Abstract
We show that the space \({\mathcal {S}}'(\Gamma )\) of Laplace transformable distributions, where \(\Gamma \subseteq {\mathbb {R}}^d\) is a non-empty convex open set, is an ultrabornological (PLS)-space. Moreover, we determine an explicit topological predual of \({\mathcal {S}}'(\Gamma )\).
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1 Introduction
Schwartz introduced the space \({\mathcal {S}}'(\Gamma )\) of Laplace transformable distributions as
where \(\Gamma \subseteq {\mathbb {R}}^d\) is a non-empty convex set [1, p. 303]. This space is endowed with the projective limit topology with respect to the mappings \({\mathcal {S}}'(\Gamma ) \rightarrow {\mathcal {S}}'({\mathbb {R}}^d)\), \(f \mapsto e^{-\xi \cdot x} f(x)\) for \(\xi \in \Gamma \). 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 \({\mathcal {S}}'(\Gamma )\) 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 \({\mathcal {S}}'(\Gamma )\).
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 \({\mathcal {O}}_C'\) is ultrabornological. He proved that \({\mathcal {O}}_C'\) is isomorphic to a complemented subspace of the sequence space \(s {\widehat{\otimes }} s'\) 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 \({\mathcal {S}}'(\Gamma )\) 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 \({\mathcal {S}}'(\Gamma )\). Once this identification has been established, we use Theorem 1.1 from [9] (see also Theorem 4.2 below) to show that \({\mathcal {S}}'(\Gamma )\) is an ultrabornological (PLS)-space and that it admits a weighted (LF)-space of smooth functions on \({\mathbb {R}}^d\) as a topological predual.
2 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 \({\mathbb {R}}^d\) defines a weighted seminorm on \(C({\mathbb {R}}^d)\) by
We endow the space
with this seminorm; it is a Banach space if v is positive and continuous. A pointwise decreasing sequence \({\mathcal {V}} = (v_N)_{N \in {\mathbb {N}}}\) of positive continuous functions on \({\mathbb {R}}^d\) is called a decreasing weight system. With this, we define the (LB)-space
We consider the following condition on a decreasing weight system \({\mathcal {V}}\), see [10, p. 114]:
The maximal Nachbin family associated with \({\mathcal {V}}\) is defined to be the family \({\overline{V}}={\overline{V}}({\mathcal {V}})\) consisting of all non-negative upper semicontinuous functions v on \({\mathbb {R}}^d\) such that
The projective hull of \({\mathcal {V}}C({\mathbb {R}}^d)\) is defined as
and endowed with the locally convex topology generated by the system of seminorms \(\{ \Vert \,\cdot \,\Vert _{v} \, | \, v \in {\overline{V}} \}\). The spaces \({\mathcal {V}}C({\mathbb {R}}^d)\) and \(C{\overline{V}}({\mathbb {R}}^d)\) always coincide as sets and, if \({\mathcal {V}}\) satisfies condition (V), also as locally convex spaces [10, Thm. 1.3 (d), p. 118].
A pointwise increasing sequence \({\mathcal {W}} = (w_N)_{N \in {\mathbb {N}}}\) of positive continuous functions on \({\mathbb {R}}^d\) 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 \({\mathcal {W}}\):
In the next lemma, we obtain a concrete representation of the \(\varepsilon \)-tensor product of weighted spaces of continuous functions.
Lemma 2.1
Let \({\mathcal {W}} = (w_N)_{N \in {\mathbb {N}}}\) be an increasing weight system and \({\mathcal {V}} = (v_n)_{n \in {\mathbb {N}}}\) a decreasing weight system satisfying (V). Then, we have the identification
where we set \(\Vert f\Vert _{w \otimes v} \mathrel {\mathop :}=\sup _{(x,\xi ) \in {\mathbb {R}}^{2d}} \left| f(x,\xi )\right| w(x) v(\xi )\) for non-negative functions w, v on \({\mathbb {R}}^d\). Moreover, \(f \in C({\mathbb {R}}^{2d}_{x,\xi })\) belongs to \({\mathcal {W}}C({\mathbb {R}}^d_x) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}C({\mathbb {R}}^d_\xi )\) if and only if \(\Vert f\Vert _{w_N \otimes v} < \infty \) for all \(N \in {\mathbb {N}}\) and \(v \in {\overline{V}}\). Consequently, the topology of \({\mathcal {W}}C({\mathbb {R}}^d_x) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}C({\mathbb {R}}^d_\xi )\) is generated by the system of seminorms \(\{ \Vert \, \cdot \, \Vert _{w_N \otimes v} \, | \, N \in {\mathbb {N}}, v \in {\overline{V}} \}\).
