Lagrangian Distributions and Fourier Integral Operators with Quadratic Phase Functions and Shubin Amplitudes

We study Fourier integral operators with Shubin amplitudes and quadratic phase functions associated to twisted graph Lagrangians with respect to symplectic matrices. We factorize such an operator as a pseudodifferential operator and a metaplectic operator. Extending the conormal distributions adapted to the Shubin calculus, we define an adapted notion of Lagrangian tempered distribution. We show that the kernels of Fourier integral operators are identical to Lagrangian distributions with respect to twisted graph Lagrangians.