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

2021 ◽  
Vol 56 (1) ◽  
pp. 61-66
Author(s):  
O. F. Aid ◽  
A. Senoussaoui
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Integral Operators ◽  
Fourier Integral ◽  
Schwartz Space ◽  
Background Information ◽  
Fourier Integral Operators ◽  
Phase Functions

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$H^s(\mathbb{R}^n)$.

H s -Boundedness of a class of Fourier Integral Operators

2021 ◽  
Vol 71 (4) ◽  
pp. 889-902
Author(s):  
Omar Farouk Aid ◽  
Abderrahmane Senoussaoui
Keyword(s):  
Sobolev Spaces ◽  
Integral Operators ◽  
Fourier Integral ◽  
Schwartz Space ◽  
Fourier Integral Operators ◽  
Decreasing Functions

Abstract In this paper, we define a particular class of Fourier Integral Operators (FIO for short). These FIO turn out to be bounded on the spaces S (ℝ n ) of rapidly decreasing functions (or Schwartz space) and S′ (ℝ n ) of temperate distributions. Results about the composition of FIO with its L 2-adjoint are proved. These allow to obtain results about the continuity on the Sobolev Spaces.

scholarly journals Regularity of Fourier integral operators with amplitudes in general Hörmander classes

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Alejandro J. Castro ◽  
Anders Israelsson ◽  
Wolfgang Staubach
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Integral Operators ◽  
Fourier Integral ◽  
Hyperbolic Partial Differential Equations ◽  
Fourier Integral Operators ◽  
Partial Differential

AbstractWe prove the global $$L^p$$ L p -boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes $$S^{m}_{\rho , \delta }(\mathbb {R}^n)$$ S ρ , δ m ( R n ) for parameters $$0\le \rho \le 1$$ 0 ≤ ρ ≤ 1 , $$0\le \delta <1$$ 0 ≤ δ < 1 . We also consider the regularity of operators with amplitudes in the exotic class $$S^{m}_{0, \delta }(\mathbb {R}^n)$$ S 0 , δ m ( R n ) , $$0\le \delta < 1$$ 0 ≤ δ < 1 and the forbidden class $$S^{m}_{\rho , 1}(\mathbb {R}^n)$$ S ρ , 1 m ( R n ) , $$0\le \rho \le 1.$$ 0 ≤ ρ ≤ 1 . Furthermore we show that despite the failure of the $$L^2$$ L 2 -boundedness of operators with amplitudes in the forbidden class $$S^{0}_{1, 1}(\mathbb {R}^n)$$ S 1 , 1 0 ( R n ) , the operators in question are bounded on Sobolev spaces $$H^s(\mathbb {R}^n)$$ H s ( R n ) with $$s>0.$$ s > 0 . This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

scholarly journals L p boundedness of rough bi-parameter Fourier integral operators

2018 ◽  
Vol 30 (1) ◽  
pp. 87-107 ◽  
Author(s):  
Qing Hong ◽  
Guozhen Lu ◽  
Lu Zhang
Keyword(s):  
Compact Support ◽  
Phase Function ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Fourier Multipliers ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Degeneracy Condition ◽  
Phase Functions

Abstract In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: T(f\/)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}% \cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta, where {x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}} and {\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}} , {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} is the amplitude, and the phase function is of the form \varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta) , with \varphi_{1},\varphi_{2}\in L^{\infty}\Phi^{2}(\mathbb{R}^{n}\times\mathbb{R}^{% n}\setminus\{0\}) , and satisfies a certain rough non-degeneracy condition (see (2.2)). The study of these operators are motivated by the {L^{p}} estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the {L^{p}} boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the {L^{p}} boundedness of the more general FIOs with amplitude {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} and non-smooth phase function {\varphi(x,\xi,\eta)} on x satisfying a rough non-degeneracy condition.

scholarly journals Generalized oscillatory integrals and Fourier integral operators

2009 ◽  
Vol 52 (2) ◽  
pp. 351-386 ◽  
Author(s):  
Claudia Garetto ◽  
Günther Hörmann ◽  
Michael Oberguggenberger
Keyword(s):  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Generalized Functions ◽  
Oscillatory Integrals ◽  
Fourier Integral Operators ◽  
Partial Differential Operators ◽  
Colombeau Algebras ◽  
Smooth Coefficients ◽  
Phase Functions

AbstractIn this paper, a theory is developed of generalized oscillatory integrals (OIs) whose phase functions and amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need for a general framework for partial differential operators with non-smooth coefficients and distribution dataffi The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wavefront sets.

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