Proof
This follows from the fact that the \(\varepsilon \)-tensor product commutes with projective limits and [10, Thm. 3.1 (c), p. 137]. \(\square \)
3 The short-time Fourier transform on \({\mathcal {D}}'({\mathbb {R}}^d)\)
The translation and modulation operators are denoted by \(T_xf(t) = f(t-x)\) and \(M_\xi f(t) = e^{2\pi i \xi \cdot t} f(t)\) for \(x, \xi \in {\mathbb {R}}^d\). The short-time Fourier transform (STFT) of a function \(f \in L^2({\mathbb {R}}^d)\) with respect to a window function \(\psi \in L^2({\mathbb {R}}^d)\) is defined as
where \((\cdot ,\cdot )_{L^2}\) denotes the inner product on \(L^2({\mathbb {R}}^d)\). We have that \(\Vert V_\psi f\Vert _{L^2({\mathbb {R}}^{2d})} = \Vert \psi \Vert _{L^2}\Vert f\Vert _{L^2}\). In particular, the mapping \(V_\psi :L^2({\mathbb {R}}^d) \rightarrow L^2({\mathbb {R}}^{2d})\) is continuous. The adjoint of \(V_\psi \) is given by the weak integral
If \(\psi \ne 0\) and \(\gamma \in L^2({\mathbb {R}}^d)\) is a synthesis window for \(\psi \), that is, \((\gamma , \psi )_{L^2} \ne 0\), then
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 set \({\mathcal {V}}_{{\text {pol}}} = ((1+ \left| \, \cdot \, \right| )^{-N})_{N \in {\mathbb {N}}}.\) Fix a window function \(\psi \in {\mathcal {D}}({\mathbb {R}}^d)\). For \(f \in {\mathcal {D}}'({\mathbb {R}}^d)\) we define
Clearly, \(V_\psi f\) is a continuous function on \({\mathbb {R}}^{2d}\). In fact,
is a well-defined continuous mapping [9, Lemma 2.2]. We define the adjoint STFT of an element \(F \in C({\mathbb {R}}^d_x) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}_{{\text {pol}}}C({\mathbb {R}}^d_\xi )\) as the distribution
Then,
is a well-defined continuous mapping by [9, Prop. 2.2]. Finally, if \(\psi \ne 0\) and \(\gamma \in {\mathcal {D}}({\mathbb {R}}^d)\) is a synthesis window for \(\psi \), then the following reconstruction formula holds [9, Prop. 2.4]:
4 Duals of inductive limits of weighted spaces of smooth functions
Let v be a non-negative function on \({\mathbb {R}}^d\) and \(n \in {\mathbb {N}}\). We define \({\mathcal {B}}^n_v({\mathbb {R}}^d)\) as the seminormed space consisting of all \(\varphi \in C^n({\mathbb {R}}^d)\) such that
As before, \({\mathcal {B}}^n_v({\mathbb {R}}^d)\) is a Banach space if v is positive and continuous. Let \({\mathcal {W}} = (w_N)_{N \in {\mathbb {N}}}\) be an increasing weight system. We define the (LF)-space
We endow the dual space \({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d) \mathrel {\mathop :}=({\mathcal {B}}_{{\mathcal {W}}^\circ }({\mathbb {R}}^d))'\) with the strong topology. If \({\mathcal {W}}\) satisfies (2.1), then \({\mathcal {D}}({\mathbb {R}})\) is densely and continuously included in \({\mathcal {B}}_{{\mathcal {W}}^\circ }({\mathbb {R}}^d)\) and therefore \({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d)\) is a vector subspace of \({\mathcal {D}}'({\mathbb {R}}^d)\).
