On the Local Smoothing for the Schrödinger Equation

We prove a family of identities that involve the solution u to the following Cauchy problem: i∂tu +∆u = 0, u(0) = f(x), (t, x) ∈ Rt × R n x , and the Ḣ 1 2 (R)-norm of the initial datum f . As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate proved in [3], [8] and [9] and a uniqueness criterion for the solutions to the Schrödinger equation. This paper is devoted to the study of the following Cauchy problem: (0.1) i∂tu+∆u = 0, u(0) = f(x), (t, x) ∈ Rt × R n x, n ≥ 1. It is well-known that the solution to (0.1) satisfies the local smoothing estimate (see [3], [8] and [9]): (0.2) sup R∈(0,∞) 1 R ∫ ∞