On the space of Laplace transformable distributions

We show that the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}'(\Gamma )$$\end{document}S′(Γ) of Laplace transformable distributions, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \subseteq {\mathbb {R}}^d$$\end{document}Γ⊆Rd is a non-empty convex open set, is an ultrabornological (PLS)-space. Moreover, we determine an explicit topological predual of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}'(\Gamma )$$\end{document}S′(Γ).


Introduction
Schwartz introduced the space S ( ) of Laplace transformable distributions as ⊆ R d is a non-empty convex set [1, p. 303]. This space is endowed with the projective limit topology with respect to the mappings S ( ) → S (R d ), f → e −ξ ·x f (x) for ξ ∈ . The second author together with Kunzinger and Ortner [2] recently presented two new proofs of Schwartz's exchange theorem for the Laplace transform of vector-valued distributions [3,Prop. 4.3,p. 186]. Their methods required them to show that S ( ) is complete, nuclear and dual-nuclear [2, Lemma 5]. Following a suggestion of Ortner, in this article, we further study the locally convex structure of the space S ( ).
In order to be able to apply functional analytic tools such as De Wilde's open mapping and closed graph theorems [4,Theorem 24.30 and Theorem 24.31] or the theory of the derived projective limit functor [5], it is important to determine when a space is ultrabornological. This is usually straightforward if the space is given by a suitable inductive limit; in fact, ultrabornological spaces are exactly the inductive limits of Banach spaces [4,Proposition 24.14]. The situation for projective limits, however, is more complicated. Particularly, this applies to the class of (PLS)-spaces (i.e., countable projective limits of (DFS)-spaces). The problem of ultrabornologicity has been extensively studied in this class, both from an abstract point of view as for concrete function and distribution spaces; see the survey article [6] of Domański and the references therein.
In the last part of his doctoral thesis [7, Chap. II, Thm. 16, p. 131], Grothendieck showed that the convolutor space O C is ultrabornological. He proved that O C is isomorphic to a complemented subspace of the sequence space s ⊗s and verified directly that the latter space is ultrabornological. Much later, a different proof was given by Larcher and Wengenroth using homological methods [8]. The first author and Vindas [9] extended this result to a considerably wider setting by studying the locally convex structure of a general class of weighted convolutor spaces. More precisely, they characterized when such spaces are ultrabornological and determined explicit topological preduals for them. One of their main tools is a topological description of these convolutor spaces in terms of the short-time Fourier transform (STFT).
In this work, we will identify S ( ) with a particular instance of the convolutor spaces considered in [9]. To this end, we make a detailed study of the mapping properties of the STFT on S ( ). Once this identification has been established, we use Theorem 1.1 from [9] (see also Theorem 4.2 below) to show that S ( ) is an ultrabornological (PLS)-space and that it admits a weighted (LF)-space of smooth functions on R d as a topological predual.

Weighted spaces of continuous functions
For formulating the mapping properties of the STFT we recall the following notions from [9,10].
Each non-negative function v on R d defines a weighted seminorm on C(R d ) by We endow the space We consider the following condition on a decreasing weight system V, see [10, p. 114]: The maximal Nachbin family associated with V is defined to be the family V = V (V) consisting of all non-negative upper semicontinuous functions v on R d such that The projective hull of VC(R d ) is defined as and endowed with the locally convex topology generated by the system of seminorms A pointwise increasing sequence W = (w N ) N ∈N of positive continuous functions on R d is called an increasing weight system. Given such a system, we define the Fréchet space We consider the following conditions on an increasing weight system W: In the next lemma, we obtain a concrete representation of the ε-tensor product of weighted spaces of continuous functions.
where we set Proof This follows from the fact that the ε-tensor product commutes with projective limits and [10, Thm. 3.1 (c), p. 137].

