On some generalizations of Gevrey classes

We present a generalization of Gevrey classes, aiming at including the inhomogeneous Gevrey functions introduced by Liess 15 and the ultradifferentiable functions in the sense of Braun et al. 4. Therefore, we treat the related dual spaces, called generalized Gevrey ultradistributions, proving also a version of the Paley-Wiener-Schwartz Theorem in our framework. Two different topologies are treated, following the lines both of Beurling 1 and of Roumieu 21, 22. We finally treat in these spaces the well-posedness of the Cauchy problem for weakly hyperbolic operators, extending the previous results of Larsson 14 and Calvo 6, 7. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.