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Refined mass-critical
Strichartz estimates for Schrödinger operators
Casey Jao |
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Vol. 13 (2020), No. 7, 1955–1994
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Abstract |
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We develop refined Strichartz estimates at regularity for a class of time-dependent Schrödinger operators. Such refinements quantify near-optimizers of the Strichartz estimate and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the -critical setting due to an enormous group of symmetries, while on the other hand, the space-time Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article. Using phase-space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao’s bilinear restriction estimate for paraboloids to more general Schrödinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large-data solutions to the -critical NLS with a harmonic oscillator potential in dimensions . This article builds on recent work of Killip, Visan, and the author in one space dimension. |
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Keywords
inverse Strichartz estimates, bilinear restriction,
Schrödinger operators
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Mathematical Subject Classification 2010
Primary: 35Q41
Secondary: 42B37
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Milestones
Received: 30 October 2017
Revised: 29 July 2018
Accepted: 6 September 2019
Published: 10 November 2020
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Authors |
| Casey Jao | |
| Department of Mathematics University of Toronto Toronto, ON Canada |
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