Some topics on the regularity of Analytic-Gevrey vectors

My aim is to give , in this talk , some topics on the question of regularity of Analytic-Gevrey vectors of partial differential operators (p.d.o.) with analytic-Gevrey coefficients . Since the results obtained in the sixties on elliptic p.d.o's , which are both hypoelliptic (Cˆ{\infty} setting) , analytic-Gevrey hypoelliptic (analytic-Gevrey setting) and satisfy the so-called Kotake-Narasimhan property , a lot of works and articles were devoted to these problems in case of non elliptic p.d.o's under suitable hypotheses (for example on the degeneracy of ellipticity). I will consider the third problem on analytic-Gevrey vectors in the three cases of global (on compact manifolds ), local ( near a point in the base-space ), microlocal (near a point in the cotangent space ) , situations , and say few words on the main two methods used in order to obtain positive (or negative) results . Finally I will focus on some new microlocal results on degenerate elliptic (also called sub-elliptic ) p.d.o's of second order , obtained in a common work with Gregorio Chinni .