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We prove scattering for the 2D cubic derivative Schr\"odinger equation withsmall data in the critical Besov space with one degree angular regularity. Themain new ingredient is that we prove a spherically averaged maximal functionestimate for the 2D Schr\"odinger equation. We also prove a globalwell-posedness result for the 2D Schr\"odinger map in the critical Besov spacewith one degree angular regularity. The key ingredients for the latter resultsare the spherically averaged maximal function estimate, null form structureobserved in \cite{Bej}, as well as the generalised spherically averagedStrichartz estimates obtained in \cite{Guo2} in order to exploit the null formstructure.

Spherically averaged maximal function and scattering for the 2D cubic derivative Schr\"odinger equation