We develop a theory of pseudodifferential operators of infinite order for the global classes Sω of ultradifferentiable functions in the sense of Björck, following the previous ideas given by Prangoski for ultradifferentiable classes in the sense of Komatsu. We study the composition and the transpose of such operators with symbolic calculus and provide several examples. IntroductionThe local theory of pseudodifferential operators grew out of the study of singular integral operators, and developed after 1965 with the systematic studies of , Hörmander [13], and others. Since then, several authors have studied pseudodifferential operators of finite or infinite order in Gevrey classes in the local sense; we mention, for instance, [12,21]. We refer to Rodino [19] for an excellent introduction to this topic, and the references therein.Gevrey classes are spaces of (non-quasianalytic) ultradifferentiable functions in between real analytic and C ∞ functions. The study of several problems in general classes of ultradifferentiable functions has received much attention in the last 60 years. Here, we will work with ultradifferentiable functions as defined by Braun, Meise and Taylor [5], which define the classes in terms of the growth of the derivatives of the functions, or in terms of the growth of their Fourier transforms (see, for example, Komatsu [15] and Björck [2], or [5], for two different points of view to define spaces of ultradifferentiable functions and ultradistributions; and [4] for a comparison between the classes defined in [5] and [15]).In [10], a full theory of pseudodifferential operators in the local sense is developed for ultradifferentiable classes of Beurling type as in [5], and it is proved that the corresponding operators are ω-pseudo-local, and the product of two operators is given in terms of a suitable symbolic calculus. In [9,11] the same authors construct a parametrix for such operators and study the action of the wave front set on them (see also [1] for a different point of view). On the other hand, very recently, Prangoski [18] studies pseudodifferential operators of global type and infinite order for ultradifferentiable classes of Beurling and Roumieu type in the sense of Komatsu, and later,in [8], a parametrix is constructed for such operators. See [18,17] and the references therein for more examples of pseudodifferential operators in global classes (e.g., in Gelfand-Shilov classes).Our aim is to study pseudodifferential operators of global type and infinite order in classes of ultradifferentiable functions of Beurling type as introduced in [5]. Hence, the right setting is the class S ω as introduced by Björck [2]. We follow the lines of Prangoski [18] and Shubin [20], but from the point of view of [10], in such a way that our proofs simplify the ones of [18]. Moreover, we clarify the role of some kind of entire functions [6,16] that become crucial throughout the text.The paper is organized as follows. First, in Section 2, we introduce our setting, we give some useful results about the class S ...

We introduce a global wave front set using Weyl quantizations of pseudodifferential operators of infinite order in the ultradifferentiable setting. We see that in many cases it coincides with the Gabor wave front set already studied by the last three authors of the present work. In this sense, we also extend, to the ultradifferentiable setting, previous work by Rodino and Wahlberg. Finally, we give applications to the study of propagation of singularities of pseudodifferential operators.

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global ω-hypoellipticity and global ω-regularity of these classes of pseudodifferential operators.

In this paper we introduce and study global wave front sets in terms of the $$\tau $$ τ -Wigner transform in global ultradifferentiable classes of Beurling type modulated with weight functions in the sense of Braun, Meise, and Taylor, and we compare it with other wave front sets existing in the literature defined by different time-frequency analysis tools, such as the short-time Fourier transform or Gabor frames. Conditions for the equality of these wave front sets are provided and some examples are given.

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