  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.
