On the boundedness in H 1/4 of the maximal square function associated with the Schrödinger equation

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.