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

Buschenhenke, Stefan; Müller, Detlef; Vargas, Ana

doi:10.2140/apde.2017.10.817

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

Stefan Buschenhenke, Detlef Müller and Ana Vargas

Vol. 10 (2017), No. 4, 817–891

DOI: 10.2140/apde.2017.10.817

Abstract

The problem of $L^{q} (ℝ^{3}) \to L^{2} (S)$ Fourier restriction estimates for smooth hypersurfaces $S$ of finite type in $ℝ^{3}$ 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 $L^{q} (ℝ^{3}) \to L^{r} (S)$ 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.

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Keywords

Fourier restriction, finite type, multilinear, bilinear

Mathematical Subject Classification 2010

Primary: 42B10

Milestones

Received: 3 February 2016

Revised: 2 September 2016

Accepted: 22 January 2017

Published: 9 May 2017

Authors

Stefan Buschenhenke
Mathematisches Seminar Christian-Albrechts-Universität Kiel Ludewig-Meyn Str. 4 D-24098 Kiel Germany

Detlef Müller
Mathematisches Seminar Christian-Albrechts-Universität Kiel Ludewig-Meyn Str. 4 D-24098 Kiel Germany

Ana Vargas
Department of Mathematics Autonomous University of Madrid 28049 Madrid Spain

Vol. 10, No. 4, 2017