Spherically averaged endpoint Strichartz estimates for the two-dimensional Schr\"odinger equation

TL;DR

This paper demonstrates that by averaging solutions in the angular variable, certain endpoint Strichartz estimates for the 2D Schrödinger equation become valid, especially for radial data, despite known failures in the standard setting.

Contribution

The paper introduces spherically averaged endpoint Strichartz estimates for the 2D Schrödinger equation, establishing their validity under angular averaging, which was previously unknown.

Findings

Homogeneous endpoint estimates hold under angular averaging.

Retarded half-endpoint estimates are valid with angular averaging.

Full retarded endpoint estimates still fail even after averaging.

Abstract

The endpoint Strichartz estimates for the Schr\"odinger equation are known to be false in two dimensions. However, if one averages the solution in $L^2$ in the angular variable, we show that the homogeneous endpoint and the retarded half-endpoint estimates hold, but the full retarded endpoint fails. In particular, the original versions of these estimates hold for radial data.