On a Class of Fourier Integral Operators on Manifolds with Boundary
  We study a class of Fourier integral operators on compact manifolds with boundary, associated with a natural class of symplectomorphisms, namely, those which preserve the boundary. A calculus of Boutet de Monvel's type can be defined for such Fourier integral operators, and appropriate continuity properties established. One of the key features of this calculus is that the local representations of these operators are given by operator-valued symbols acting on Schwartz functions or temperate distributions. Here we focus on properties of the corresponding local phase functions, which allow to prove this result in a rather straightforward way. 
Ubertino Battisti and Sandro Coriasco and Elmar Schrohe
math.OA math.FA