Multiversion of the Hausdorff--Young inequality

TL;DR

This paper develops a unified framework extending hypercontractivity and related inequalities to multivariable Gaussian settings, leading to new versions of classical inequalities like Hausdorff--Young and log-Sobolev, with applications in covariance analysis.

Contribution

It introduces a multiversion approach that generalizes hypercontractivity and classical inequalities to multi-function Gaussian contexts, including complex and real noise operators.

Findings

Derived multiversions of Hausdorff--Young inequality.

Extended log-Sobolev and Jensen inequalities to Gaussian noise settings.

Provided a covariance-based characterization of Brascamp--Lieb inequality.

Abstract

We consider a family of jointly Gaussian random vectors $\xi_j \in \mathbb{R}^{k_j}$, each standard normal but possibly correlated, and investigate when\[ \mathbb{E}\, F\!\Bigl(B\bigl(|T_{z_1} f_1(\xi_1)|,\dots,|T_{z_n} f_n(\xi_n)|\bigr)\Bigr) \;\;\le\;\; F\!\Bigl(\,\mathbb{E}\,B\bigl(|f_1(\xi_1)|,\dots,|f_n(\xi_n)|\bigr)\Bigr) \] 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.