Multi-scale bilinear restriction estimates for general phases

Abstract

We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to high-low frequency interactions for solutions to partial differential equations, as well as to the linear restriction problem for surfaces with degenerate curvature. As a consequence, we obtain new bilinear restriction estimates for elliptic phases and wave/Klein-Gordon interactions in the full bilinear range, and give a refined Strichartz inequality for the Klein-Gordon equation. In addition, we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

“…After distributing our preprint though the arXiv, we learned from Candy that the bilinear estimate in Theorem could also be deduced from his more general bilinear estimates in [, Theorem 1.4], after applying the crucial scaling in that we use in Subsection . The convexity assumptions on the sets in his theorem is not really necessary, as he pointed out to us.…”

Extremizers for Fourier restriction on hyperboloids

“…For the conic case 1 < r ≤ N , we can follow a similar path, invoking either Wolff's bilinear estimates for the cone [20] or a variant on Tao's estimates for the paraboloid noted in [11]. We choose to take a shortcut, noting that Candy's recent work [2] on bilinear restriction estimates for general phases already implies the adequate rescaled substitute of (4.10) in the conic regime. More precisely, [2, Theorem 1.10] specializes to the inequality…”

“…Main theorem. The first result to address the sharp form of (1.5) is due to Quilodrán [12], in which he computes the exact values of H d,p in the endpoint cases (d, p) = (2, 4), (2,6) and (3,4), and establishes the non-existence of extremizers in these cases. 2 A crucial element of his proof is the fact that the Lebesgue exponents p under consideration are even integers, a fact that allows one to use the convolution structure of the problem via an application of Plancherel's theorem.…”

“…The main new ingredient of the proof, when compared to that of [3,Theorem 2], is the use of machinery from bilinear restriction theory to obtain a refined version of inequality (1.5). As in [2,3], we exploit the fact that the hyperboloid is well approximated by the paraboloid and the cone. The geometric construction underlying the bilinear restriction machinery accounts for this fact: in some sense, it interpolates between the two endpoint cases, which we will refer to as the elliptic and the conic regimes, respectively.…”

“…Several authors have investigated the interface between bilinear restriction theory and these extremal questions, both from the restriction side and the partial differential equations point of view. Here we mention the works [1,2,5,6,7,10,14], all of which deal with these connections. Many other authors have contributed to the development of the area, and we refer the reader to [3] for an exposition of related literature on sharp Fourier restriction theory.…”