The optimal trilinear restriction estimate for a class of hypersurfaces with curvature

Abstract

Abstract. In [3] nearly optimal L 1 trilinear restriction estimates in R n+1 are established under transversality assumptions only. In this paper we show that the curvature improves the range of exponents, by establishing L p estimates, for any p > 2(n+4) 3(n+2) in the case of double-conic surfaces. The exponent 2(n+4) 3(n+2) is shown to be the universal threshold for the trilinear estimate. IntroductionFor n ≥ 1, let U ⊂ R n be an open, bounded and connected neighborhood of the origin and let Σ : U → R n+1 be a smooth parametrization of an n-dimensional submanifold of R n+1 (hypersurface), which we denote by S = Σ(U). To this we associate the operator E defined byGiven k smooth, compact hypersurfaces S i ⊂ R n+1 , i = 1, .., k, where 1 ≤ k ≤ n + 1, the k-linear restriction estimate is the following inequality. In a more compact format this estimate is abbreviated as follows:The fundamental question regarding the above estimate is the value of the optimal p for which it holds true. Given that the estimate R * (2 × ... × 2 → ∞) is trivial, the optimality is translated into the smallest p for which the estimate holds true. In [3] Bennett, Carbery and Tao clarified the role of transversality between the surfaces involved and established that under a transversality condition between S 1 , .., S k the optimal exponent is p = 2 k−1 ; the actual result in [3] is near-optimal, the optimal problem is currently open. The optimality can be easily revealed by taking S i to be transversal hyperplanes, in which case the estimate becomes the classical Loomis-Whitney inequality.It is also known, in some cases (precisely when k ≤ 2), or expected, in most of the others, that curvature assumptions improve the range of exponents in (1.1), except for the case k = n + 1. In [3], the authors state that "simple heuristics suggest that the optimal k-linear restriction theory requires at least n + 1 − k non-vanishing principal curvatures, but that further curvature assumptions have no further effect". However, this aspect of the theory is left as an open problem in [3]. We detail below what is known for each k.2010 Mathematics Subject Classification. 42B15 (Primary); 42B25 (Secondary).

The multilinear restriction estimate: a short proof and a refinement

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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].

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“…In the end of the argument, δ will be chosen in terms of absolute constants and ǫ, but not R, and the power of δ −1 will be absorbed into C(ǫ). This idea originates from the work of Guth in [9] and the author later used it in [1,2,3].…”

“…42B15 (Primary); 42B25 (Secondary). 1 A natural condition to impose on the hypersurfaces is the standard transversality condition: there exists ν > 0 such that…”

“…An alternative and shorter proof for the near-optimal multilinear Kakeya estimate was provided by Guth in [9]. An alternative and shorter proof for the multilinear restriction estimate that bypasses the use of the multlinear Kakeya estimate was provided by the author in [1].…”

“…In proving Theorem 1.3, we use an induction on scale argument, similar to the one from [1] used for proving a weaker version of this result. Dealing with the localization properties during the induction process requires a complete change in the approach (relative to [1]) that is more robust and general.…”