Endpoint Strichartz estimates with angular integrability and some applications
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...
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| Main Authors | , , |
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| Format | Journal Article |
| Language | English |
| Published |
29.12.2019
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| Subjects | |
| Online Access | Get full text |
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| Summary: | 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. |
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| DOI: | 10.48550/arxiv.1912.12784 |