FBI transforms in Gevrey classes

In this work we develop the FBI Transform tools in Gevrey classes. Our goal is to extend to a Gevrey-s obstacle withs < 3 the localization of poles result obtained by Sjöstrand 10] in the analytic class. In that work, the author proved that the pole-free zone is controlled by a constantC _0,a (which was only implicit in Bardos-Lebeau-Rauch 1]), improving the constantC _0,∞ of the results of Hargé-Lebeau 13] and Sjöstrand-Zworski 13] valid in C^∞ The works 3], 13] and 10] feature an adapted complex scaling for convex obstacles, but in 10] there is the addition of a small complex “G³ deformation”. The study of such Gevrey deformations for operators with symbols in Gevrey classes is the central point of this work.