Abstract
AbstractWe consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ R 3 , which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method.
Highlights
Let S ⊂ Rn be a smooth hypersurface
The sharp range in dimension n = 2 for curves with non-vanishing curvature was determined through work by Fefferman et al [13,38]
The sharp L p −L2 result for hypersurfaces with non-vanishing Gaussian curvature was obtained by Stein and Tomas [27,34]
Summary
Let S ⊂ Rn be a smooth hypersurface. The Fourier restriction problem, introduced by E. For the case of hypersurfaces of non-vanishing Gaussian curvature but principal curvatures of different signs, besides Tomas-Stein type Fourier restriction estimates, until recently the only case which had been studied successfully was the case of the hyperbolic paraboloid (or “saddle”) in R3: in 2015, independently Lee [22] and Vargas [35] established results analogous to Tao’s theorem [30] on elliptic surfaces (such as the 2 -sphere), with the exception of the end-point, by means of the bilinear method. And the same reasoning as in [10] allows to prove Fourier restriction to surfaces given as the graph of φ(x, y) := x y + h(y), where the function h is smooth and of finite type at the origin, in the same range of p’s and q’s as in Theorem 1.1. An -removal theorem (Theorem 5.3 in [21]) gives Theorem 1.1
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