Strichartz Estimates for the Schrödinger Equation

El objetivo de este trabajo es reportar los progresos recientes sobre estimativas de Strichartz para la ecuación de Schrödinger y presentar el estado de arte. Estas estimativas han sido obtenidas en espacios de Lebesgue, espacios de Sobolev, y recientemente, en espacios de Wiener amalgamados y de modulación. Presentamos y comparamos los diferentes aspectos técnicos envueltos. Ilustramos los resultados con aplicaciones a buena colocación.
这项工作的目标是报告最近在 Schrödinger 方程估计方面取得的进展，并介绍现有技术水平。这些估计是在勒贝格空间、Sobolev 空间以及最近的 Wiener 混合空间和调制空间中获得的。我们展示并比较了涉及的不同技术方面。我们通过应用于良好放置来说明结果。

Key words and phrases: Dispersive estimates, Strichartz estimates, Wiener amalgam spaces, Modulation spaces, Schrödinger equation.

Math. Subj. Class.: 42B35,35B65, 35J10, 35B40.

In this note, we focus on the Cauchy problem for Schrödinger equations. To begin with, the Cauchy problem for the free Schrödinger equation reads as follows
在这个笔记中，我们关注的是薛定谔方程的柯西问题。首先，自由薛定谔方程的柯西问题如下所示。

with and . In terms of the Fourier transform, we can write the solution as follows
用 和 。就傅立叶变换而言，我们可以将解写成如下形式

where the Fourier multiplier is known as Schrödinger propagator. The corresponding inhomogeneous equation is
傅立叶乘子 被称为薛定谔传播子。相应的非齐次方程是

with and . By Duhamel's principle and (2), the integral version of (3) has the form
通过杜哈默尔原理和(2)，(3)的积分版本具有以下形式：

The study of space-time integrability properties of the solution to (2) and (4) has been pursued by many authors in the last thirty years. The matter of fact is given by the Strichartz estimates, that have become a fundamental and amazing tool for the study of PDE's. They have been studied in the framework of different function/distribution spaces, like Lebesgue, Sobolev, Wiener amalgam and modulation spaces and have found applications to well-posedness and scattering theory for nonlinear Schrödinger equations , .
过去三十年来，许多作者一直在研究解(2)和(4)的时空可积性质。事实是由 Strichartz 估计给出的，这已成为研究偏微分方程的基本和令人惊奇的工具。它们已在不同的函数/分布空间框架中进行了研究，如勒贝格、索博列夫、维纳混合和调制空间，并已应用于非线性薛定谔方程的良定性和散射理论。

In this paper we exhibit these problems. First, in Section 3, we introduce the dispersive estimates and show how can be carried out for these different spaces. The classical dispersive estimates read as follows
在本文中，我们展示了这些问题。首先，在第 3 节中，我们介绍了色散估计，并展示了如何可以针对这些不同空间进行计算。经典的 色散估计如下所示。

Section 4 is devoted to the study of Strichartz estimates. The nature of these estimates is highlighted and the results among different kinds of spaces are compared with each others. Historically, the spaces [19, 24, 26, 39, 49] were the first to be looked at. The celebrated homogeneous Strichartz estimates for the solution read
第 4 节致力于研究 Strichartz 估计。这些估计的性质得到了突出，并且不同空间中的结果进行了比较。从历史上看，首先研究的是 空间[19, 24, 26, 39, 49]。对于解的著名齐次 Strichartz 估计如下

for , with , i.e., for Schrödinger admissible (see Definition 4.1). Here, as usual, we set
对于 ，使用 ，即对于 Schrödinger 可接受（见定义 4.1）。在这里，像往常一样，我们设置

