Fourier integral operators and inhomogeneous Gevrey classes
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 calculus of Fourier integral operators of the reduced form \[ Au(x)=\int \exp (i\omega (x,\eta))\chi (\eta)a(x,\eta)\hat u(\eta)d\eta \] and refers to the products \(A^*\circ B\), \(B^*\circ A\) and \(A\circ p(x,D)\) with A and B Fourier integral operators associated with the same phase function \(\omega\) and \[ p(x,D)u(x)=\int \exp (i<x,\eta >)\chi '(\eta)p(x,\eta)\hat u(\eta)d\eta,\quad p\in \tilde S^ m_{\psi}(X,\Gamma,\epsilon). \] In part IV the results of parts II, III are applied to the constructions of parametrices and the study of propagation of singularities for the model operator \[ D=D_ t-a_ 1(t,x,D_ x)-a_ 0(t,x,D_ x,D_ t), \] where \(a_ 1(t,x,\xi)\) is an analytic family of real-valued symbols of order one, \[ D_ t=-\sqrt{- 1}\partial /\partial t,\quad D_{x_ j}=-\sqrt{-1}\partial /\partial x_ j \] and \(a_ 0(t,x,\tau,\xi)\) is a symbol of order zero for a certain class in \(R^{n+1}\). The result is: the set of the inhomogeneous Gevrey singularities of the solutions of the equation \(Pu=0\) is invariant under the action of the Hamiltonian flow associated to \(p_ 1(t,x,\tau,\xi)=\tau -a_ 1(t,x,\xi).\)
scientific article; zbMATH DE number 4105482
scientific article