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.

翻译：在本文中,我们为波兰空间的分布获得了尖锐的非线性非维度无Brascamp-Lieb的不平等(包括超合同性不平等),这加强了经典的Brascamp-Lieb的不平等,申请包括将Gerber夫人的lemma的lemma扩展至波兰空间的R'enyi差异和分布,加强小型扩张理论,以及确定美元稳定性的出处特征。我们在本文中的证据基于信息理论和组合技术。