A sharp Schrödinger maximal estimate in $\mathbb{R}^2$

Du, Xiaojiang; Guth, Larry; Li, Xiaochun

doi:10.4007/annals.2017.186.2.5

Cited by 139 publications

(168 citation statements)

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“…Theorem 4.2 is closely related to the refined Strichartz estimates from [5], [6] and [7], and we record a corollary in a similar form. To set up the statement, we need to set up a little notation.…”

Section: {2mentioning

confidence: 69%

“…The proof of Proposition 2.2 is based on Liu's framework and on decoupling. We will prove and then use a refinement of the decoupling theorem (Theorem 4.2) which is related to the refined Strichartz estimates that appear in [5], [6], and [7]. This refinement of decoupling was proven independently by Xiumin Du and Ruixiang Zhang (personal communication).…”

Section: Figurementioning

confidence: 97%

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On Falconer’s distance set problem in the plane

et al. 2019

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If E Ă R 2 is a compact set of Hausdorff dimension greater than 5{4, we prove that there is a point x P E so that the set of distances t|x´y|u yPE has positive Lebesgue measure. 1 arXiv:1808.09346v1 [math.CA] 28 Aug 2018 the complications of replacing the upper Minkowski dimension by the Hausdorff dimension in the claim above.Falconer's distance problem can be thought of as a continuous analogue of a combinatorial problem raised by Erdős in [11]: given a set P of N points in R d , what is the smallest possible cardinality of ∆pP q. A grid is the best known example in all dimensions. In two dimensions, Guth and Katz [15] proved a lower bound for |∆pP q| which nearly matches the grid example (up to a factor of log 1{2 N ). In higher dimensions, there is a larger gap, and the best known result is due to Solymosi and Vu [35]. The Erdős distinct distance problem also makes sense for general norms and much less is known about it. In the planar case, if K is smooth and has strictly positive curvature, the best known bound says that if |P | " N , then |∆ K pP q| Á N 3{4 , with stronger estimates established by Garibaldi in special cases ([13]). There is a conversion mechanism to go from Falconer-type results to Erdőstype results that was developed by the second author together with Hoffman ([17]), Laba ([19]), and Rudnev and Uriarte-Tuero ([18]). It gives estimates for point sets that are fairly spread out. Applying the conversion mechanism to Theorem 1.3 we get the following corollary:Corollary 1.5. Let K be a symmetric convex body in R 2 whose boundary BK is C 8 smooth and has strictly positive curvature. Let P be a set of N points in r0, 1s 2 so that the distance between any two points is Á N´1 {2 . Then there exists x P P such that (1.1) |∆ K,x pP q| Ç N 4 5 .

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Section: {2mentioning

confidence: 69%

Section: Figurementioning

confidence: 97%

On Falconer’s distance set problem in the plane

et al. 2019

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“…Now the estimate (19) is an immediate consequence of (22). Letting p = 2 d+2 d in (19) and interpolating it with · X 0,0…”

Section: Preliminariesmentioning

confidence: 99%

Pointwise Convergence of the Schrödinger Flow

Compaan

Lucà

Staffilani

2020

International Mathematics Research Notices

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In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schrödinger flow to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain smoothing effects for the non-homogeneous part of the solution can be used to upgrade to an uniform convergence to zero of that part and we discuss the sharpness of the results obtained. We also use randomization techniques to prove almost everywhere convergence with much less regularity of the initial data, hence showing how more generic results can be obtained.

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“…When n = 1, Carleson [5] proved the convergence for the sharp range s ≥ 1/4 through the maximal estimate K loc (L 1 x L ∞ t ; −1/4). When n = 2, Bourgain [4], Moyua-Vargas-Vega [20], Tao-Vargas [26] and S. Lee [17] made improvements, and recently Du-Guth-Li [10] have obtained the convergence for the sharp range s > 1/3 by showing the local maximal K loc (L 3…”

Section: Introductionmentioning

confidence: 99%

“…For any small ǫ > 0 we set r = 1/ǫ andr = 2/ǫ. By the local maximal estimate K loc (L 3 x L ∞ t ; −1/3 − ǫ/2) of Du-Guth-Li [10] and Höler's inequality, one has K loc (L 3 x L r t ; −1/3 − ǫ/2). From this and (i) of Theorem 1.1 we have the global-in-time estimate K glb (L 3…”

Section: Introductionmentioning

confidence: 99%

Global Kato Type Smoothing Estimates via Local Ones for Dispersive Equations

Lee

2020

J Fourier Anal Appl

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In this paper we show that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schrödinger equation. From this we immediately have two results as follows. One is that the known local Kato smoothing estimates are sharp. The sharp regularity ranges of the global Kato smoothing estimates are already known, but those of the local Kato smoothing estimates are not. Sun, Trélat, Zhang and Zhong [24] have shown it only in spacetime R × R. Our result resolves this issue in higher dimensions. The other one is the sharp global-intime maximal Schrödinger estimates. Recently, the pointwise convergence conjecture of the Schrödinger equation has been settled by Du-Guth-Li-Zhang [11] and Du-Zhang [12]. For this they proved related sharp local-in-time maximal Schrödinger estimates. By our result, these lead to the sharp global-in-time maximal Schrödinger estimates.D α u L r t (I;L q x (B n )) ≤ C α,m f L 2

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A sharp Schrödinger maximal estimate in $\mathbb{R}^2$

Abstract: We study the almost everywhere pointwise convergence of the solutions to Schrödinger equations in R 2 . It is shown that lim t→0

Cited by 139 publications

References 20 publications

On Falconer’s distance set problem in the plane

On Falconer’s distance set problem in the plane

Pointwise Convergence of the Schrödinger Flow

Global Kato Type Smoothing Estimates via Local Ones for Dispersive Equations

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