Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates
Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates
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DOI：
10.1017/fms.2020.64
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
N. Bez;Sanghyuk Lee;Shohei Nakamura
中科院分区：
文献类型：
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作者：
N. Bez;Sanghyuk Lee;Shohei Nakamura
摘要在波动方程、Klein-Gordon方程和分数阶薛定谔方程的情况下，我们建立了初值的正交族的新的Strichartz估计。我们的估计在波和Klein-Gordon方程的情况下推广了Frank-Sabin的估计，推广了Frank等人的工作。和Frank-Sabin的薛定谔方程。由于一定的技术障碍，除了经典的薛定谔方程外，关于初值的正交族的Strichartz估计以前还没有建立到所有可允许对的尖锐可和指数。我们得到了各种重要情形下的最优估计，改进了以前的结果。本文的主要创新之处在于我们得到了加权振荡积分的估计，并将其与Frank和Sabin的方法相结合。在某种意义上，我们的加权振荡积分估计是已知离散估计的相当精致的端点形式，其具有形式为$|\xi|^{-\lambda}$或$(1+|\xi|^2)^{-\lambda/2}$，其中$\lambda\in\mathbb{R}$。我们通过考虑这样的权重和适当的$\lambda\in\mathbb{C}$来获得最优衰减率。对于波动方程和Klein-Gordon方程，我们的加权振荡积分估计是新的。对于分数薛定谔方程，我们的结果与Kenig-Ponce-Vega在一定区域内的工作重叠。我们对加权振荡积分理论的贡献也受到了Carberg-Ziesler，Cowling等人和Sogge-Stein早期工作的影响。最后，我们给出了新的正交族Strichartz估计在Hartree型无穷系统理论中的一些应用，动力学输运方程的加权速度平均引理，以及Besov空间中数据的精化Strichartz估计。
Abstract 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.