 
Summary: THE HARDY UNCERTAINTY PRINCIPLE REVISITED
M. COWLING, L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA
Abstract. We give a realvariable proof of the Hardy uncertainty principle.
The method is based on energy estimates for evolutions with positive viscosity,
convexity properties of free waves with Gaussian decay at two different times,
elliptic L2estimates and the invertibility of the Fourier transform on L2(Rn)
and S (Rn).
1. Introduction
There are different ways of stating uncertainty principles for the Fourier trans
form: a function f and its Fourier transform
^f() =
1
2 R
ei·x
f(x) dx, R,
can not be highly concentrated unless f is zero. Among them one finds the Hardy
uncertainty principle (A1) [10] (see also [16, pp.131]), its extension (A2) established
in [4] and the BeurlingH¨ormander result (B) in [11]:
(A1) If f(x) = O(ex2
