Microlocal analytic smoothing effects for operators of real principal type

Abstract

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.