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On the local smoothing for the Schrödinger equation

Authors:	Luis Vega Nicola Visciglia

Institution:	Departamento de Matemáticas, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain ; Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy

Abstract:	We prove a family of identities that involve the solution to the following Cauchy problem: $\displaystyle {\bf i} \partial_t u + \Delta u=0, u(0)=f(x), (t, x)\in {\mathbf R}_t\times {\mathbf R}^n_x,$ and the $\dot H^\frac 12({\mathbf R}^n)$ -norm of the initial datum . As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate proved by Constantin and Saut (1989), Sjölin (1987) and Vega (1988) and a uniqueness criterion for the solutions to the Schrödinger equation.

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