--Lieb inequalities

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 nondegenerate cases of the inverse Brascamp–Lieb inequality. Here we build on work of Chen, Dafnis and 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 inequalities 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 Verdú.