 
Summary: HARDY'S UNCERTAINTY PRINCIPLE, CONVEXITY AND
SCHR¨ODINGER EVOLUTIONS
L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA
Abstract. We prove the logarithmic convexity of certain quantities, which
measure the quadratic exponential decay at infinity and within two charac
teristic hyperplanes of solutions of Schr¨odinger evolutions. As a consequence
we obtain some uniqueness results that generalize (a weak form of) Hardy's
version of the uncertainty principle. We also obtain corresponding results for
heat evolutions.
1. Introduction
In this paper we continue the study initiated in [11] and [2] on unique continua
tion properties of solutions of Schr¨odinger evolutions
(1.1) itu + u = V (x, t)u , in Rn
× [0, 1].
The goal is to obtain sufficient conditions on a solution u, the potential V and the
behavior of the solution at two different times, t0 = 0 and t1 = 1, which guarantee
that u 0 in Rn
× [0, 1].
One of our motivations comes from a well known result due to G. H. Hardy [16,
pp. 131] (see also [1] for a recent survey on this topic), which concerns the decay
