Anisotropic uncertainty principles for metaplectic operators

TL;DR

This paper develops anisotropic uncertainty principles for metaplectic operators, revealing directional uncertainty phenomena linked to symplectic degeneracies and extending classical results to a broader, geometric setting.

Contribution

It introduces new anisotropic uncertainty principles for metaplectic operators, characterizes extremizers, and extends classical theorems to the symplectic and metaplectic framework.

Findings

Uncertainty phenomena are directionally confined to an effective phase-space dimension.

Complete characterization of extremizers as partially Gaussian functions.

Extension of classical uncertainty principles to the metaplectic setting with geometric insights.

Abstract

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$. Building on this framework, we extend the Beurling-H\"ormander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.