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

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

Stefan Buschenhenke, Detlef Müller and Ana Vargas

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors