Vol. 11, No. 5, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system

Timothy Candy and Sebastian Herr

Vol. 11 (2018), No. 5, 1171–1240
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.

Keywords
bilinear Fourier restriction, adapted function spaces, quadratic variation, atomic space, Dirac–Klein–Gordon system, resonance, global well-posedness, scattering
Mathematical Subject Classification 2010
Primary: 42B37, 35Q41
Secondary: 42B20, 42B10, 81Q05
Milestones
Received: 9 May 2017
Revised: 19 October 2017
Accepted: 29 November 2017
Published: 11 April 2018
Authors
Timothy Candy
Universität Bielefeld
Fakultät für Mathematik
Bielefeld
Germany
Sebastian Herr
Universität Bielefeld
Fakultät für Mathematik
Bielefeld
Germany