Hans Triebel
The composition of the Fourier transform in $\mathbb{R}^n$ with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on mapping properties of the Fourier transform as developed in a preceding paper and related assertions for pseudodifferential operators.
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We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik in [24] to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators [11], which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrö…
Compact pseudodifferential and Fourier integral operators via localization
本文提出局部化算子的一般框架，证明其在调制空间之间的有界性与紧致性，并用于统一推广伪微分和傅里叶积分算子理论，得到三参数伪微分算子的新结果。
We consider the operator algebra $\mathscr A$ on $\mathscr S(\mathbb R^n)$ generated by the Shubin type pseudodifferential operators, the Heisenberg-Weyl operators and the lifts of the unitary operators on $\mathbb C^n$ to metaplectic operators. With the help of an auxiliary operator in the Shubin calculus, we find trace expansions for these operators in the spirit of Grubb and Seeley. Moreover, we can define a noncommutative residue generalizing that for the Shubin pseudodifferential operators and obtain a class of localized equivariant traces on the algebra.
Pseudodifferential operators on periodic graphs
研究了一类作用在无限度量图$Γ$上的加权Lebesgue空间上的有界线性算子的Fredholm性质，该度量图$Γ$关于群${\mathbb Z}^n$的作用是周期的.所考虑的算子通过它们的局部行为来区分：它们在图的每个开边上充当$OPS^0$类中的（傅立叶）伪微分算子，并且它们可以在$Γ$的每个顶点的邻域上表示为矩阵Mellin伪微分算子。应用这些结果研究了图上一类奇异积分算子和某些局部紧算子的Fredholm性质.
Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential operators.
In this paper we focus on the almost-diagonalization properties of $τ$-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any $τ$-pseudodifferential operator, for $τ\in [0,1]$, also with symbol in weighted Wiener amalgam s…
Characterization of Non-Smooth Pseudodifferential Operators
$\mathbb{R}^n$上的光滑伪微分算子可以由Beals和Ueberberg给出的$L^p-$索伯列夫空间之间的映射性质来刻画.在应用中，这样的特征在非光滑情况下也是有用的，例如显示偏微分方程的解的正则性。因此，我们将证明，每个线性算子$P$，它满足一些特定的连续性假设，是一个非光滑的伪微分算子的符号类$C^τ S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$。主要的新困难是具有非光滑符号的伪微分算子的有限映射性质。
Hörmander type pseudodifferential calculus on homogeneous groups
我们生产，一般的齐次群，一个类似的通常Hörmander pseudodifferential微积分的欧几里得空间，至少就产品和伴随而言。与早期的工作相比，我们不局限于经典符号的类似物，也不局限于海森堡群。关键技术是理解任意给定阶数j的“乘法器”，以及卷积算子及其傅里叶逆变换，我们在这里称之为j阶的卷积算子（这里“乘法器”是霍曼德型符号a（x，n）的类似物，它与x无关）。具体地说，我们刻画了任意阶j乘子的傅里叶逆变换空间，并利用这个刻画证明了j_1和j_2阶的卷积算子的组合是j_1+j_2阶的卷积算子.
