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We consider the initial value problem for the (generalized) periodic Kortewegde Vries equationThere is extensive literature on the IVP for the KDV-equation in the nonperiodic case, based on various methods such as the inverse scattering method or the fixpoint argument applied to the corresponding integral equation /0' (7) -We establish a local result by Picard's theorem, verifying a contraction property for the transformation associated to (7.4) in a suitable space, mainly using Fourier Analysis estimates.

In this paper we continue some investigations on the periodic NLSE ίu t + u xx + u\u\ p~2 = 0 (p ^ 6) started in [LRS]. We prove that the equation is globally wellposed for a set of data φ of full normalized Gibbs measure e -βff(Φ)Hdφ(x) 9 H(φ) = \ / \φ'\ 2 -± / \φ\ p (after suitable L 2 -truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [Bl] on periodic NLS and KdV type equations.

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...

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