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

Candy, Timothy; Herr, Sebastian

doi:10.2140/apde.2018.11.1171

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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

DOI: 10.2140/apde.2018.11.1171

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

Vol. 11, No. 5, 2018