Problems on Pointwise Convergence of Solutions to the Schrödinger Equation

Cho, Chu-Hee; Lee, Sanghyuk; Vargas, Ana

doi:10.1007/s00041-012-9229-2

Cited by 54 publications

(70 citation statements)

References 27 publications

(38 reference statements)

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“…Our novelty here is the use of a temporal localization (see Lemma 3.2) which is available only after frequency and spatial localizations. This kind of localization was first observed by the second author [21] and similar idea was used to study the space time estimates for the Schrödinger equation [22,23]. This make it possible to reduce the time local estimate to that of the same scale in spatial space so that it suffices to work on the estimate which is local in both time and spatial spaces, and this localization also plays a role in obtaining precise estimates for general ω.…”

Section: Introductionmentioning

confidence: 85%

Strichartz estimates in spherical coordinates

Cho¹,

Lee²

2013

Indiana Univ. Math. J.

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Section: Introductionmentioning

confidence: 85%

Strichartz estimates in spherical coordinates

Cho¹,

Lee²

2013

Indiana Univ. Math. J.

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“…Here we always use the term “decreasing” as synonymous with “nonincreasing.” Given such a sequence we seek to find the precise range of s such that

\lim_{n \to \infty} S^{a} f false(x, t_{n} false) = f false(x false) a . e .

holds for every

f \in H^{s}

. This is partially motivated by the work on approach regions for pointwise convergence for solutions of the Schrödinger equation, and also by the work on the pointwise convergence of spherical means of

L^{p}

functions (although the mathematical issues and expected outcomes for the latter problem are different).…”

Section: Introductionmentioning

confidence: 99%

On Pointwise Convergence of Schrödinger Means

Dimou

Seeger

2020

Mathematika

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For functions in the Sobolev space H s and decreasing sequences t n → 0 we examine convergence almost everywhere of the generalized Schrödinger means on the real line, given byhere a > 0, a = 1. For decreasing convex sequences we obtain a simple characterization of convergence a.e. for all functions in H s when 0 < s < min{a/4, 1/4} and a = 1. We prove sharp quantitative local and global estimates for the associated maximal functions. We also obtain sharp results for the case a = 1.

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“…Cho, Lee and Vargas [5] showed that the non-tangential convergence holds if s > β(Θ)+1 4 when a = 2 and n = 2. β(Θ) denotes the upper Minkowski dimension of the upper cover of the cone which will be given soon. Cho, Lee and Vargas [5] deal with estimating the boundary of the operator along the restricted direction and time localization argument. Shiraki [17] extended result of [5] to a > 1.…”

Section: Introductionmentioning

confidence: 99%

“…Cho, Lee and Vargas [5] deal with estimating the boundary of the operator along the restricted direction and time localization argument. Shiraki [17] extended result of [5] to a > 1. In this paper, we will deal with the case of 0 < a < 1, n = 1.…”

Section: Introductionmentioning

confidence: 99%