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

$H^{s}$ boundedness of $h$-Fourier integral operators

  • O. F. Aid Laboratoire de Mathématiques Fondamentales et Appliquées d’Oran (LMFAO), Uni- versité Oran1, Ahmed Ben Bella. B.P. 1524 El M’naouar, Oran, Algeria.
  • A. Senoussaoui Laboratoire de Mathématiques et Aplliquées, Université Oran1
Keywords: h-Fourier integral operators;, symbol and phase;, Sobolev spaces

Abstract

We introduce the relevant background information that
will be used throughout the paper.
Following that, we will go over some fundamental concepts from the
theory of a particular class of semiclassical Fourier integral
operators (symbols and phase functions), which will serve as the
starting 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 decreasing
functions (or Schwartz space) and its dual
$S^{\prime}\left(\mathbb{R}^{n}\right)$ the space of temperate
distributions.

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 integral
operators with its $L^{2}$-adjoint are proved. These allow to obtain
results about the boundedness on the Sobolev spaces
$H^s(\mathbb{R}^n)$.

Author Biographies

O. F. Aid, Laboratoire de Mathématiques Fondamentales et Appliquées d’Oran (LMFAO), Uni- versité Oran1, Ahmed Ben Bella. B.P. 1524 El M’naouar, Oran, Algeria.

Laboratoire de Mathématiques Fondamentales et Appliquées dOran (LMFAO), Uni- versité Oran1, Ahmed Ben Bella. B.P. 1524 El Mnaouar, Oran, Algeria.

A. Senoussaoui, Laboratoire de Mathématiques et Aplliquées, Université Oran1

Laboratoire de Mathématiques et Aplliquées, niversité Oran1

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Published
2021-10-23
How to Cite
1.
Aid OF, Senoussaoui A. The boundedness of a class of semiclassical Fourier integral operators on Sobolev space $H^{s}$: $H^{s}$ boundedness of $h$-Fourier integral operators . Mat. Stud. [Internet]. 2021Oct.23 [cited 2022Jan.20];56(1):61-6. Available from: http://www.matstud.org.ua/ojs/index.php/matstud/article/view/140
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