Understanding of linear operators through Wigner analysis

IF 1.2 3区 数学 Q1 MATHEMATICS Journal of Mathematical Analysis and Applications Pub Date : 2024-10-11 DOI:10.1016/j.jmaa.2024.128955
Elena Cordero , Gianluca Giacchi , Edoardo Pucci
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Abstract

In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators belong to (weighted) modulation spaces, particularly in Sjöstrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes.
Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schrödinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations SSp(d,R). The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator T on Rd into a pseudodifferential operator K on R2d. This transformation involves a symbol σ well-localized around the manifold defined by z=Sw.
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通过维格纳分析了解线性算子
在这项工作中,我们扩展了维格纳的原始框架,通过研究线性算子的维格纳核与施瓦茨核之间的关系来分析线性算子。我们的方法包括引入傅里叶积分算子(FIO)(准)代数,其中包括 I 型和 II 型 FIO。这些算子的符号属于(加权)调制空间,特别是 Sjöstrand 类调制空间,该类调制空间因其在时频分析中的有利特性而闻名。我们研究的重要成果之一是证明了这些符号类的反封闭性。我们的分析包括与薛定谔型方程相关的伪微分算子和傅里叶积分算子等基本例子。这些例子的典型特征是受线性交映变换 S∈Sp(d,R) 控制的经典哈密顿流。我们方法的核心思想是利用维格纳核将 Rd 上的傅里叶积分算子 T 变换成 R2d 上的伪微分算子 K。这种变换涉及一个在由 z=Sw 定义的流形周围很好定位的符号 σ。
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来源期刊
CiteScore
2.50
自引率
7.70%
发文量
790
审稿时长
6 months
期刊介绍: The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.
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Editorial Board Editorial Board Editorial Board Editorial Board Bivariate homogeneous functions of two parameters: Monotonicity, convexity, comparisons, and functional inequalities
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