Spherically averaged maximal function and scattering for the 2D cubic derivative Schrodinger equation

We prove scattering for the 2D cubic derivative Schrodinger equation with small data in the critical Besov space with one degree angular regularity. The main new ingredient is that we prove a spherically averaged maximal function estimate for the 2D Schrodinger equation. We also prove a global well-posedness result for the 2D Schrodinger map in the critical Besov space with one degree angular regularity. The key ingredients for the latter results are the spherically averaged maximal function estimate, null form structure observed in cite{Bej}, as well as the generalised spherically averaged Strichartz estimates obtained in cite{Guo2} in order to exploit the null form structure.