The multilinear restriction estimate: a short proof and a refinement

Abstract

The first result in this paper provides a very general ǫ-removal argument for the multilinear restriction estimate. The second result provides a refinement of the multilinear restriction estimate in the case when some terms have appropriate localization properties; this generalizes a prior result of the author in [1].

Sharp Bounds for Multilinear Curved Kakeya, Restriction and Oscillatory Integral Estimates Away From the Endpoint

“…By restricting attention to maps φ j of the affine-linear form (1.3), we see that Theorem 1.3 implies a local version of Theorem 1.1 in which the L p norm is restricted to (say) B d (0, 1), and δ is also restricted to be at most 1; these restrictions can then be easily lifted by a scaling argument, so that the full strength of Theorem 1.1 becomes a corollary of Theorem 1.3. This implication also shows that the estimate (1.10) is sharp except for the factors A O (1)…”

“…Thus, B 0 j can be factored as B 0 j = π jB 0 j for some matrixB 0 j ∈ R d−1×d−1 , where π j ∈ R d×d−1 is the matrix from (4.2). Since B 0 j is of full rank with all non-trivial singular values A O(1) , we conclude from (1.8) thatB 0 j is invertible, with all singular values A O (1) . If we then replace each φ j with the map x → φ j (x)(B 0 j ) −1 (and adjust C 0 and A as necessary), we see that we may assume without loss of generality thatB 0 i is the identity, thus by (4.4) we now have…”

“…In [3] (using the optimised quantitative analysis from [2, § 4.3]), the multilinear Kakeya estimate was used to give 5 a local version of the above theorem in which the L p (R d ) norm was replaced by an L p (B(0, R)) norm for any R 2, and an additional loss of log O(1) R appeared on the right-hand side. An alternate proof of such a local estimate (with a loss of O ε (R ε ) for any ε > 0, rather than a polylogarithmic loss) was given in [1], using (a discrete form of) the Loomis-Whitney inequality in place of the multilinear Kakeya estimate, but now requiring a high degree of regularity M. Remark 1.10. In the case where the surfaces S j have curvature uniformly bounded away from zero (so that the Fourier transform of the surfaces exhibits some decay at infinity), Theorem 1.7 (with a worse dependence of constants on p) was essentially obtained in [8, Lemma A3], adapting the "epsilon removal" argument from [13].…”

“…In particular, by Cramer's rule, we can find an invertible matrix T ∈ R d×d with singular values A O (1) such that n j 0 T = e j for all j ∈ [d]. Applying this transformation (and adjusting C 0 , A, t as necessary), replacing φ j with the map (x, ω j ) → φ j (xT −1 , ω j ) and U with U T , we may assume without loss of generality that n 0 j = e j for all j.…”

“…LEMMA 5.1. There exist matrices B j ∈ R d−1×d−1 for j ∈ [d] with all singular values A O (1) with the following property: if for any ξ j ∈ B(ξ 0 j , 2ε) we define the matrix B j (ξ j ) ∈ R d×d−1 by…”