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- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-022-00797-4
- Publisher site
- See Article on Publisher Site
Abstract
The endpoint Strichartz estimate ‖eitΔf‖Lt2Lx∞≲‖f‖L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert e^{it\Delta } f\Vert _{L_t^2 L_x^\infty } \lesssim \Vert f\Vert _{L^2}$$\end{document} is known to be false in two space dimensions. Taking averages spherically on the polar coordinates x=ρω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x=\rho \omega $$\end{document}, ρ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\rho >0$$\end{document}, ω∈S1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega \in {{\mathbb {S}}}^1$$\end{document}, Tao showed a substitute of the form ‖eitΔf‖Lt2Lρ∞Lω2≲‖f‖L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert e^{it\Delta } f\Vert _{L_t^2L_\rho ^\infty L_\omega ^2} \lesssim \Vert f\Vert _{L^2}$$\end{document}. 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 L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document} data.
Journal
Journal of Evolution Equations – Springer Journals
Published: Jun 1, 2022
Keywords: Weighted estimates; Well-posedness; Nonlinear Schrödinger equations; Primary: 35B45; 35A01; Secondary: 35Q55