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
Abstract

The problem of ${L}^{q}\left({ℝ}^{3}\right)\to {L}^{2}\left(S\right)$ 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 ${L}^{q}\left({ℝ}^{3}\right)\to {L}^{r}\left(S\right)$ 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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