Weighted Strichartz estimates with angular integrability and their applications

Jungkwon Kim, Yoonjung Lee, Ihyeok Seo

Published 2019-12-30Version 1

The endpoint Strichartz estimate $\|e^{it\Delta} f\|_{L_t^2 L_x^\infty} \lesssim \|f\|_{L^2}$ in dimension $2$ is known to be false. In this case, Tao showed an alternative of the form $\|e^{it\Delta} f\|_{L_t^2L_\rho^\infty L_\omega^2} \lesssim \|f\|_{L^2}$ by introducing the mixed norms on the polar coordinates $x=\rho\omega$ with $\rho>0$, $\omega\in\mathbb{S}^1$. Motivated by this, we study the Strichartz estimates with angular integrability in higher dimensions. More generally, we consider a weighted mixed norm in the polar coordinates. As an application, the existence of solutions for the inhomogeneous nonlinear Schr\"odinger equation $i \partial_{t} u + \Delta u =\lambda |x|^{-\alpha} |u|^{\beta} u$ for $L^2$ data is shown up to the $L^2$-critical case which has been left unsolved until quite recently. Our result here will provide more information on the solution such as the angular integrability.