Schrödinger type propagators, pseudodifferential operators and modulation spaces
我们证明连续性结果的傅里叶积分算子与调制空间中的符号，调制空间之间的作用。相函数属于一类非退化广义二次型，其中包括薛定谔传播算子和伪微分算子。作为一个副产品，我们得到的调制空间的所有指数$p,q,r_1,r_2,t_1,t_2 \in [1,\infty]$的特性，使得在$M^{p,q}(\mathbb R^{2d})$的符号给出了一个伪微分算子，是连续的从$M^{r_1,r_2}(\mathbb R^d)$到$M^{t_1,t_2}(\mathbb R^d)$。
We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation \emph{Short-time Fourier Transform} (STFT) is replaced by the $\mathcal{A}$-\emph{Wigner distribution} defined by $W_{\mathcal A} (f)=μ({\mathcal A})(f\otimes\bar{f})$, where ${\mathcal A}$ is a $4d\times 4d$ symplectic matrix and $μ({\mathcal A})$ is an associate metaplectic operator. Basic examples are given by the so-called $τ$-Wigner distributions. Such representations provide a new characterization for modulation spaces when $τ\in (0,1)$. Furthermore, they can be efficiently employed in the …
In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results …
We introduce multilinear localization operators in terms of the short-time Fourier transform, and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. For such interpretation we use the kenrel theorem for the Gelfand-Shilov space. Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when res…
Quantum Time-Frequency Analysis and Pseudodifferential Operators
本文提出量子时频分析，利用双相空间射影表示构造极化 Cohen 类与算子调制空间，并将其联系到伪微分算子符号类及经典结果。
Boundedness of metaplectic operators within $L^p$ spaces, applications to pseudodifferential calculus, and time-frequency representations
本文推广了 Housdorff-Young 不等式，刻画了所有 metaplectic 算子在 $L^p-L^q$ 空间的有界性条件，并基于此推导了伪微分算子的有界性结果及 metaplectic Wigner 分布在调制空间中的性质。
Magnetic Fourier Integral Operators
在以前的一些论文中，我们已经定义和研究了“磁”伪微分作为规范协变推广的Weyl演算时，磁场是存在的。本文将标准的Fourier积分算子理论推广到有磁场的情形，证明了“磁”Sobolev空间中的复合定理、连续性定理和Egorov型定理。主要的应用是代表的发展群所产生的一阶“磁”伪微分算子（特别是相对论薛定谔运营商与磁场）作为这样一个“磁”傅立叶积分算子。作为这种表示的结果，我们得到了一些估计的分布核的这个演化群和传播的奇点的结果。
For operators belonging either to a class of global bisingular pseudodifferential operators on $R^m \times R^n$ or to a class of bisingular pseudodifferential operators on a product $M \times N$ of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain associated homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to the even larger classes of Toeplitz type operators.
Pseudodifferential operators on manifolds with linearization
本文给出了光滑非紧流形上的伪微分的构造，它与全局定义的坐标无关的完备符号演算相关联，推广了$\R^n$上的标准伪微分.我们考虑流形$M$的情况下，线性化的Bokobza-Haggiag的意义，使相关的（抽象的）指数映射提供全球的$M$与$\R^n$在任何一点的同胚。Cartan-Hadamard流形是这类流形的特例。抽象的指数映射编码的概念无穷的流形上，允许，模一些假设的$S_σ$有界几何，以定义施瓦茨空间的快速衰减功能，全球定义的傅立叶变换和类的符号与均匀和衰减控制的$x$变量。给出了流形上的线性化及无穷远点控制的一些性质，构造了符号映射和$\la$ -量子化，余切丛上的显式Moyal星积，以及一类伪微分算子.我们表明，这些类是稳定的组合下，和$\la$ -量化映射给出了符号和伪微分算子之间的代数同构（这取决于线性化）。我们研究，在我们的设置，$L^2$ -连续性，并给出了一些例子。我们特别表明，双曲2-空间$\HH$有$S_1$有界的几何，允许建设一个全球性的符号演算的伪微分算子$§(\HH)$。
This is the direct continuation of the paper "Mapping properties of Fourier transforms" (arXiv:2112.04896) using the same notation as there without further explanations. It deals with continuous and compact mappings of the Fourier transform $F$ between some weighted function spaces on $\mathbb{R}^n$.
Mapping properties of the Fourier transform in spaces with dominating mixed smoothness
该论文聚焦控制混合光滑函数空间，探究傅里叶变换的连续及紧映射特性，为相关函数空间理论提供映射性质层面的研究支撑。
Boundedness of multilinear pseudo-differential operators on modulation spaces
基于多线性伪微分算子符号的短时傅里叶变换的有序可积性条件，得到了多线性伪微分算子在调制空间乘积上的有界性结果.通过对双线性和三线性希尔伯特变换的应用，证明了所介绍方法的灵活性和强度。
The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on $\mathbb{R}^n$. The degree of compactness will be measured in terms of related entropy numbers. We are more interested in the interplay of already available ingredients than in generality.
The Modulation Mapping for Magnetic Symbols and Operators
我们扩展的巴格曼变换的磁伪微分，使用规范协变家庭的相干态。我们还介绍了调制映射，第一步适应调制空间的磁性情况。
Fourier transform decay of distributions in Hardy-Morrey spaces
本文研究 Hardy-Morrey 空间及其可局部化版本上分布的 Fourier 变换衰减，结合逐点估计与消去矩条件建立估计，并用于刻画 Fourier 乘子及伪微分算子连续性的最优性。
We investigate so-called Laplace--Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev- and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff--Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.
