Fourier integral operators with complex-valued phase function and the Cauchy problem for hyperbolic operators

1991 ◽  
pp. 1-70 ◽  
Author(s):  
Kunihiko Kajitani
Keyword(s):  
Cauchy Problem ◽  
Phase Function ◽  
Integral Operators ◽  
Fourier Integral ◽  
Fourier Integral Operators ◽  
Hyperbolic Operators ◽  
The Cauchy Problem ◽  
Complex Valued

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 GEOMETRIC APPROACH TO THE HAMILTON–JACOBI EQUATION AND GLOBAL PARAMETRICES FOR THE SCHRÖDINGER PROPAGATOR

2011 ◽  
Vol 23 (09) ◽  
pp. 969-1008 ◽  
Author(s):  
SANDRO GRAFFI ◽  
LORENZO ZANELLI
Keyword(s):  
Phase Function ◽  
Jacobi Equation ◽  
Semiclassical Approximation ◽  
Integral Operators ◽  
Geometric Approach ◽  
Fourier Integral ◽  
Fourier Integral Operators ◽  
Hamiltonian Flow ◽  
The Real ◽  
Hamilton Jacobi Equation

We construct a family of global Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton–Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.

On a particular family of Fourier integral operators

2015 ◽  
Vol 6 (3) ◽  
pp. 407-412
Author(s):  
Mohsen Alimohammady ◽  
Mohammad Habibi
Keyword(s):  
Integral Operators ◽  
Fourier Integral ◽  
Fourier Integral Operators
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