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, 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.

Partitions of Flat One-Variate Functions and a Fourier Restriction Theorem for Related Perturbations of the Hyperbolic Paraboloid

“…Our main result, which generalizes Theorem 1.1 in [7], allows to treat also the latter case under a monotonicity assumption on h : 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 (1.2) of the above phase function φ(x, y) := x y + h(y), where the function h is smooth and satisfies (1.1). Assume further that either the function h is of finite type at the origin, or flat and such that h is monotonic.…”

“…In this article we continue our study of Fourier restriction to hyperbolic hypersurfaces that we had begun in [7,8].…”

“…Moreover, Cho and Lee [9], and Kim [16], improved the range by adapting ideas by Guth [11,12] 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 [8].…”

“…Note that if h is of finite type at the origin, then this assumption implies that there is some m ≥ 1 such that h (m+2) (0) = 0, i.e., h(y) = y m+2 a(y), with a(0, 0) = 0. This case had already been treated in [7], so what remained open is the case where h is flat at 0, i.e., where h (k) (0) = 0 for every k ∈ N.…”

Waves, Spheres, and Tubes

“…Whereas perturbed balls (such as arise in the elliptic case) still look roughly like balls, perturbed rectangles may curve, leading to new difficulties. Some of these issues are described in [8] and the references therein.…”