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Anisotropic uncertainty principles for metaplectic operators

Elena Cordero, Gianluca Giacchi, Edoardo Pucci

TL;DR

This work extends classical uncertainty principles to arbitrary metaplectic operators, including degenerate cases where the $B$ block of the associated symplectic matrix has a nontrivial kernel. By exploiting a phase-space decomposition aligned with $\ker(B)$ and its orthogonal complement, the authors prove sharp anisotropic Heisenberg–Pauli–Weyl inequalities in $r=\mathrm{rank}(B)$ effective directions and characterize extremizers as partially Gaussian along null directions. They further develop metaplectic Beurling– Hörmander and Morgan-type UPs, revealing a precise polynomial–Gaussian phase-space structure and a sharp, metaplectic-invariant threshold that governs admissible functions. The results recover the classical Fourier case and free metaplectic transforms as special instances and illuminate the geometric and anisotropic nature of uncertainty in the presence of symplectic degeneracies, with connections to dispersive PDEs and evolution equations.

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-Hörmander 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.

Anisotropic uncertainty principles for metaplectic operators

TL;DR

This work extends classical uncertainty principles to arbitrary metaplectic operators, including degenerate cases where the block of the associated symplectic matrix has a nontrivial kernel. By exploiting a phase-space decomposition aligned with and its orthogonal complement, the authors prove sharp anisotropic Heisenberg–Pauli–Weyl inequalities in effective directions and characterize extremizers as partially Gaussian along null directions. They further develop metaplectic Beurling– Hörmander and Morgan-type UPs, revealing a precise polynomial–Gaussian phase-space structure and a sharp, metaplectic-invariant threshold that governs admissible functions. The results recover the classical Fourier case and free metaplectic transforms as special instances and illuminate the geometric and anisotropic nature of uncertainty in the presence of symplectic degeneracies, with connections to dispersive PDEs and evolution equations.

Abstract

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on , including degenerate cases associated with symplectic matrices whose -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 . First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to , 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 . Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both 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.
Paper Structure (7 sections, 15 theorems, 139 equations, 1 figure)

This paper contains 7 sections, 15 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The symplectic group
  4. The metaplectic group
  5. Anisotropic tensor products
  6. Uncertainty principle of Heisenberg-Pauli-Weyl
  7. Beurling and Morgan-type UPs

Key Result

Theorem 1.1

Let $i = 1, \dots, d$ and $f \in L^2(\mathbb{R}^d)$. Then Moreover, (1) is an equality if and only if $f$ is of the form where $\Theta$ is a function in $L^2(\mathbb{R}^{d-1})$, $\gamma > 0$, and $a, b$ are real constants for which the two infimum in e00 are realized.

Figures (1)

  • Figure 1: A schematic representations of the action of the block $B$, its Moore-Penrose inverse $B^+$, and the parametrization $V$ of $\ker(B)^\perp$, chosen so that $V^+=V^T$.

Theorems & Definitions (28)

  • Theorem 1.1: Heisenberg’s Inequality
  • Theorem 1.2
  • Theorem 1.3: Gel’fand-Shilov type UP
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • ...and 18 more