In the sequel, the estimates for Sobolev spaces were essentially derived from the Lebesgue ones. Recently, several authors ( ) have turned their attention to fixed time and space-time estimates for the Schrödinger propagator between spaces widely used in time-frequency analysis, known as Wiener amalgam spaces and modulation spaces. The first appearance of amalgam spaces can be traced to Wiener in his development of the theory of generalized harmonic analysis [46, 47, 48] (see [22] for more details). In this setting, Cordero and Nicola have discovered that the pattern to obtain dispersive and Strichartz estimates is similar to that of Lebesgue spaces. The main idea is to show that the fundamental solution (see (20) below) lies in the Wiener amalgam space (see Section 2 for the definition) which generalizes the classical space and, consequently, provides a different information between the local and global behavior of the solutions. Beside the similar arguments, we point out also some differences, mainly in proving the sharpness of dispersive estimates and Strichartz estimates. Indeed, dilation arguments in Wiener amalgam and modulation spaces don't work as in the classical spaces.
在续篇中，Sobolev 空间的估计基本上是从 Lebesgue 空间推导出来的。最近，一些作者（ ）已经将注意力转向了在时频分析中广泛使用的空间之间的固定时间和时空估计，即维纳混合空间和调制空间的薛定谔传播子。混合空间的首次出现可以追溯到维纳在他发展广义谐波分析理论时[46, 47, 48]（更多细节请参见[22]）。在这种情况下，Cordero 和 Nicola 发现获得色散和 Strichartz 估计的模式类似于 Lebesgue 空间。主要思想是表明基本解 （见下面的（20））位于维纳混合空间 （有关定义，请参见第 2 节），该空间推广了经典 空间，并因此在解的局部和全局行为之间提供了不同的信息。除了类似的论点，我们还指出了一些差异，主要是在证明色散估计和 Strichartz 估计的尖锐性方面。 事实上，在 Wiener 合并和调制空间中的扩张论证与经典 空间中的情况不同。

Modulation spaces were introduced by Feitchinger in 1980 and then were also redefined by Wang [44] using isometric decompositions. The two different definitions allow to look at the problem in two different manners. As a result, in [45], a beautiful use of interpolation theory on modulation spaces allows to combine the estimates obtained by means of the classical definition in [1, 2] and the isometric definition in [44, 45], to obtain more general fixed time estimates in this framework. In order to control the growth of singularity at , we usually have the restriction ; cf. [5, 26]. By using the isometric decomposition in the frequency space, as in [44, 45], one can remove the singularity at and preserve the decay at in certain modulation spaces.
调制空间是由 Feitchinger 于 1980 年引入的，然后也被 Wang [44]使用等距分解重新定义。这两种不同的定义允许以两种不同的方式看待这个问题。因此，在[45]中，对调制空间的插值理论的精彩运用使得可以结合通过经典定义[1, 2]和等距定义[44, 45]获得的估计，以在这个框架中获得更一般的固定时间估计。为了控制 处奇异性的增长，通常我们有限制 ；参见[5, 26]。通过在频率空间中使用等距分解，如在[44, 45]中，可以消除 处的奇异性并在某些调制空间中保持 处的衰减。

The Strichartz estimates can be applied, e.g., to the well-posedness of non-linear Schrödinger equations or of linear Schrödinger equations with time-dependent potentials. We shall show examples in the last Section 5.
Strichartz 估计可以应用于非线性薛定谔方程或具有时间相关势能的线性薛定谔方程的适定性，我们将在最后的第 5 节中展示示例。

Notation. We define , for , where is the inner product on . The space of smooth functions with compact support is denoted by , the Schwartz class by , the space of tempered distributions by . The Fourier transform is normalized to be . Translation and modulation operators (time and frequency shifts) are defined, respectively, by
符号。我们定义 ，对于 ，其中 是 上的内积。具有紧支撑的光滑函数空间表示为 ，Schwartz 类表示为 ，调和分布空间表示为 。傅里叶变换被归一化为 。翻译和调制算子（时间和频率移位）分别定义为

We have the formulas , and . The notation means for a suitable constant , whereas means , for some . The symbol denotes the continuous embedding of the linear space into .
我们有公式 ，和 。符号 表示适当常数 ，而 表示 ，对于一些 。符号 表示线性空间 连续嵌入到 中。

In this section we present the function/distribution spaces we work with, and the properties used in our study.
在本节中，我们介绍我们所使用的函数/分布空间以及在研究中使用的属性。

([34,35]). We recall that the Lorentz space on is defined as the space of tempered distributions such that
([34,35]). 我们回顾洛伦兹空间 在 上定义为调和分布空间 ，使得

when , and

when . Here, as usual, denotes the distribution function of and .
当 。在这里，如往常一样， 表示 和 的分布函数。

One has if , and . Moreover, for and , is a normed space and its norm is equivalent to the above quasi-norm .
如果 ，并且 ，则 。此外，对于 和 ， 是一个赋范空间，其范数 等价于上述准范数 。

The function lives in but observe that this function doesn't live in any . We now recall the following classical Hardy-Littlewood-Sobolev fractional integration theorem (see e.g. [33, Theorem 1, pag 119] and [34]), which will be used in the sequel
函数 存在于 ，但请注意，该函数不存在于任何 。我们现在回顾以下经典的 Hardy-Littlewood-Sobolev 分数积分定理（见例如[33，定理 1，第 119 页]和[34]），这将在接下来的部分中使用。

