  We prove scattering for the 2D cubic derivative Schr\"odinger 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 Schr\"odinger equation. We also prove a global
well-posedness result for the 2D Schr\"odinger 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.