We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces both in the quasi-analytic and in the non-quasi-analytic case. Moreover, we prove composition theorems and certain invariance properties of these classes.
Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
设$P$是$n$维光滑流形$Ω_1$上的一个经典的复阶伪微分算子.对于截断$P_Ω$到一个光滑子集$Ω$有一个著名的理论边值问题时，$P_Ω$的传输性能（保留$C^\infty (\barΩ)$），是整数阶; Boutet德蒙维尔演算。许多有趣的运营商，如例如复杂的权力拉普拉斯$(-Δ)^μ$与非整数μ，不包括在内。它们具有在Hörmander的书中定义的mu传输属性，将$x_n^μC^\infty (\barΩ)$映射到$C^\infty (\barΩ)$。在1965年的一份未发表的讲义中，Hörmander描述了一个$L_2$ -μ-传输算子的可解性理论，与Vishik和Eskin的结果不同。我们在这里发展的理论在$L_p$ Sobolev空间（$1<p<\infty$）在现代的设置。它不仅导致Fredholm可解性声明，但也正则性的结果，在满标度的Sobolev空间（$s\to \infty$）。解空间在边界处有一个奇异点，我们将详细描述。此外，我们在Hölder空间中得到的结果，从根本上改善最近的分数拉普拉斯正则性结果。
Exponentially sparse representations of Fourier integral operators
研究相位为二次增长的傅里叶积分算子的Gabor矩阵表示的稀疏性。已知这样的无限矩阵是稀疏的并且组织良好，实际上沿着对应的正则变换的图集中。在这里，我们表明，如果相位和符号有一个规律的Gevrey型的顺序$s>1$或解析（$s=1$），上述衰减实际上是次指数或指数，分别。我们还通过一个反例证明了超解析正则性（$s<1$）不给出超指数衰减。这是在形成鲜明对比的情况下，更有利的伪微分算子，甚至（广义）metaplectic运营商，这是治疗以及。
Mapping properties of Fourier transforms, revisited
本文利用小波展开、对偶性、空间插值和扩展的 Hausdorff-Young 不等式，研究了 Fourier 变换在 Besov 与 Sobolev 空间间的连续与紧映射性质，并通过熵数和逼近数度量紧致度。
Quantization procedures play an essential role in microlocal analysis, time-frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, $a(x,ξ)$ an operator $\mathrm{Op}(a)$, according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol $a(x,ξ)=f(x)$ depends only on $x$ or $a(x,ξ)=g(ξ)$ depends only on $ξ$: \[ \mathrm{Op}(f\otimes1)u=fu,\quad\mathrm{Op}(1\otimes g)u=\mathcal{F}% ^{-1}(g\mathcal{F}u) \] where $\mathc…
On the torus, it is possible to assign a global symbol to a pseudodifferential operator using Fourier series. In this paper we investigate the relations between the local and global symbols for the operators in the classical Hörmander calculus and describe the principal symbols, the non-commutative residue and the canonical trace of an operator in terms of its global symbol. We also generalise these results to any compact Lie group.
The classical Hausdorff-Young inequalities for the Fourier transform acting between appropriate $L_p$ spaces are cornerstones of Fourier analysis. Here we extend it to weighted spaces of Besov or Sobolev type where the weight has the form $w(x)=(1+|x|^2)^{α/2}$. This note is not a paper or draft but a sketchy complement to some earlier results where we dealt with mapping properties of the Fourier transform.
