The boundedness of a class of semiclassical Fourier integral operators on Sobolev space $H^{s}$

Q3 Mathematics Matematychni Studii Pub Date : 2021-10-23 DOI:10.30970/ms.56.1.61-66
O. F. Aid, A. Senoussaoui
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Abstract

We introduce the relevant background information thatwill be used throughout the paper.Following that, we will go over some fundamental concepts from thetheory of a particular class of semiclassical Fourier integraloperators (symbols and phase functions), which will serve as thestarting point for our main goal. Furthermore, these integral operators turn out to be bounded on$S\left(\mathbb{R}^{n}\right)$ the space of rapidly decreasingfunctions (or Schwartz space) and its dual$S^{\prime}\left(\mathbb{R}^{n}\right)$ the space of temperatedistributions. Moreover, we will give a brief introduction about$H^s(\mathbb{R}^n)$ Sobolev space (with $s\in\mathbb{R}$).Results about the composition of semiclassical Fourier integraloperators with its $L^{2}$-adjoint are proved. These allow to obtainresults about the boundedness on the Sobolev spaces$H^s(\mathbb{R}^n)$.
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Sobolev空间H^{s}$上一类半经典傅里叶积分算子的有界性
我们将介绍相关的背景信息,这些信息将在整个论文中使用。接下来,我们将讨论一类特定的半经典傅立叶积分算子(符号和相位函数)理论中的一些基本概念,这些概念将作为我们主要目标的起点。此外,这些积分算子在快速递减函数空间(或Schwartz空间)的$S\left(\mathbb{R}^{n}\right)$和其对偶温度分布空间的$S^{\prime}\left(\ mathbb{R}^{n}\right)$上是有界的。此外,我们还将简要地介绍$H^s(\mathbb{R}^n)$Sobolev空间(带有$s\in\mathbb{R}$).证明了具有$L^{2}$伴随的半经典傅立叶积分算子的组成结果。这些结果允许得到关于Sobolev空间$H^s(\mathbb{R}^n)$上有界性的结果。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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