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2017 A Fourier restriction theorem for a two-dimensional surface of finite type
Stefan Buschenhenke, Detlef Müller, Ana Vargas
Anal. PDE 10(4): 817-891 (2017). DOI: 10.2140/apde.2017.10.817

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.

Citation

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Stefan Buschenhenke. Detlef Müller. Ana Vargas. "A Fourier restriction theorem for a two-dimensional surface of finite type." Anal. PDE 10 (4) 817 - 891, 2017. https://doi.org/10.2140/apde.2017.10.817

Information

Received: 3 February 2016; Revised: 2 September 2016; Accepted: 22 January 2017; Published: 2017
First available in Project Euclid: 19 October 2017

zbMATH: 1364.42010
MathSciNet: MR3649369
Digital Object Identifier: 10.2140/apde.2017.10.817

Subjects:
Primary: 42B10

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.10 • No. 4 • 2017
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