Schrödinger equations: pointwise convergence to the initial data

Source abstract

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