Multi-products of Fourier integral operators and the fundamental solution for a hyperbolic system with involutive characteristics
type: Item
instance of: scholarly article
title: Multi-products of Fourier integral operators and the fundamental solution for a hyperbolic system with involutive characteristics
zbMATH Open document ID: 0546.35079
published in: Osaka Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: In [\textit{H. Kumano-go} and the author, Funkc. Ekvacioj, Ser. Int. 22, 161-196 (1979)] we have constructed the fundamental solution for a hyperbolic system with variable multiplicity by solving eikonal equations and using the successive approximation. In this method multi-products of Fourier integral operators appear. The first purpose of the present paper is to show an estimate of their symbols. The method used here is the reduction of a multi-product of Fourier integral operators to a multi- product of pseudo-differential operators multiplied by a Fourier integral operator insert. Then, we get the desired estimate. The key point of this reduction is the use of the inverse operator of the operator I-P, where P is a pseudo-differential operator with small semi-norms. We give its inverse operator by the Neuman series \(\sum^{\infty}_{\nu =0}P^{\nu}\) by deriving a sharper estimate for \(P^{\nu}\) than the estimate given by \textit{H. Kumano-go} and \textit{C. Tsutsumi} [Osaka J. Math. 10, 147-174 (1973; Zbl 0264.35019)]. Then, we can obtain the inverse operator of the operator I-P only by the symbol calculus. The latter half of the present paper is devoted to the representation of the fundamental solution for a hyperbolic system with involutive characteristics as a finite sum of Fourier integral operators and to get a estimate of wave front sets for a solution of the Cauchy problem. The key point of the representation is to obtain a commutative law for the {\#}-product of phase functions. Then, we get an information of the propagation of wave front sets easily.
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 35L40
Mathematics Subject Classification ID: 47Gxx
zbMATH DE Number: 3869880
zbMATH Keywords: fundamental solution
zbMATH Keywords: hyperbolic system
zbMATH Keywords: variable multiplicity
zbMATH Keywords: eikonal equations
zbMATH Keywords: successive approximation
zbMATH Keywords: multi-products of Fourier integral operators
zbMATH Keywords: involutive characteristics
zbMATH Keywords: estimate of wave front sets
zbMATH Keywords: Cauchy problem
author: Kazuo Taniguchi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3869880