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In this paper we study regularity of partial differential equations withpolynomial coefficients in non isotropic Beurling spaces of ultradifferentiablefunctions of global type. We study the action of transformations of Gabor andWigner type in such spaces and we prove that a suitable representation ofWigner type allows to prove regularity for classes of operators that do nothave classical hypoellipticity properties.

Regularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods