Strong Brascamp-Lieb Inequalities

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 Rényi 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.

68 pages, 4 figures. Properties of the two-parameter entropy were added in the introduction, the range of the parameter $(p,\hat{p},q)$ for the single-function version of Brascamp--Lieb inequalities was corrected, many proof details were added, and some typos and minor errors were fixed

References in corpus (2)

• Graphs of Joint Types, Noninteractive Simulation, and Stronger Hypercontractivity
• The Convexity and Concavity of Envelopes of the Minimum-Relative-Entropy Region for the DSBS

Cited by in corpus (3)

• The Convexity and Concavity of Envelopes of the Minimum-Relative-Entropy Region for the DSBS
• Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Renyi entropy
• On the $\Phi$-Stability and Related Conjectures