Prescribing Gevrey singularities for solutions of pseudodifferential operators

The problem of the wave front sets of solutions of differential and pseudodifferential operators has been widely investigated. In particular, several authors have been interested to the question whether one can find a distribution-solution $u$ of a given differential operator or pseudodifferential operator $P$ so that $Pu$ is smooth or analytic or Gevrey of some order $s>1$ and $u$ has prescribed singularities or prescribed wave front sets. This question is closely related to the problems of hypoellipticity and solvability, local or global, smooth or analytic or Gevrey of some order $s>1$. In fact, in order to prove that the differential or pseudodifferential operator under study is not hypoelliptic in the category of some function space, the corresponding method consists in showing the existence of singular solutions. The purpose of this paper is to give a criterion interms of a priori estimates to establish the existence of distribution or smooth-solutions with prescribed Gevrey wave front sets for pseudodifferential operators of finite order.