Multiversion of the Hausdorff--Young inequality

Abstract

We consider a family of jointly Gaussian random vectors $ξ_j \in \mathbb{R}^{k_j}$, each standard normal but possibly correlated, and investigate when\[ \mathbb{E}\, F\!\Bigl(B\bigl(|T_{z_1} f_1(ξ_1)|,\dots,|T_{z_n} f_n(ξ_n)|\bigr)\Bigr) \;\;\le\;\; F\!\Bigl(\,\mathbb{E}\,B\bigl(|f_1(ξ_1)|,\dots,|f_n(ξ_n)|\bigr)\Bigr) \] holds, where $T_{z}$ is either a Mehler transform $(z \in \mathbb{C})$ or a noise operator $(z \in \mathbb{R})$. This framework unifies and extends real and complex hypercontractivity to multi-function settings, yielding multiversions of the sharp Hausdorff--Young inequality, the log-Sobolev inequality, and a noisy Gaussian--Jensen inequality. Applications include a new covariance-based characterization of the Brascamp--Lieb inequality in the presence of noise.