  A long standing conjecture for the linear Schroedinger equation states that
1/4 of 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 0. 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.
