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

We extend Hardy’s uncertainty principle for a square integrable function ψ and its Fourier transform to the n-dimensional 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 Wψ of ψ. 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.