A mixed norm variant of Wolff’s inequality for paraboloids

Garrigós, Gustavo; Seeger, Andreas

doi:10.1090/conm/505/09923

Cited by 34 publications

(13 citation statements)

References 9 publications

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“…Theorem 1.1 has been proved in [20] for p > 2 + 8 n−1 − 4 n(n−1) . A standard construction is presented in [20] to show that, up to the δ −ǫ term, the exponent of δ is optimal.…”

Section: The L 2 Decoupling Theoremmentioning

confidence: 97%

“…Theorem 1.1 has been proved in [20] for p > 2 + 8 n−1 − 4 n(n−1) . 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.…”

Section: The L 2 Decoupling Theoremmentioning

confidence: 97%

Section: The L 2 Decoupling Theoremmentioning

confidence: 99%

“…An argument similar to the one in [20] was used in [18] to prove Theorem 1.1 for p > 2(n+2) n−1 , by interpolating Wolff's machinery with the estimate p = 2n n−1 from [11]. This range is better than the one in [20] due to the use of multilinear theory as opposed to bilinear theory. 1 We mention briefly that there is a stronger form of decoupling, sometimes referred to as square function estimate, which predicts that…”

Section: The L 2 Decoupling Theoremmentioning

confidence: 99%

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The proof of the l^2 Decoupling Conjecture

2015

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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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“…Theorem 1.1 has been proved in [20] for p > 2 + 8 n−1 − 4 n(n−1) . A standard construction is presented in [20] to show that, up to the δ −ǫ term, the exponent of δ is optimal.…”

Section: The L 2 Decoupling Theoremmentioning

confidence: 97%

Section: The L 2 Decoupling Theoremmentioning

confidence: 97%

Section: The L 2 Decoupling Theoremmentioning

confidence: 99%

Section: The L 2 Decoupling Theoremmentioning

confidence: 99%

See 2 more Smart Citations

The proof of the l^2 Decoupling Conjecture

2015

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“…Note that by Minkowski, (2.7) implies estimates with the same constant over any spatial cubes with side length larger than δ −1 , and thus we shall always use the weight associated to the largest spatial cube throughout our iteration. We apply the iteration argument sketched in [1], [3] and [5]. Namely, we have…”

Section: Introductionmentioning

confidence: 99%

ℓ² decoupling in ℝ² for curves with vanishing curvature

Biswas

Gilula

et al. 2020

Proc. Amer. Math. Soc.

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Sharp Local Smoothing Estimates for Fourier Integral Operators

Beltran

Hickman²,

Sogge

2021

Springer INdAM Series

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The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.

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A mixed norm variant of Wolff’s inequality for paraboloids

Abstract: Abstract. We adapt the proof for ℓ p (L p ) Wolff inequalities in the case of plate decompositions of paraboloids, to obtain stronger ℓ 2 (L p ) versions. These are motivated by the study of Bergman projections for tube domains.

Cited by 34 publications

References 9 publications

The proof of the l^2 Decoupling Conjecture

The proof of the l^2 Decoupling Conjecture

ℓ² decoupling in ℝ² for curves with vanishing curvature

Sharp Local Smoothing Estimates for Fourier Integral Operators

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