The FBI transform on compact 𝒞^{∞} manifolds

Source abstract

We present a geometric theory of the Fourier-Bros-Iagolnitzer transform on a compact C ∞ {\mathcal {C}^\infty } manifold M M . The FBI transform is a generalization of the classical notion of the wave-packet transform. We discuss the mapping properties of the FBI transform and its relationship to the calculus of pseudodifferential operators on M M . We also describe the microlocal properties of its range in terms of the “scattering calculus” of pseudodifferential operators on the noncompact manifold T ∗ M T^* M .