Abstract
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.
Citation
Casey Jao. "Refined mass-critical Strichartz estimates for Schrödinger operators." Anal. PDE 13 (7) 1955 - 1994, 2020. https://doi.org/10.2140/apde.2020.13.1955
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