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 Mr. and Mrs. Gerbers lemmas to the cases of Renyi 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.
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