Multiversion of the Hausdorff--Young inequality

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$$\mathbb{E}F\text{\hspace{0.17em}⁣}\left(B\right(\mathrm{\mid }{T}_{z_1}{f}_1({\xi }_1)\mathrm{\mid },\dots ,\mathrm{\mid }{T}_{z_n}{f}_n({\xi }_n)\mathrm{\mid }\left)\right)\le F\text{\hspace{0.17em}⁣}\left(\mathbb{E}B\right(\mathrm{\mid }{f}_1({\xi }_1)\mathrm{\mid },\dots ,\mathrm{\mid }{f}_n({\xi }_n)\mathrm{\mid }\left)\right)$$ 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.