Fourier integral operators and inhomogeneous Gevrey classes
type: Item
instance of: scholarly article
title: Fourier integral operators and inhomogeneous Gevrey classes
zbMATH Open document ID: 0675.35090
DOI: 10.1007/BF01761469
author: Otto Liess
author: Luigi Rodino
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: Let X be some open set in \(R^ n\). Consider a Fourier distribution of type \[ K(u)=\iint \exp (i\omega (x,\eta))a(x,\eta)u(x)dx d\eta,\quad u\in C_ 0^{\infty}(X). \] The amplitude a(x,\(\eta)\) and the phase \(\omega\) (x,\(\eta)\) are assumed to be \(C^{\infty}\) on \(X\times R^ n\) for large \(\eta\) and a(x,\(\eta)\) satisfies an estimate \[ | D_ x^{\alpha}D^{\beta}_{\eta}a(x,\eta)| \leq C^{| \alpha | +| \beta | +1} \alpha !\beta !\phi (\eta)^{m-| \beta |} \] and the real-valued function \(\omega\) satisfies the same inequality with \(m=1\). \(\phi\) (\(\eta)\), a suitable weight-function, being uniformly Lipschitzian on \(R^ n\) and fulfils \(\phi (\eta)\geq c(1+(\eta))^{\delta}.\) If the phase is essentially nonhomogeneous, then it is necessary to use microlocalization with respect to some structure adapted to the inhomogenity. For this purpose fix \(\Gamma \subset R^ n\), set \[ \Gamma_{\epsilon \phi}=\{\xi \in R^ n:\quad dist(\xi,\Gamma)<\epsilon \phi (\xi)\}, \] assume that a and \(\omega\) are defined only in \(X\times \Gamma_{\epsilon \phi}\) and consider the integral \[ K_ 1(u)=\iint \exp (i\omega (x,\eta))a(x,\eta)\chi (\eta)u(x)dx d\eta, \] where \(\chi\) is some \(C^{\infty}\) function in \(R^ n\) such that \(\chi (\eta)=0\) if \(\eta \not\in \Gamma_{\epsilon '\phi}\), \(\chi (\eta)=1\) if \(\eta \in \Gamma_{\epsilon ''\phi}\) where \(0<\epsilon ''<\epsilon '<\epsilon\) and moreover \(| D^{\alpha}\chi (\eta)\leq c_{\alpha}\) for all \(\eta \in R^ n\). Part I of this paper consists mainly of a number of definitions and technical results concerning weight-functions, symbols, phases and partial integration. The microlocal analysis of \(K_ 1\) is carried out and its wave front is evaluated in part II of the paper. It is especially proved that the microlocal singularities of \(K_ 1\) are not affected (under some assumptions) by changing \(\chi\) or replacing the functions \(\alpha\) and \(\omega\) with equivalent ones. Part III develops the symbolic calcul…
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 4105482
zbMATH Keywords: oscillatory integrals
zbMATH Keywords: Fourier distribution
zbMATH Keywords: amplitude
zbMATH Keywords: phase
zbMATH Keywords: weight- function
zbMATH Keywords: uniformly Lipschitzian
zbMATH Keywords: microlocalization
zbMATH Keywords: symbols
zbMATH Keywords: partial integration
zbMATH Keywords: microlocal analysis
zbMATH Keywords: wave front
zbMATH Keywords: microlocal singularities
zbMATH Keywords: symbolic calculus
zbMATH Keywords: Fourier integral operators
zbMATH Keywords: parametrices
zbMATH Keywords: propagation of singularities
zbMATH Keywords: Gevrey singularities
zbMATH Keywords: Hamiltonian flow
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4105482