Schrödinger equations: pointwise convergence to the initial data

Let be the solution of the Schrödinger equation with initial data in the Sobolev space <!-- MATH ${H^s}({{\mathbf{R}}^n})$ --> with <!-- MATH $s > \frac{1}{2}$ --> \frac{1}{2}$">. The a.e. convergence of to follows from a weighted estimate of the maximal function <!-- MATH $u * (x,t) = {\text{su}}{{\text{p}}_{t > 0}}|u(x,t)|$ --> 0}}\vert u(x,t)\vert$">.