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 generalization of Mr. and Mrs. Gerber's lemmas to R\'enyi entropies 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.

翻译：在本文中,我们从波兰空间的分布中获得尖锐的非线性非维度无Brascamp-Lieb的不平等(包括超分率不平等),这强化了经典的Brascamp-Lieb不平等,应用包括将Gerber夫妇的lemmas对R'enyi的物种和波兰空间的分布进行概括化,加强小型扩展理论,以及描述Boolean功能的可兑换性。我们在本文中的证据基于信息理论和组合技术。