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

We study the composition of an arbitrary number of Fourier integral oper- ators A_j , j = 1, . . . , M, M ≥ 2, defined through symbols belonging to the so-called SG classes. We give conditions ensuring that the composition A_1 ◦ · · · ◦ 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 in SG classes, by constructing the associated fundamental solutions. These results expand the existing theory for the study of the properties “at infinity” of the solutions to hyperbolic Cauchy problems on Rn with polynomially bounded coefficients.