Proposition 2.1. Let and such that

Then the following estimate

holds for all .
对于所有 。

Potential and Sobolev spaces. For , we define the Fourier multipliers , and . Then, for , the potential space [4] is defined by
潜在和 Sobolev 空间。对于 ，我们定义傅立叶乘子 和 。然后，对于 ，势空间[4]由以下定义：

with norm . The homogeneous potential space [4] is defined by
使用规范 。同质势空间[4]由以下定义：

with norm . 使用规范 。

For the previous spaces are called Sobolev spaces and homogeneous Sobolev spaces , respectively.
对于 ，先前的空间分别称为 Sobolev 空间 和齐次 Sobolev 空间 。

( . Let be a test function that satisfies . We will refer to as a window function. For , recall the spaces, defined by
（ . 令 是一个满足 的测试函数。我们将 称为窗口函数。对于 ，回想一下 空间，定义为

they are Banach spaces equipped with the norm
它们是配备有范数的巴拿赫空间

In the same way, for , the Banach spaces are defined by
以同样的方式，对于 ，Banach 空间 是由定义的

equipped with the norm
配备标准

Let one of the following Banach spaces: , valued in a Banach space, or also spaces obtained from these by real or complex interpolation. Let be the space, , scalar-valued. For any given function which is locally in (i.e. ), we set .
让 是以下巴拿赫空间之一： ，取值于巴拿赫空间，或者通过实数或复数插值得到的空间。让 是 空间， ，标量值。对于任何给定的局部在 （即 ）的函数 ，我们设定 。

The Wiener amalgam space with local component and global component is defined as the space of all functions locally in such that . Endowed with the norm is a Banach space. Moreover, different choices of generate the same space and yield equivalent norms.
韦纳混合空间 ，具有局部分量 和全局分量 ，被定义为所有在 局部的函数 的空间，使得 。赋予范数 ，是一个 Banach 空间。此外，不同选择的 会生成相同的空间并产生等价的范数。

If (the Fourier algebra), the space of admissible windows for the Wiener amalgam spaces can be enlarged to the so-called Feichtinger algebra . Recall that the Schwartz class is dense in .
如果 （傅立叶代数），则威纳混合空间 的可接受窗口空间可以扩展到所谓的费希廷格代数 。回想一下，施瓦茨类 在 中是稠密的。

We use the following definition of mixed Wiener amalgam norms. Given a measurable function of the two variables we set
我们使用混合维纳拼接范数的以下定义。给定两个变量 的可测函数 ，我们设置

Observe that [6]

The following properties of Wiener amalgam spaces will be frequently used in the sequel.
威纳拼接空间的以下性质将在接下来的部分中经常使用。

Lemma 2.1. Let , be Banach spaces such that are well defined. Then,
引理 2.1. 设 ，是 Banach 空间，使得 被很好地定义。那么，

(i) Convolution. If and , we have
(i) 卷积。如果 和 ，我们有

In particular, for every , we have
特别是，对于每一个 ，我们有

(ii) Inclusions. If and ,
(ii) 包含物。如果 和 ，

Moreover, the inclusion of into need only hold "locally" and the inclusion of into "globally". In particular, for , we have
此外，将 包含到 中只需要在“局部”保持，将 包含到 中需要在“全局”保持。特别是对于 ，我们有

(iii) Complex interpolation. For , we have
(iii) 复杂插值。对于 ，我们有

if or has absolutely continuous norm.
如果 或 具有绝对连续范数。

(iv) Duality. If are the topological dual spaces of the Banach spaces respectively, and the space of test functions is dense in both and , then
(iv) 对偶性。如果 分别是 Banach 空间 的拓扑对偶空间，并且测试函数空间 在 和 中都是稠密的，则

The proof of all these results can be found in ([12, 14, 15, 22]).
所有这些结果的证明可以在([12, 14, 15, 22])中找到。

Finally, let us recall the following lemma [8, Lemma 6.1], that will be used in the last Section 5 .
最后，让我们回顾以下引理[8，引理 6.1]，它将在最后的第 5 节中使用。

Lemma 2.2. Let . If

then

with norm inequality .
带有规范不等式 。

([13, 21]). Let be a non-zero window function and consider the so-called short-time Fourier transform (STFT) of a function/tempered distribution with respect to the the window :
([13, 21]). 让 是一个非零窗口函数，并考虑所谓的短时傅立叶变换（STFT） 与窗口 相关的函数/调和分布 ：