On paracomposition and change of variables in Paradifferential operators
本文推广仿复合算子，去掉微分同胚假设，给出Sobolev和Zygmund空间的全局估计，并证明变量变换下仿微分算子的性质及截断的优化。
Sparsity properties for phase-space representations of several types of operators have been extensively studied in recent papers, including pseudodifferential, Fourier integral and metaplectic operators, with applications to time-frequency analysis of Schrödinger-type evolution equations. It has been proved that such operators are approximately diagonalized by Gabor wave packets. While the latter are expected to undergo some spreading phenomenon, there is no record of this issue in the aforementioned results. In this paper we prove refined estimates for the Gabor matrix of metaplectic operator…
Differentiability of scalar functions applied to Hermitian operators - a Fourier transform approach
设g是实数上的一个（即充分可微的）函数。我们知道如何将g应用于C* 代数的埃尔米特元素A。然而，映射A到g（A）的可微性问题并不是微不足道的，因为一般来说“A和dA不交换”。然而，由于该映射线性地依赖于g，因此可以通过傅里叶变换将一般g的情况简化为指数函数的情况。对于后者，有一个显式的n阶导数公式（比标量情况更复杂-仍然是“A和dA不对易”）。通过这种方式，可以通过涉及g在[-r，r]上的（n+1）阶导数的Sobolev范数来限制r球上（A to g（A））的n阶导数的范数。
Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations
研究了作用在紧支撑分布空间$(\mathcal{F}L^p)_{comp}$，$1\leq p\leq\infty$上的H{<$}rmander型傅里叶积分算子，其傅里叶变换在$L^p$中.我们表明，这样的运营商是有界的这些空间上的导数的急剧损失有关的秩$r$的Hessian的相位$Φ(x,η)$相对于空间变量$x$。事实上，我们表明，运营商的顺序$m=-r|1/2-1/p|$有界$(\mathcal{F}L^p)_{comp}$，如果映射$x\longmapsto\nabla_xΦ(x,η)$是恒定的纤维，余维$r$，仿射纤维化。
The paper deals with the problem under which conditions for the parameters $s_1,s_2\in\mathbb{R}$, $1\leq p,q_1,q_2\leq\infty$ the Fourier transform $\mathcal{F}$ is a nuclear mapping from $A^{s_1}_{p,q_1}(\mathbb{R}^n)$ into $A^{s_2}_{p,q_2}(\mathbb{R}^n)$, where $A\in\{B,F\}$ stands for a space of Besov or Triebel-Lizorkin type, and $n\in\mathbb{N}$. It extends the recent paper arXiv:2112.04896 where the compactness of $\mathcal{F}$ acting in the same type of spaces was studied.
Global L2-boundedness theorems for a class of Fourier integral operators
在Hörmander \cite{H}和Eskin \cite{E}中建立了傅里叶积分算子的局部$L^2$ -映射性质.在本文中，我们处理的整体$L^2$ -有界的一类算子，自然出现在许多问题。因此，我们将改进几类伪微分算子和傅里叶积分算子的整体结果，并推广Asada和Fujiwara \cite{AF}或Kumano-go \cite{Ku}的结果.作为应用，我们给出了广义薛定谔方程的全局光滑估计，推广了Ben-Artzi和Devinatz的\cite{BD}，Walther \cite{Wa}和\cite{Wa2}的结果.
Calculus, continuity and global wave-front properties for Fourier integral operators on $\mathbf{R}^d$
我们说明了一个扩展的SG傅里叶积分算子族的复合性质。我们证明了这类算子关于$L^2$和加权调制空间的连续性结果，并讨论了$\mathscr{S}$，$\mathscr{S}^\prime$和加权索伯列夫空间上的连续性.我们研究了在这些傅里叶积分算子作用下的整体波前集的映射性质。我们将经典的结果推广到更一般的情况。例如，对于相位函数没有均匀性的要求。最后，我们将所得结果应用于调制空间中相应线性双曲算子Cauchy问题解的奇性传播的研究.
Fourier Integral Operators of Boutet de Monvel Type
给定两个带边界的紧致流形$X,Y,$和一个保边界辛同构$χ:T^*Y\setminus0\to T^*X\setminus0$，它在纤维中是单齐次的且满足透射条件，我们引入了与$χ$相联系的Boutet de Monvel型傅里叶积分算子.我们研究了它们在索伯列夫空间之间的映射性质，发展了一个微积分，证明了一个Egorov型定理。我们还介绍了一个概念的椭圆性，这意味着Fredholm属性。最后，我们展示了如何-在精神上的一个经典的建设，由A。Weinstein --这种类型的弗雷德霍姆算子可以与$χ$和Maslov丛的一部分相关联。如果$\dim Y>2$或Maslov丛是平凡的，则索引与截面无关，因此是辛同胚的不变量。
Microlocal analysis of double fibration transforms with conjugate points
本文研究了具有共轭点的双纤维变换的正常算子结构，揭示其分解为椭圆伪微分算子和傅里叶积分算子的机制，并拓展了测地线X射线变换的研究成果。
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