Maximal estimates for the Schrödinger equation with orthonormal initial data

Bez, Neal; Lee, Sanghyuk; Nakamura, Shohei

doi:10.1007/s00029-020-00582-6

Cited by 6 publications

(22 citation statements)

References 36 publications

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“…Roughly speaking, Strichartz estimates of the form (1.1) (and their close cousins) have played a key role in the recent breakthroughs in giving a rigorous meaning to solutions of (1.2) when the initial data γ 0 is not of trace class ( [11,12,27,42,43]). Further recent developments in this direction include the study of ground states [25,30] and a version of Carleson's pointwise convergence problem [5].…”

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confidence: 99%

On the Strichartz estimates for orthonormal systems of initial data with regularity

Bez

Hong

Lee

et al. 2019

Advances in Mathematics

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The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the formfor orthonormal systems (f j ) j of initial data in L 2 , firstly in work of Frank-Lewin-Lieb-Seiringer and later by Frank-Sabin. The primary objective is identifying the largest possible α as a function of p and q, and in contrast to the classical case, for such estimates the critical case turns out to be (p, q) = ( d+1 d , d+1 d−1 ). We consider the case of orthonormal systems (f j ) j in the homogeneous Sobolev spacesḢ s for s ∈ (0, d 2 ) and we establish the sharp value of α as a function of p, q and s, except possibly an endpoint in certain cases, at which we establish some weak-type estimates. Furthermore, at the critical case (p, q) = ( d+1 d−2s , 2s) ) for general s, we show the veracity of the desired estimates when α = p if we consider frequency localised estimates, and the failure of the (non-localised) estimates when α = p; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting.Date: August 21, 2017. 1 A B means A ≤ CB for an appropriate constant C 1 arXiv:1708.05588v1 [math.FA]

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confidence: 99%

On the Strichartz estimates for orthonormal systems of initial data with regularity

Bez

Hong

Lee

et al. 2019

Advances in Mathematics

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“…Let be a domain in the complex plane that contains the strip { ∈ C : − /2 ≤ ℜ ≤ 0}. Suppose that (Θ ) is an analytic family of functions for ∈ 6 that satisfies the following: for some constants 0 , 1 and > 0, 6 This means the map ↦ → Θ ( ) is an analytic function on for each ∈ R .…”

Section: Strichartz Estimates For Orthonormal Families: the Sharp Admmentioning

confidence: 99%

“…When = 1, we have ( , ) = (4, ∞). We refer the reader to[6] for some recent results in such a case.https://www.cambridge.org/core/terms. https://doi.org/10.1017/fms.2020.64Downloaded from https://www.cambridge.org/core.…”

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confidence: 99%

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

Bez

Lee

Nakamura

2021

Forum of Mathematics, Sigma

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We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form $|\xi |^{-\lambda }$ or $(1 + |\xi |^2)^{-\lambda /2}$ , where $\lambda \in \mathbb {R}$ . We achieve optimal decay rates by considering such weights with appropriate $\lambda \in \mathbb {C}$ . For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein. Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.

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“…At the endpoint q = ∞ in d = 1 it is known that the bound in (b) does not hold for β ≥ 2 (see [28] and also [32,Proposition 1]) and one may wonder whether it holds for β < 2. In [3] it is shown that the bound holds for β ≤ 4/3 and that the slightly weaker inequality with the…”

Section: Strichartz Inequality For Orthonormal Functions the Strichar...mentioning

confidence: 99%

“…x -norm holds for all β < 2. Strichartz inequalities for orthonormal system have been proved for more general operators than −∆ (see, e.g., [31,4]) and for more regular functions (see, e.g., [2,3,4]).…”

Section: Strichartz Inequality For Orthonormal Functions the Strichar...mentioning

confidence: 99%

Lieb-Thirring inequalities and other functional inequalities for orthonormal systems

Frank

2021

Preprint

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Maximal estimates for the Schrödinger equation with orthonormal initial data

Cited by 6 publications

References 36 publications

On the Strichartz estimates for orthonormal systems of initial data with regularity

On the Strichartz estimates for orthonormal systems of initial data with regularity

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

Lieb-Thirring inequalities and other functional inequalities for orthonormal systems

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