#### Vol. 13, No. 7, 2020

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Refined mass-critical Strichartz estimates for Schrödinger operators

### Casey Jao

Vol. 13 (2020), No. 7, 1955–1994
##### Abstract

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 $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.

##### Keywords
inverse Strichartz estimates, bilinear restriction, Schrödinger operators
Primary: 35Q41
Secondary: 42B37