A Fourier restriction theorem for a perturbed hyperbolic paraboloid

Abstract

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.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

“…Our main result, which generalizes Theorem 1.1 in [BMV17], is the following Theorem 1.1. Assume that r > 10/3 and 1/q ′ > 2/r, and let E denote the Fourier extension operator associated to the graph S in (1.1) of the above phase function φ(x, y) := xy + h(y), where the function h is smooth and of finite type at the origin.…”

“…where P 2 (κ, y) denotes the Taylor polynomial of H κ (y) of degree 2 centered at y = 0. As in our previous paper [BMV17], we may then write…”

“…As in [BMV17], it will be particularly important to look at the expression (3.7) when z = z 1 ∈ U 1 , and z = z 2 ∈ U 2 , so that the two "transversalities"…”

“…Moreover, C. H. Cho and J. Lee [ChL17], and J. Kim [K17], improved the range by adapting ideas by Guth [Gu16], [Gu17] which are based on the polynomial partitioning method. For further information on the history of the restriction problem, we refer the interested reader to our previous paper [BMV17].…”

“…For the proof of this result, we shall strongly build on the approach devised for the special case where h(y) = y 3 /3. In many arguments, we shall be able to basically follow [BMV17]. Therefore, we shall concentrate on explaining the new ideas and modifications that are needed to handle more general finite type perturbations.…”