On the Strichartz estimates for orthonormal systems of initial data with regularity
On the Strichartz estimates for orthonormal systems of initial data with regularity
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DOI：
10.1016/j.aim.2019.106736
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发表时间：
2017-08
影响因子：
1.7
通讯作者：
N. Bez;Younghun Hong;Sanghyuk Lee;Shohei Nakamura;Y. Sawano
中科院分区：
文献类型：
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作者：
N. Bez;Younghun Hong;Sanghyuk Lee;Shohei Nakamura;Y. Sawano
自由薛定谔传播子的经典薛定谔估计最近已基本上推广到形式为λ j的估计|e i t Δ f j| 2 λ α，对于L2中初始数据的标准正交系（f j）j，首先在Frank-Lewin-Lieb-Seiringer的工作中，后来由Frank-Sabin工作。主要目标是确定最大可能的α作为p和q的函数，与经典情况相反，对于这样的估计，临界情况是（p，q）=（d+ 1 d，d+ 1 d− 1）。考虑了齐次Sobolev空间H stecs（s∈（0，d2））中的正交系统（fj）j的情形，并建立了α作为p，q和s的函数的尖值，除了在某些情况下可能有端点.此外，在临界情况下（p，q）=（d+ 1 d− 2 s，d（d+ 1）（d− 1）（d− 2 s））对于一般s，如果我们考虑频率局部估计，我们证明了当α= p时期望估计的准确性，以及α= p时的（非局部）估计值;再次与经典设置相反，这显示了在该上下文中从频率局部化估计升级的困难。
The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the form‖∑ j λ j| e i t Δ f j| 2‖ L p t L x q≲‖ λ‖ ℓ α for 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 spaces H˙ 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. Furthermore, at the critical case (p, q)=(d+ 1 d− 2 s, d (d+ 1)(d− 1)(d− 2 s)) 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.