Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 32, 2007, 425-435

GLOBAL ESTIMATES FOR THE SCHRÖDINGER MAXIMAL OPERATOR

Keith M. Rogers and Paco Villarroya

Universidad Autónoma de Madrid, Departamento de Matemáticas
28049 Madrid, Spain; keith.rogers 'at' uam.es

University of California
Los Angeles, CA 90095-1555, USA; paco.villarroya 'at' uv.es

Abstract. The Schrödinger equation, i\partial_tu + \Delta u = 0, with initial datum f contained in a Sobolev space H^s(Rⁿ), has solution e^it\Deltaf. We give sharp conditions under which sup_t|e^it\Deltaf| is bounded from H^s(R) to L^q(R) for all q, and give sharp conditions under which sup_0<t<1|e^it\Deltaf| is bounded from H^s(R) to L^q(R) for all q \neq 2. In higher dimensions, we show that sup_t|e^it\Deltaf| and sup_0<t<1|e^it\Deltaf| are bounded from H^s(Rⁿ) to L^q(Rⁿ) only if s \ge 1/2 - 1/2(n+1).

2000 Mathematics Subject Classification: Primary 35Q55, 42B25.

Key words: Schrödinger equation, pointwise convergence.

Reference to this article: K.M. Rogers and P. Villarroya: Global estimates for the Schrödinger maximal operator. Ann. Acad. Sci. Fenn. Math. 32 (2007), 425-435.

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