Strong Brascamp-Lieb Inequalities

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 lemmas 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 q-stability of Boolean functions. Our proofs in this paper are based on information-theoretic and coupling techniques.