Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

Abstract

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 ...

Global Wave Front Sets in Ultradifferentiable Classes

“…The first author in [4] studies the change of quantization in the class of global pseudodifferential operators introduced in [6] and gives sufficient conditions to obtain parametrices for any quantization. This is the starting point to define a new wave front set in terms of Weyl quantizations for S ω (R d ).…”

“…This is the starting point to define a new wave front set in terms of Weyl quantizations for S ω (R d ). The purpose of the present paper is twofold: on the one hand, we define a Weyl wave front set and study when it coincides with the (continuous version of the) Gabor wave front set of [12] for the ultradifferentiable setting; on the other hand, we give applications of this set to the regularity of pseudodifferential operators in the very general setting of [6].…”

Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

“…In our setting of ultradifferentiable functions, this fact helps, for instance, to study the behaviour (propagation of singularities or wave front sets) of a differential or pseudodifferential operator when acting on a distribution. See, for example, [1,7,16,17,33,38] and the references therein.…”

“…Theorem 18(1)], we have that (M) is distinguished, and then, from [4, Corollary 8(f)] and (5.1) we get Since a Fréchet space is nuclear if and only if its dual is nuclear[34, pg. 78], it is enough to prove that (M) is nuclear; from[10, Theorem 3.1] this is true if and only if By Lemma 5, we can now use(3.15) with = k and with a fixed N > 2d ; since…”

Global pseudodifferential operators in spaces of ultradifferentiable functions

“…The proof of (4) follows proceeding as in (3). Formula ( 5) is shown in [7, Identity 132], and with this, we can show (6). Formula ( 7) is [55, (0.3.16)], and then, we obtain (8):…”