A note on pointwise convergence for the Schrödinger equation

Lucà, Renato; Rogers, Keith M.

doi:10.1017/s0305004117000743

Cited by 45 publications

(39 citation statements)

References 30 publications

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“…We then have the following theorem. the best positive and negative results to date are in [18] and [28,29], respectively.…”

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confidence: 99%

“…Using these facts and the Sobolev embedding H 1 2 + (Ω) ֒→ L ∞ (Ω), we see that the (1d analog of) property (8) is satisfied by initial data in H s (Ω) with s > 1/6. On the other hand, in [36] (see also [29]) it has been observed that for any s < 1/4 there are initial data such that lim sup t→0 |e it∆ f (x)| = ∞ for x in a set of strictly positive measure. This construction, done for Ω = R, is based on the Dahlberg-Kenig counterexample and can be repeated also in the periodic setting.…”

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confidence: 99%

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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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“…We then have the following theorem. the best positive and negative results to date are in [18] and [28,29], respectively.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Pointwise Convergence of the Schrödinger Flow

Compaan

Lucà

Staffilani

2020

International Mathematics Research Notices

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“…In higher dimensions, Sjölin [22], Vega [27], Bourgain [3] and Du-Guth-Li-Zhang [11] made progresses, and recently, Du-Zhang [12] have proven the convergence for the sharp range s > n/2(n + 1) by showing the local maximal K loc (L 2 x L ∞ t ; α) for α < −n/2(n + 1). (For sharpness of the regularity range s, see [2,9,18,19].) (When n = 1 the maximal estimate K loc (L 2 x L ∞ t ; α) for α ≤ −1/4 was already obtained by Kenig-Ruiz [16].…”

Section: Introductionmentioning

confidence: 97%

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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“…almost everywhere whenever f ∈ H s (R n ). Many authors put a lot of effort on the development of this subject and obtained sharp or partial pointwise convergence results for various P (ξ), see the related works [1][2][3], [5], [8], [14][15][16][17][18], [19,20], [22][23][24][25][26][27][28][29][30], [32], [34][35][36][37][38][39][40][41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

On convergence properties for generalized Schrödinger operators along tangential curves

Li¹,

Wang²

2021

Preprint

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In this paper, we consider convergence properties for generalized Schrödinger operators along tangential curves with less smoothness than curves with Lipschitz condition.Firstly, we obtain sharp convergence rate for generalized Schrödinger operators with polynomial growth along tangential curves. Secondly, it was open until now on pointwise convergence of solutions to the Schrödinger equation along non-C 1 curves in higher dimensional case (n ≥ 2), we obtain the corresponding results along a class of tangential curves in R 2 by the broadnarrow argument and polynomial partitioning. Then the corresponding convergence rate will follow. Thirdly, we get the convergence result in R along a family of tangential curves. As a consequence, we obtain the sharp upper bound for p in L p -Schrödinger maximal estimates along tangential curve, when smoothness of the function and the curve are fixed.

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A note on pointwise convergence for the Schrödinger equation

Cited by 45 publications

References 30 publications

Pointwise Convergence of the Schrödinger Flow

Pointwise Convergence of the Schrödinger Flow

Global Kato Type Smoothing Estimates via Local Ones for Dispersive Equations

On convergence properties for generalized Schrödinger operators along tangential curves

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