  The paper describes a new approach to global smoothing problems for
inhomogeneous dispersive evolution equations based on an idea of canonical
transformation. In our previous papers, we introduced such a method to show
global smoothing estimates for homogeneous dispersive equations. It is
remarkable that this method allows us to carry out a global microlocal
reduction of equations to some low dimensional model cases. The purpose of this
paper is to pursue the same treatment for inhomogeneous equations. Especially,
time-global smoothing estimates for the operator $a(D_x)$ with lower order
terms are the benefit of our new method.
