The problem of L p (R 3 ) → L 2 (S) 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 p (R 3 ) → L q (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 non-vanishing 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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In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non‐vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2‐surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or ‘saddle’ z=xy. The bilinear method gave here sharp results for p>10/3, and this result was recently improved to p>3.25. This paper aims to be the first step in extending those results to more general hyperbolic surfaces. We consider a specific cubic perturbation of the saddle and obtain the sharp result, up to the endpoint, for p>10/3. In the application of the bilinear method, we show that the behavior at small scales in our surface is drastically different from the saddle. Indeed, as it turns out, in some regimes the perturbation term assumes a dominant role, which necessitates the introduction of a number of new techniques that should also be useful for the study of more general hyperbolic surfaces. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations.

In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2-surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or "saddle" z = xy. The bilinear method gave here sharp results for p > 10/3 ([L05], [V05], [Sto17]), and this result was recently improved to p > 3. 25 [ChL17] [K17]. This paper aims to be a first step in extending those results to more general hyperbolic surfaces. We consider a specific cubic perturbation of the saddle and obtain the sharp result, up to the end-point, for p > 10/3. In the application of the bilinear method, we show that the behavior at small scales in our surface is drastically different from the saddle. Indeed, as it turns out, in some regimes the perturbation term assumes a dominant role, which necessitates the introduction of a number of new techniques that should also be useful for the study of more general hyperbolic surfaces. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations. Contents 1. Introduction 2. Transversality conditions and admissible pairs of sets 2.1. Admissible pairs of sets U 1 , U 2 on which transversalities are of a fixed size: an informal discussion 2.2. Precise definition of admissible pairs within Q × Q 2.3. The exact transversality conditions 2.4. A prototypical admissible pair in the curved box case and the crucial scaling transformation 2.5. Reduction to the prototypical case 3. Statements of the bilinear estimates. The proofs. 3.1. The bilinear argument: proof of Theorem 3.3 4. The Whitney-type decomposition and its overlap 4.1. The Whitney-type decomposition of V 1 × V 2 4.2. Handling the overlap in the Whitney-type decomposition of V 1 × V 2 5. Passage to linear restriction estimates and proof of Theorem 1.1 2010 Mathematical Subject Classification. 42B25.

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