Calculus for Fourier Integral Operators in generalized SG classes

We construct a calculus for generalized $\mathbf{SG}$ Fourier integral operators, extending known results to a broader class of symbols of $\mathbf{SG}$ 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 $\mathbf{SG}$ Fourier integral operators having regular phase functions and amplitudes uniformly bounded on $\mathbf{R}^{2d}$.