Characterization of nuclearity for Beurling–Björck spaces

Abstract

We characterize the nuclearity of the Beurling-Björck spaces S (ω) (η) (R d ) and S {ω} {η} (R d ) in terms of the defining weight functions ω and η.A weight function ω is called radially increasing if ω(x) ≤ ω(y) whenever |x| ≤ |y|. Given a weight function ω and a parameter λ > 0, we introduce the family of norms ϕ ω,λ = sup x∈R d |ϕ(x)|e λω(x) .

The nuclearity of Gelfand–Shilov spaces and kernel theorems

“…As a corollary of our more general considerations, we shall also obtain in Section 5 an analogue of Theorem 1.2 for spaces of Beurling-Björck type [3] with ultradifferentiable component defined through a Braun-Meise-Taylor weight function [7]; in fact, Theorem 5.8 considerably improves recent results by Boiti et al [4,5,6]. In this regard, we mention our recent article [11], whose results apply to Beurling-Björck spaces not necessarily arising from Braun-Meise-Taylor weight functions.…”

“…We refer to the monograph [21] and the survey article [14] for accounts on applications of Gelfand-Shilov spaces; see also [9,26] for global pseudo-differential calculus in this setting. The study of nuclearity for spaces of type S goes back to Mityagin [20], and has recently captured much attention [4,5,6,11,25]; particularly, in connection with applications to microlocal analysis of pseudo-differential operators and the convolution theory for generalized functions. We mention that in some cases nuclearity becomes a straightforward consequence of sequence space representations provided by eigenfunction expansions with respect to various PDO [8,18,31].…”

Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

“…Hence, taking unions (Roumieu setting) or intersections (Beurling setting) in in the situation (A) gives different classes of functions than in the situation (B) in general. On the other hand, it is a very difficult problem to determine when the classes treated in this work are nontrivial, a question not considered in [14,15]. We characterize in a very general way (Propositions 2 and 3) when the Hermite functions are contained in our classes and this fact is closely related to classes being non-trivial.…”

“…In both [9] and [10], we use (different) isomorphisms between that space S ( ) (ℝ d ) and some sequence space and prove that S ( ) (ℝ d ) is nuclear by an application of the Grothendieck-Pietsch criterion [32,Theorem 28.15]. Very recently, Debrouwere, Neyt and Vindas [14,15] (cf. [27] for related results about local spaces), using different techniques have extended our previous results in a very general framework.…”

“…[27] for related results about local spaces), using different techniques have extended our previous results in a very general framework. In [14], they characterize when mixed spaces of Björck [3] of Beurling type or of Roumieu type are nuclear under very mild conditions on the weight functions. In [15], using weight matrices in the sense of [37], the same authors characterize the nuclearity of generalized Gel'fand-Shilov classes which extend their previous work [14] and treat also many other mixed classes defined by sequences.…”