On traces of Fourier integral operators on submanifolds

Given a smooth embedding $i\text{\hspace{0.17em}⁣}:X\hookrightarrow M$ of manifolds and a Fourier integral operator $\mathrm{\Phi }=\mathrm{\Phi }(\mathrm{\Lambda })$ on $M$ associated with a Lagrangian submanifold $\mathrm{\Lambda }\subset T^{\ast }(X\times X)\setminus \{0\}$, we consider its trace $i^{!}(\mathrm{\Lambda })$ on the submanifold $X$, i.e. the composition $i^{\ast }\mathrm{\Phi }{i}_{\ast }$, where $i^{\ast }$ and $i_{\ast }$ are the boundary and coboundary operators, respectively. We establish the conditions under which the trace $i^{!}(\mathrm{\Phi })$ is also a Fourier integral operator and calculate its amplitude in canonical local coordinates.