Decouplings for three-dimensional surfaces in $$\mathbb {R}^{6}$$R6

Oh, Changkeun

doi:10.1007/s00209-017-2022-9

Cited by 11 publications

(19 citation statements)

References 16 publications

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“…In our applications, we will take p = 14 3 . The forthcoming discussion is following very closely the arguments from [13]. This is a variant of the induction on scales that was used in [14] and then in [5] to prove the sharp decoupling for the cone.…”

Section: From Planes To Arbitrary Surfacesmentioning

confidence: 91%

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Sharp decouplings for three dimensional manifolds in $\mathbb R^5$

2019

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Section: From Planes To Arbitrary Surfacesmentioning

confidence: 91%

“…Instead, it turns out that we can control the lower dimensional contribution clustered near each 2-variety, once we can do it for planes. This follows via an approximation argument very similar to the one from [13], that we describe in Section 5.…”

Section: Introductionmentioning

confidence: 97%

Sharp decouplings for three dimensional manifolds in $\mathbb R^5$

2019

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“…This follows from Lemma 3.7. In the case

d = n

considered in [7, 19], it might have previously seemed important that only certain specific varieties can obstruct transversality. Thanks to Corollary 2.18, we can afford not to keep track of which varieties may or may not arise here.…”

Section: General Surfacesmentioning

confidence: 99%

“…Remark The notion of transversality in Definition 2.21 goes back to [12]. In the case

d = n

M = 2

, it specializes to the notions used in [7, 19], where transversality means that

V false(t_{1} false), V false(t_{2} false)

do not share common directions. In the case

n = 1

P_{1}

positive definite,

M = d + 1

, it specializes to the notion used in [6, 10], because the associated Brascamp–Lieb inequality is the Loomis–Whitney inequality, and the best constant in that inequality is the reciprocal of the volume of the parallelepiped spanned by the normal directions of functions

V_{j}

.…”

Section: General Surfacesmentioning

confidence: 99%

“…In the end of the overview, let us make a few comments on the differences among proofs in [6, 7, 9, 14, 19]. The proofs in [6, 9] use a

(d + 1)

‐linear argument, based on the

(d + 1)

‐linear Loomis–Whitney inequality, while the proofs in [7, 19] use a bilinear argument, based on certain change of variables, and the proof in [14] uses an

M

‐linear argument with

M

ranging in an interval of integers (the same as the current paper). The bilinear argument of [7, 19] is specific to the case

d = n

, that is the dimension of the surface of a half of the dimension of the total space.…”

Section: Introductionmentioning

confidence: 99%

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Decoupling for certain quadratic surfaces of low co‐dimensions

Guo

Zorin-Kranich

2020

J. London Math. Soc.

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Decoupling Inequality for Paraboloid Under Shell Type Restriction and Its Application to the Periodic Zakharov System

Kinoshita,

Nakamura,

Sanwal

2023

Commun. Math. Phys.

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In this paper, we establish local well-posedness for the Zakharov system on $${\mathbb {T}}^d$$ T d , $$d\ge 3$$ d ≥ 3 in a low regularity setting. Our result improves the work of Kishimoto (J Anal Math 119:213–253, 2013). Moreover, the result is sharp up to $$\varepsilon $$ ε -loss of regularity when $$d=3$$ d = 3 and $$d\ge 5$$ d ≥ 5 as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain–Demeter’s range of exponent for the linear decoupling inequality for paraboloid (Bourgain and Demeter in Ann Math 182:351–389) under the constraint that the input space-time function f satisfies $$\textrm{supp}\, {\hat{f}} \subset \{ (\xi ,\tau ) \in {\mathbb {R}}^{d+1}: 1- \frac{1}{N} \le |\xi | \le 1 + \frac{1}{N},\; |\tau - |\xi |^2| \le \frac{1}{N^{2}} \} $$ supp f ^ ⊂ { ( ξ , τ ) ∈ R d + 1 : 1 - 1 N ≤ | ξ | ≤ 1 + 1 N , | τ - | ξ | 2 | ≤ 1 N 2 } for large $$N\ge 1$$ N ≥ 1 .

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Decouplings for three-dimensional surfaces in $$\mathbb {R}^{6}$$R6

Cited by 11 publications

References 16 publications

Sharp decouplings for three dimensional manifolds in $\mathbb R^5$

Sharp decouplings for three dimensional manifolds in $\mathbb R^5$

Decoupling for certain quadratic surfaces of low co‐dimensions

Decoupling Inequality for Paraboloid Under Shell Type Restriction and Its Application to the Periodic Zakharov System

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