Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on R^n

TL;DR

This paper investigates the algebraic properties of Fourier integral operators with SG class symbols and constructs fundamental solutions for hyperbolic systems with polynomially bounded coefficients, ensuring well-posedness in weighted Sobolev spaces.

Contribution

It establishes conditions under which compositions of SG class Fourier integral operators remain in the same class and constructs fundamental solutions for hyperbolic PDE systems with such coefficients.

Findings

Composition of SG class Fourier integral operators remains in the same class under certain conditions.

Fundamental solutions for hyperbolic systems with polynomially bounded coefficients are constructed.

Well-posedness in weighted Sobolev spaces is achieved for these hyperbolic systems.

Abstract

We study the composition of an arbitrary number of Fourier integral operators $A_j$, $j=1,\dots,M$, $M\ge 2$, defined through symbols belonging to the so-called SG classes. We give conditions ensuring that the composition $A_1\circ\cdots\circ A_M$ of such operators still belongs to the same class. Through this, we are then able to show well-posedness in weighted Sobolev spaces for first order hyperbolic systems of partial differential equations with coefficients in SG classes, by constructing the associated fundamental solutions.