Pseudodifferential operators of infinite order in spaces of tempered ultradistributions

Abstract

Specific global symbol classes and corresponding pseudodifferential operators of infinite order that act continuously on the space of tempered ultradistributions of Beurling and Roumieu type are constructed. For these classes, symbolic calculus is developed.Mathematics Subject Classification 47G30, 46F05

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

“…The next result is crucial for the proof of Theorem 5.4. We observe that it is stronger than the ones given in [17,Theorem 6.3.3] and [18,Proposition 5]. satisfies: We denote by A n the operator associated to the kernel K n .…”

“…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].…”

“…symbols are different from the ones of [18]. In Section 4 we develop the symbolic calculus and we state some previous results needed to compose two pseudodifferential operators.…”

Asymptotic boundedness and moment asymptotic expansion in ultradistribution spaces

“…Our considerations naturally lead to introduce the ultradistribution spaces K 1˚p R d q and K 1: pR d q, which are intimately connected with the MAE and UMAE. We note that in even dimension our space K 1: pR 2d q arises as the dual of one of the spaces of symbols of 'infinite order' pseudo-differential operators from [26]. The plan of the article is as follows.…”

Pseudo-Differential Calculus in Anisotropic Gelfand–Shilov Setting

“…These operators are commonly known as operators of infinite order and they have been studied in [2] in the analytic class and in [12,24,34] in the Gevrey spaces where the symbol has an exponential growth only with respect to ξ and applied to the Cauchy problem for hyperbolic and Schrödinger equations in Gevrey classes, see [12,13,15,23]. Parallel results have been obtained in Gelfand-Shilov spaces for symbols admitting exponential growth both in x and ξ, see [3,4,7,8,11,27]. We stress that the above results concern the non-quasi-analytic isotropic case s = σ > 1.…”