On traces of Fourier integral operators on submanifolds

Given a smooth embedding $i\colon X \hookrightarrow M$ of manifolds and a Fourier integral operator $\Phi = \Phi(\Lambda)$ on $M$ associated with a Lagrangian submanifold $\Lambda \subset T^*(X\times X) \setminus \{0\}$, we consider its trace $i^!(\Lambda)$ on the submanifold $X$, i.e. the composition $i^* \Phi i_*$, where $i^*$ and $i_*$ are the boundary and coboundary operators, respectively. We establish the conditions under which the trace $i^!(\Phi)$ is also a Fourier integral operator and calculate its amplitude in canonical local coordinates.