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We prove scattering for the two-dimensional 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 two-dimensional Schrödinger equation. We also prove a global well-posedness result for the two-dimensional 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 [ 2 ], as well as the generalized spherically averaged Strichartz estimates obtained in [ 10 ] in order to exploit the null form structure.

Spherically Averaged Maximal Function and Scattering for the Two-Dimensional Cubic Derivative Schrödinger Equation