Strong Brascamp-Lieb Inequalities

TL;DR

This paper establishes sharp, nonlinear, dimension-free Brascamp-Lieb inequalities for distributions on Polish spaces, extending classical results and enabling new applications in information theory and analysis.

Contribution

It introduces strengthened Brascamp-Lieb inequalities for Polish space distributions, broadening their scope and applicability with novel proof techniques.

Findings

Extension of Mrs. Gerber's lemma to Rènyi divergences

Strengthening of small-set expansion theorems

Characterization of the q-stability exponent

Abstract

In this paper, we derive sharp nonlinear dimension-free Brascamp--Lieb inequalities (including hypercontractivity inequalities) for distributions on Polish spaces, which strengthen the classic Brascamp--Lieb inequalities. Applications include the extension of Mrs. Gerber's lemma to the cases of R\'enyi divergences and distributions on Polish spaces, the strengthening of small-set expansion theorems, and the characterization of the exponent of the $q$-stability. Our proofs in this paper are based on information-theoretic and coupling techniques.