Strichartz estimate for the free Schrödinger equation

In two dimensions, the endpoint spacetime estimate and its dual include

If a global cubic Schrödinger solution has finite $L_{t,x}^4$ norm on a time ray, the dual Strichartz estimate makes its interaction representation Cauchy in $L^2$. It therefore converges to scattering data $u_{\infty }$, and $u(t)-e^{it\Delta }{u}_{\infty }\to 0$ in $L^2$.