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

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We prove scattering for the 2D cubic derivative Schrödinger 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 Schrödinger equation. We also prove a global well-posedness result for the 2D Schrödinger 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 Bej, as well as the generalised spherically averaged Strichartz estimates obtained in Guo2 in order to exploit the null form structure.