Abstract
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.
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Lascar, B., Lascar, R. FBI transforms in Gevrey classes. J. Anal. Math. 72, 105–125 (1997). https://doi.org/10.1007/BF02843155
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DOI: https://doi.org/10.1007/BF02843155