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arXiv:0803.0910

The Multi-Dimensional Hardy Uncertainty Principle and its Interpretation in Terms of the Wigner Distribution; Relation With the Notion of Symplectic Capacity

Maurice A. de Gosson, Luef, Franz

Published Mar 6, 2008• 5 citations
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Abstract

We extend Hardy's uncertainty principle for a square integrable function and its Fourier transform to the multidimensional case using a symplectic diagonalization. We use this extension to show that Hardy's uncertainty principle is equivalent to a statement on the Wigner distribution of the function. We give a geometric interpretation of our results in terms of the notion of symplectic capacity of an ellipsoid. Furthermore, we show that Hardy's uncertainty principle is valid for a general Lagrangian frame of the phase space. Finally, we discuss an extension of Hardy's theorem for the Wigner distribution for exponentials with convex exponents.

DOI
10.48550/arxiv.0803.0910

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We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on L2(Rd)L^2(\mathbb{R}^d)L2(Rd), including degenerate cases associated with symplectic matrices whose BBB-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 rank(B)\mathrm{rank}(B)rank(B). First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to ker⁡(B)⊥\ker(B)^\perpker(B)⊥, 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 BBB. 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 fff 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.