Vol. 10, No. 4, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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
Abstract

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.

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