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Journal of the London Mathematical Society Advance Access originally published online on November 20, 2007
Journal of the London Mathematical Society 2008 77(1):51-68; doi:10.1112/jlms/jdm087
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© 2007 London Mathematical Society

On the boundedness in H1/4 of the maximal square function associated with the Schrödinger equation

Giacomo Gigante

Dipartimento di Ingegneria dell’Informazione e Metodi Matematici
Università di Bergamo
Viale Marconi 5
24044 Dalmine (BG)
Italy

Fernando Soria

Departamento de Matemáticas
Facultad de Ciencias, C-XV
Universidad Autónoma de Madrid
28049 Madrid
Spain
fernando.soria{at}uam.es

A long-standing conjecture for the linear Schrödinger equation states that 1/4 of the derivative in L2, in the sense of Sobolev spaces, suffices in any dimension for the solution to that equation to converge almost everywhere to the initial datum as the time goes to zero. This is only known to be true in dimension 1, by work of Carleson. In this paper we show that the conjecture is true on spherical averages. To be more precise, we prove the L2 boundedness of the associated maximal square function on the Sobolev class H1/4(Rn) in any dimension n.

2000 Mathematics Subject Classification 42B15, 42B25

This research was partially supported by the European Commission via the Harmonic Analysis Network ‘HARP’ and by grant MTM2004-00678.

Received October 2, 2006; published online November 20, 2007.


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