Vol. 10, No. 4, 2017

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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

The problem of Lq(3) L2(S) Fourier restriction estimates for smooth hypersurfaces S of finite type in 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 Lq(3) 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.

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Fourier restriction, finite type, multilinear, bilinear
Mathematical Subject Classification 2010
Primary: 42B10
Received: 3 February 2016
Revised: 2 September 2016
Accepted: 22 January 2017
Published: 9 May 2017
Stefan Buschenhenke
Mathematisches Seminar
Christian-Albrechts-Universität Kiel
Ludewig-Meyn Str. 4
D-24098 Kiel
Detlef Müller
Mathematisches Seminar
Christian-Albrechts-Universität Kiel
Ludewig-Meyn Str. 4
D-24098 Kiel
Ana Vargas
Department of Mathematics
Autonomous University of Madrid
28049 Madrid