  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.
