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

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.