Endpoint Strichartz estimates with angular integrability and some applications

TL;DR

This paper explores spherically averaged endpoint Strichartz estimates with angular integrability, establishing new weighted estimates and applying them to prove existence results for inhomogeneous nonlinear Schrödinger equations with L^2 data.

Contribution

It introduces weighted spherically averaged Strichartz estimates and demonstrates their application to nonlinear Schrödinger equations.

Findings

Established weighted spherically averaged Strichartz estimates.

Proved existence of solutions for inhomogeneous nonlinear Schrödinger equations.

Extended endpoint estimates to include angular integrability.

Abstract

The endpoint Strichartz estimate $\|e^{it\Delta} f\|_{L_t^2 L_x^\infty} \lesssim \|f\|_{L^2}$ is known to be false in two space dimensions. Taking averages spherically on the polar coordinates $x=\rho\omega$, $\rho>0$, $\omega\in\mathbb{S}^1$, Tao showed a substitute of the form $\|e^{it\Delta} f\|_{L_t^2L_\rho^\infty L_\omega^2} \lesssim \|f\|_{L^2}$. Here we address a weighted version of such spherically averaged estimates. As an application, the existence of solutions for the inhomogeneous nonlinear Schr\"odinger equation is shown for $L^2$ data.