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On the boundedness in H 1/4 of the maximal square function associated with the Schrödinger equation
Author(s) -
Gigante Giacomo,
Soria Fernando
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm087
Subject(s) - mathematics , sobolev space , almost everywhere , dimension (graph theory) , conjecture , square (algebra) , function (biology) , schrödinger equation , square integrable function , mathematical analysis , maximal function , pure mathematics , combinatorics , geometry , evolutionary biology , biology
A long‐standing conjecture for the linear Schrödinger equation states that 1/4 of the derivative in L 2 , 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 L 2 boundedness of the associated maximal square function on the Sobolev class H 1/4 (ℝ n ) in any dimension n .