Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 5 (2018), 1171-1240.

Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system

Timothy Candy and Sebastian Herr

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Abstract

Firstly, bilinear Fourier restriction estimates — which are well known for free waves — are extended to adapted spaces of functions of bounded quadratic variation, under quantitative assumptions on the phase functions. This has applications to nonlinear dispersive equations, in particular in the presence of resonances. Secondly, critical global well-posedness and scattering results for massive Dirac–Klein–Gordon systems in dimension three are obtained, in resonant as well as in nonresonant regimes. The results apply to small initial data in scale-invariant Sobolev spaces exhibiting a small amount of angular regularity.

Article information

Source
Anal. PDE, Volume 11, Number 5 (2018), 1171-1240.

Dates
Received: 9 May 2017
Revised: 19 October 2017
Accepted: 29 November 2017
First available in Project Euclid: 17 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.apde/1523930418

Digital Object Identifier
doi:10.2140/apde.2018.11.1171

Mathematical Reviews number (MathSciNet)
MR3785603

Zentralblatt MATH identifier
06866546

Subjects
Primary: 42B37: Harmonic analysis and PDE [See also 35-XX] 35Q41: Time-dependent Schrödinger equations, Dirac equations
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein- Gordon and other equations of quantum mechanics

Keywords
bilinear Fourier restriction adapted function spaces quadratic variation atomic space Dirac–Klein–Gordon system resonance global well-posedness scattering

Citation

Candy, Timothy; Herr, Sebastian. Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system. Anal. PDE 11 (2018), no. 5, 1171--1240. doi:10.2140/apde.2018.11.1171. https://projecteuclid.org/euclid.apde/1523930418


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