Calculus for Fourier Integral Operators in generalized SG classes

We construct a calculus for generalized $\mathbf{S}\mathbf{G}$ Fourier integral operators, extending known results to a broader class of symbols of $\mathbf{S}\mathbf{G}$ type. In particular, we do not require that the phase functions are homogeneous. We also prove the $L^2({\mathbf{R}}^d)$-boundedness of the generalized $\mathbf{S}\mathbf{G}$ Fourier integral operators having regular phase functions and amplitudes uniformly bounded on ${\mathbf{R}}^{2d}$.