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

Abstract

We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid z = x y which are of the form z = x y + h(y), where h(y) is a smooth function which is flat at the origin. The case of perturbations of finite type had already been handled before, but the flat case imposes several new obstacles. By means of a decomposition into intervals on which |h | is of a fixed size λ, we can apply methods devised in preceding papers, but since we lose control on higher order derivatives of h we are forced to rework the bilinear method for wave packets that are only slowly decaying. Another problem lies in the passage from bilinear estimates to linear estimates, for which we need to require some monotonicity of h .

Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

“…In [15], Stovall proved certain endpoint cases when d D 3 that do not follow from arguments in [14] and [19]. See also the paper [13] by Kim. For other recent progress on restriction estimates for perturbations of the hyperbolic paraboloid in dimension 3, see the recent papers of Buschenhenke-Müller-Vargas [5] and Guo-Oh [8].…”

A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

“…In our previous paper [9], we considered a one variable perturbation of the hyperbolic paraboloid, and applied the bilinear method, obtaining results analogous to [22,35]. Further results for more general classes of one-variate finite type, respectively flat, perturbations based on the bilinear method were obtained in [10,11]. Bilinear estimates are also key elements in the results obtained with the polynomial partitioning method for the non-negative curvature case.…”