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

TL;DR

This paper establishes scattering and global well-posedness for the 2D cubic derivative Schrödinger equation and Schrödinger map in critical Besov spaces, introducing a novel spherically averaged maximal function estimate.

Contribution

The main novelty is the proof of a spherically averaged maximal function estimate for the 2D Schrödinger equation, enabling new scattering and well-posedness results.

Findings

Proved scattering for small data in critical Besov space.

Established global well-posedness for 2D Schrödinger map.

Developed spherically averaged maximal function estimate.

Abstract

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.