Schrödinger equations: pointwise convergence to the initial data

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