Advances in Differential Equations

Bilinear Fourier restriction estimates related to the 2d wave equation

Sigmund Selberg

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We study bilinear $L^2$ Fourier restriction estimates which are related to the 2d wave equation in the sense that we restrict to subsets of thickened null cones. In an earlier paper we studied the corresponding 3d problem, obtaining several refinements of the Klainerman-Machedon-type estimates. The latter are bilinear generalizations of the $L^4$ estimate of Strichartz for the 3d wave equation. In 2d there is no $L^4$ estimate for solutions of the wave equation, but, as we show here, one can nevertheless obtain $L^2$ bilinear estimates for thickened null cones, which can be viewed as analogs of the 3d Klainerman-Machedon estimates. We then prove a number of refinements of these estimates. The application we have in mind is the Maxwell-Dirac system.

Article information

Adv. Differential Equations, Volume 16, Number 7/8 (2011), 667-690.

First available in Project Euclid: 17 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L05: Wave equation 42B37: Harmonic analysis and PDE [See also 35-XX]


Selberg, Sigmund. Bilinear Fourier restriction estimates related to the 2d wave equation. Adv. Differential Equations 16 (2011), no. 7/8, 667--690.

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