Strichartz estimate for the free Schrödinger equation

In two dimensions, the exponents $(q,r)=(4,4)$ give the Strichartz estimate for the free Schrödinger equation below, since $2/q+2/r=1$. The homogeneous estimate and its dual areThe second operator is the adjoint of the first, so the bound follows by Lp duality. It also defines the time integral by continuous extension when an ordinary absolutely convergent integral is unavailable.

The pair $(4,4)$ is the critical cubic spacetime pair. The excluded two-dimensional endpoint is $(2,\infty )$, so the displayed bound should not be called that endpoint estimate.^[ref]

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 $L^2$.