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

Smoothing properties of inhomogeneous equations via canonical transforms