## Analysis & PDE

• Anal. PDE
• Volume 10, Number 4 (2017), 817-891.

### A Fourier restriction theorem for a two-dimensional surface of finite type

#### Abstract

The problem of $Lq(ℝ3) → L2(S)$ Fourier restriction estimates for smooth hypersurfaces $S$ of finite type in $ℝ3$ is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up the study of more general $Lq(ℝ3) → Lr(S)$ Fourier restriction estimates, by studying a prototypical model class of two-dimensional surfaces for which the Gaussian curvature degenerates in one-dimensional subsets. We obtain sharp restriction theorems in the range given by Tao in 2003 in his work on paraboloids. For high-order degeneracies this covers the full range, closing the restriction problem in Lebesgue spaces for those surfaces. A surprising new feature appears, in contrast with the nonvanishing curvature case: there is an extra necessary condition. Our approach is based on an adaptation of the bilinear method. A careful study of the dependence of the bilinear estimates on the curvature and size of the support is required.

#### Article information

Source
Anal. PDE, Volume 10, Number 4 (2017), 817-891.

Dates
Revised: 2 September 2016
Accepted: 22 January 2017
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.apde/1508432240

Digital Object Identifier
doi:10.2140/apde.2017.10.817

Mathematical Reviews number (MathSciNet)
MR3649369

Zentralblatt MATH identifier
1364.42010

#### Citation

Buschenhenke, Stefan; Müller, Detlef; Vargas, Ana. A Fourier restriction theorem for a two-dimensional surface of finite type. Anal. PDE 10 (2017), no. 4, 817--891. doi:10.2140/apde.2017.10.817. https://projecteuclid.org/euclid.apde/1508432240

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