Optimal Bilinear Restriction Estimates for General Hypersurfaces and the Role of the Shape Operator

Abstract

Abstract. It is known that under some transversality and curvature assumptions on the hypersurfaces involved, the bilinear restriction estimate holds true with better exponents than what would trivially follow from the corresponding linear estimates. This subject was extensively studied for conic and parabolic surfaces with sharp results proved by Wolff [28] and Tao [23, 24], and with later generalizations [16,17,8,20]. In this paper we provide a unified theory for general hypersurfaces and clarify the role of curvature in this problem, by making statements in terms of the shape operators of the hypersurfaces involved.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid

“…Recall also from (1.9) that Γ φ z1 (z 1 , z 2 ) = 2(y 2 − y 1 )τ z1 (z 1 , z 2 ) and Γ φ z2 (z 1 , z 2 ) = 2(y 2 − y 1 )τ z2 (z 1 , z 2 ). Thus, by Lemma 4.1 and (2.11), we have that for z 1…”

“…For more details on this condition, we refer to the corresponding literature dealing with bilinear estimates, for instance [1,21,22,35]. In particular, according to [21, Theorem 1.1], transversality is achieved if the modulus of the following quantity…”

Multi-scale bilinear restriction estimates for general phases

“…In particular, if Φ 1 and Φ 2 are elliptic phases with curvature Financial support by the DFG through the CRC "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" is acknowledged. 1 at different scales, then (1.1) was obtained by Stovall [20] with an almost sharp dependence on the scale. If n = 2, and Φ 1 = Φ 2 = ξ m1 1 + ξ m2 2 are surfaces of finite type with rectangular sets Λ j ⊂ {0 < ξ 1 , ξ 2 < 1}, then the essentially optimal dependence of the constant C on the rectangles Λ j and parameters m j 2 has been obtained in recent work of Buschenhenke-Müller-Vargas [3].…”

“…On the other hand, in the case of the Schrödinger equation, Φ 1 = Φ 2 = |ξ| 2 , if Λ 1 is a ball of radius 1, and Λ 2 is a ball of radius λ 1 such that the sets Λ 1 and Λ 2 are separated by a distance λ, then work of Tao [22] shows that for q = r > n+3 n+1 the bilinear estimate (1.1) holds with C ≈ λ 1 r −1+ǫ for every ǫ > 0. In the case of general phases which are both at unit scale, under suitable transversality and curvature assumptions, and Bejenaru [1] have shown that (1.1) again holds in the non-endpoint range q = r > n+3 n+1 . For general phases which are not at unit scale, only in certain special cases is the dependence of C in (1.1) on the phases Φ j and sets Λ j known.…”

“…The key assumption (A1) is essentially equivalent to the conditions appearing previously in the literature [14, 1,4], except that we require a normalised version to correctly determine the dependence of the constant on the phases Φ j . To gain a better geometric understanding of (A1), we observe that letting ξ ′ → ξ and taking η = h − ξ, since ∇Φ j (ξ) − ∇Φ k (h − ξ) is normal to Σ j (h), (A1) implies the local condition…”