Regularized Brascamp--Lieb inequalities

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 Verdú.

50 pages. Title changed, Theorem 2.2 and an appendix added. To appear in Analysis & PDE

References in corpus (12)

• On a reverse form of the Brascamp-Lieb inequality
• A direct proof of the functional Santalo inequality
• Partitions and functional Santalo inequalities
• Regularization under diffusion and anti-concentration of the information content
• On the nonlinear Brascamp-Lieb inequality
• Behaviour of the Brascamp--Lieb constant
• On integer solutions of Parsell-Vinogradov systems
• Decoupling for moment manifolds associated to Arkhipov--Chubarikov--Karatsuba systems
• Short probabilistic proof of the Brascamp-Lieb and Barthe theorems
• Stability of the Shannon-Stam inequality via the Föllmer process
• Kakeya-Brascamp-Lieb inequalities
• Decoupling inequalities for quadratic forms