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2020 Refined mass-critical Strichartz estimates for Schrödinger operators
Casey Jao
Anal. PDE 13(7): 1955-1994 (2020). DOI: 10.2140/apde.2020.13.1955


We develop refined Strichartz estimates at L 2 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 L 2 -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 L 2 -critical NLS with a harmonic oscillator potential in dimensions 2 . This article builds on recent work of Killip, Visan, and the author in one space dimension.


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Casey Jao. "Refined mass-critical Strichartz estimates for Schrödinger operators." Anal. PDE 13 (7) 1955 - 1994, 2020.


Received: 30 October 2017; Revised: 29 July 2018; Accepted: 6 September 2019; Published: 2020
First available in Project Euclid: 19 November 2020

MathSciNet: MR4175818
Digital Object Identifier: 10.2140/apde.2020.13.1955

Primary: 35Q41
Secondary: 42B37

Keywords: bilinear restriction , inverse Strichartz estimates , Schrödinger operators

Rights: Copyright © 2020 Mathematical Sciences Publishers


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Vol.13 • No. 7 • 2020
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