Abstract. We prove the l 2 Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which implies the full range of expected L p x,t Strichartz estimates for both the rational and (up to N ǫ losses) the irrational torus. Another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere. Various applications to Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory. The l 2 Decoupling TheoremLet S be a compact C 2 hypersurface in R n with positive definite second fundamental form. Examples include the sphere S n−1 and the truncated (elliptic) paraboloidUnless specified otherwise, we will implicitly assume throughout the whole paper that n ≥ 2. We will write A ∼ B if A B and B A. The implicit constants hidden inside the symbols and ∼ will in general depend on fixed parameters such as p, n and sometimes on variable parameters such as ǫ, ν. We will not record the dependence on the fixed parameters.Let N δ be the δ neighborhood of P n−1 and let P δ be a finitely overlapping cover of N δ with curved regions θ of the formwhere C θ runs over all cubes cIt is also important to realize that the normals to these boxes are ∼ δ 1/2 separated. A similar decomposition exists for any S as above and we will use the same notation P δ for it. We will denote by f θ the Fourier restriction of f to θ.Our main result is the proof of the following l 2 Decoupling Theorem.Key words and phrases. discrete restriction estimates, Strichartz estimates, additive energy. The first author is partially supported by the NSF grant DMS-1301619. The second author is partially supported by the NSF Grant DMS-1161752. and ǫ > 0Theorem 1.1 has been proved in [20] for p > 2 +. A standard construction is presented in [20] to show that, up to the δ −ǫ term, the exponent of δ is optimal. We point out that Wolff [36] has initiated the study of l p decouplings, p > 2 in the case of the cone. His work provides part of the inspiration for our paper.A localization argument and interpolation between p = 2(n+1) n−1 and the trivial bound for p = 2 proves the subcritical estimate. Estimate (3) is false for p < 2. This can easily be seen by testing it with functions of the form f θ (x) = g θ (x + c θ ), where supp( g θ ) ⊂ θ and the numbers c θ are very far apart from each other.Inequality (3) has been recently proved by the first author for p = We mention briefly that there is a stronger form of decoupling, sometimes referred to as square function estimate, which predicts thatin the slightly smaller range 2 ≤ p ≤ 2n n−1. When n = 2 this easily follows via a geometric argument. Minkowski's inequality shows that (4) is indeed stronger than (3) in the range 2 ≤ p ≤ 2n n−1 . This is also confirmed by the lack of any results for (4) when n...
We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves.Given g : [0, 1] → C and an interval J ⊂ [0, 1], we define the extension operator E J = E (n) J in R n as followsx = (x 1 , . . . , x n ). Theorem 1.1 will follow from the following general result. The connection is explained in Section 4.Theorem 1.2. Let n ≥ 2 and 0 < δ ≤ 1. For each ball B ⊂ R n with radius δ −n and each g : [0, 1] → C we haveThe implicit constant is independent of δ, B, g.A sum such as J⊂[0,1] |J|=δ will always be understood when the reciprocal of δ is an integer, and J ranges over the intervals [jδ, (j + 1)δ], 0 ≤ j ≤ δ −1 − 1.Acknowledgment. The authors thank T. Wooley for stimulating discussions over the last two years and they thank T. Tao for a careful reading of an earlier version of the manuscript and for pointing out a few typos. They also thank G. Liu for some mumerics
Abstract. We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from [3]. As a consequence of this we obtain sharp (up to ǫ losses) Strichartz estimates for the hyperbolic Schrödinger equation on the torus.Our second main result is an l 2 decoupling for non degenerate curves which has implications for Vinogradov's mean value theorem.
We prove Bourgain's Return Times Theorem for a range of exponents p and q that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain's averages.
Abstract. We establish multilinear L p bounds for a class of maximal multilinear averages of functions of one variable, reproving and generalizing the bilinear maximal function bounds of Lacey (2000). As an application we obtain almost everywhere convergence results for these averages, and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system.
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