We are interested in the microlocal smoothing effect for operators
of real principal type. On the initial value problem for
a dispersive evolution equation, we study the fact that the
sufficient decay of the initial data gives the smoothness
of the solution. We develop the theory of the FBI transform
in order to transform our operator of real principal type
into a simple operator of first order. Since the smoothing
effect is of global nature, our transformation is realized
globally along the bicharacteristics defined from the principal
symbol of the operator.