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Abstract. We introduce pseudodifferential operators (of infinite order) in the framework of non-quasianalytic classes of Beurling type. We prove that such an operator with (distributional) kernel in a given Beurling class D (ω) is pseudo-local and can be locally decomposed, modulo a smoothing operator, as the composition of a pseudodifferential operator of finite order and an ultradifferential operator with constant coefficients in the sense of Komatsu, both operators with kernel in the same class D (ω) . We also develop the corresponding symbolic calculus. The study of several problems in classes of (non-quasianalytic) ultradifferentiable functions has also received much attention recently. These are intermediate classes between real analytic functions and C ∞ functions. There are essentially two ways to introduce them: the theory of Komatsu [16], in which one looks at the growth of the derivatives on compact sets, and the theory developed by Björk [2] in 1966, following the ideas previously announced by Beurling, in which one pays attention to the growth of the Fourier transforms. We will work with ultradifferentiable functions as defined by Braun, Meise and Taylor [8]. Their point of view permits a unified treatment of both theories, contains the most relevant cases of Komatsu's theory and is strictly broader than Beurling-Björk's.Pseudodifferential operators (of finite or infinite order) on Gevrey classes have been extensively studied by many authors ([5]

We investigate the action of pseudodifferential operators of Beurling type on the wave front sets. More precisely, we show that these operators are microlocal, that is, preserve or reduce wave front sets. Some consequences on micro-hypoellipticity are derived.

Abstract. We consider the counter images J (R d ) and J 0 (R d ) of entire functions with exponential and almost exponential bounds, respectively, under the Bargmann transform, and we characterize them by estimates of powers of the harmonic oscillator. We also consider the Pilipović spaces S s (R d ) and Σ s (R d ) when 0 < s < 1/2 and deduce their images under the Bargmann transform. IntroductionThe aim of the paper is to characterize the images of the Pilipović spaces Σ s (R d ) and S s (R d ) under the Bargmann transform when s < 1/2, as well as the test function spaces, in terms of estimates of powers of the harmonic oscillator. The set J (R d ) consists of all f ∈ S (R d ) such that their Hermite series expansions are given bywhere

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