Endpoint Strichartz estimates with angular integrability and some applications

Abstract

The endpoint Strichartz estimate $\Vert e^{it\Delta }f{\Vert }_{L_t^2{L}_x^{\infty }}\lesssim \Vert f{\Vert }_{L^2}$ 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 $\Vert e^{it\Delta }f{\Vert }_{L_t^2{L}_{\rho }^{\infty }{L}_{\omega }^2}\lesssim \Vert f{\Vert }_{L^2}$. Here we address a weighted version of such spherically averaged estimates. As an application, the existence of solutions for the inhomogeneous nonlinear Schrödinger equation is shown for $L^2$ data.

arXiv commentTo appear in J. Evol. Equ.,14 pages