Multi-products of Fourier integral operators and the fundamental solution for a hyperbolic system with involutive characteristics
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.
multi-products of Fourier integral operators
scientific article; zbMATH DE number 3869880
scientific article