The short-time Fourier transform on D (R d )
The translation and modulation operators are denoted by where (·, ·) L 2 denotes the inner product on In particular, the mapping V ψ : We refer to [11] for further properties of the STFT. Next, we explain how the STFT can be extended to the space of distributions; see [9, Sect. 2] for details and proofs. We Clearly, V ψ f is a continuous function on R 2d . In fact, is a well-defined continuous mapping [9, Lemma 2.2]. We define the adjoint STFT of an Then, is a well-defined continuous mapping by [ (3.1)

Duals of inductive limits of weighted spaces of smooth functions
Let v be a non-negative function on R d and n ∈ N. We define B n v (R d ) as the seminormed space consisting of all ϕ ∈ C n (R d ) such that is a Banach space if v is positive and continuous. Let W = (w N ) N ∈N be an increasing weight system. We define the (LF)-space We endow the dual space with the strong topology. If W satisfies (2.1), then D(R) is densely and continuously included in On the other hand, we define the convolutor space is continuous, as follows from the closed graph theorem. We endow O C,W (R d ) with the topology induced via the embedding where β denotes the topology of uniform convergence on bounded sets. In [9] the structural and topological properties of the spaces B W (R d ) and O C,W (R d ) are discussed. We now present the main results of this paper and refer to [9] for more details and proofs. 1

2) and (2.3) and let ψ ∈ D(R d ). Then, the mappings
and are well-defined and continuous.

1), (2.2) and (2.3). Then, B W (R d ) = O C,W (R d ) as sets and the inclusion mapping
is continuous. Moreover, the following statements are equivalent:

Remark 4.3 Condition (4.1) is closely connected with D.
Vogt's condition ( ) that plays an essential role in the structure and splitting theory for Fréchet spaces.

The space S (0)
Our next goal is to characterize S ( ) in terms of the STFT. Let ∅ = ⊆ R d be open and convex. We denote by CCS( ) the family of all non-empty compact convex subsets of and by B(S(R d )) the family of all bounded subsets of S(R d ). The topology of S ( ) can easily be described by a system of concrete seminorms which essentially is due to Schwartz [1, p. 301]; for this, note that the system of convex hulls of finite sets is cofinal in CCS( ): 1 To be precise, the spaces considered in [9], denoted there by Moreover, the topology of S ( ) is generated by the system of seminorms We need to introduce some additional terminology. Given a non-empty compact convex subset K of R d , we define its supporting function as It is clear from the definition that h K is subadditive and positive homogeneous of degree one. In particular, h K is convex. Supporting functions have the following elementary properties.  where B(0, r ) denotes the closed ball in R d centered at the origin with radius r . Next, let K be a non-empty compact convex subset of R d and ε > 0. We set K ε = K + B(0, ε). Lemma 5.2 and the above yield and are well-defined and continuous.
We need some preparation for the proof of Proposition 5.4. Firstly, Lemma 2.1 implies that the topology of is generated by the system of seminorms For k, n ∈ N we write The topology of S(R d ) is generated by the system of seminorms { · S n k | k, n ∈ N}. We now give two technical lemmas.
Proof Choose r > 0 such that supp ψ ⊆ B(0, r ) and R ≥ 1 such that K ⊆ B(0, R). For all k, n ∈ N we have that

Lemma 5.6
Let ψ ∈ D(R d ) and η ∈ R d . Then, for all k, n ∈ N and ϕ ∈ S(R d ), In particular, sup η∈K C η,k,n,ψ < ∞ for all K ⊂ R d compact.
Proof We have that Proof of Proposition 5.4 (i) V ψ : is well-defined and continuous: Let K ∈ CCS( ) and v ∈ V (V pol ) be arbitrary. Choose ε > 0 so small that K ε ∈ CCS( ) and pick, for x ∈ R d fixed, η x ∈ K such that h −K (x) ≤ (−η x · x) + 1. Example 5.3 implies that, for all f ∈ S ( ) and (x, ξ) ∈ R 2d , . Lemma 5.6 implies that, for all η ∈ , is a well-defined continous linear functional on S(R d ).