On the other hand, we define the convolutor space
For \(f \in {\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) fixed, the mapping
is continuous, as follows from the closed graph theorem. We endow \({\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) with the topology induced via the embedding
where \(\beta \) denotes the topology of uniform convergence on bounded sets.
In [9] the structural and topological properties of the spaces \({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d)\) and \({\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) are discussed. We now present the main results of this paper and refer to [9] for more details and proofs.Footnote 1
Proposition 4.1
[9, Prop. 4.2] Let \({\mathcal {W}}\) be an increasing weight system satisfying (2.1), (2.2) and (2.3) and let \(\psi \in {\mathcal {D}}({\mathbb {R}}^d)\). Then, the mappings
and
are well-defined and continuous.
Theorem 4.2
[9, Thm. 3.4, Thm. 4.6 and Thm. 4.15] Let \({\mathcal {W}} = (w_N)_{N \in {\mathbb {N}}}\) be an increasing weight system satisfying (2.1), (2.2) and (2.3). Then, \({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d) = {\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) as sets and the inclusion mapping \({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d) \rightarrow {\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) is continuous. Moreover, the following statements are equivalent:
- (i):
-
\({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d) = {\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) as locally convex spaces.
- (ii):
-
\({\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) is an ultrabornological (PLS)-space.
- (iii):
-
The (LF)-space \({\mathcal {B}}_{{\mathcal {W}}^\circ }({\mathbb {R}}^d)\) is complete.
- (iv):
-
\({\mathcal {W}}\) satisfies
$$\begin{aligned}&\forall N \in {\mathbb {N}}\, \exists M \ge N \, \forall P \ge M \, \exists \theta \in (0,1) \, \exists C > 0 \, \forall x \in {\mathbb {R}}^d: \nonumber \\&{w_N(x)}^{1-\theta }{w_P(x)}^{\theta } \le Cw_M(x). \end{aligned}$$(4.1)
Remark 4.3
Condition (4.1) is closely connected with D. Vogt’s condition \((\Omega )\) that plays an essential role in the structure and splitting theory for Fréchet spaces.
5 The space \({\mathcal {S}}'(\Gamma )\)
Our next goal is to characterize \({\mathcal {S}}'(\Gamma )\) in terms of the STFT.
Let \(\emptyset \ne \Gamma \subseteq {\mathbb {R}}^d\) be open and convex. We denote by \({\text {CCS}}(\Gamma )\) the family of all non-empty compact convex subsets of \(\Gamma \) and by \({\mathfrak {B}}({\mathcal {S}}({\mathbb {R}}^d))\) the family of all bounded subsets of \({\mathcal {S}}({\mathbb {R}}^d\)). The topology of \({\mathcal {S}}'(\Gamma )\) 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 \({\text {CCS}}(\Gamma )\):
Lemma 5.1
[1, p. 301] Let \(\emptyset \ne \Gamma \subseteq {\mathbb {R}}^d\) be open and convex. For all \(K \in {\text {CCS}}(\Gamma )\) and \(B \in {\mathfrak {B}}({\mathcal {S}}({\mathbb {R}}^d))\) we have that
Moreover, the topology of \({\mathcal {S}}'(\Gamma )\) is generated by the system of seminorms \(\{p_{K,B} \, | \, K \in {\text {CCS}}(\Gamma ), B \in {\mathfrak {B}}({\mathcal {S}}({\mathbb {R}}^d))\}\).
We need to introduce some additional terminology. Given a non-empty compact convex subset K of \({\mathbb {R}}^d\), 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.
Lemma 5.2
[12, Cor. 1.8.2 and Prop. 1.8.3] Let \(K_1\) and \(K_2\) be non-empty compact convex subsets of \({\mathbb {R}}^d\).
- (a):
-
\(K_1 \subseteq K_2\) if and only if \(h_{K_1}(x) \le h_{K_2}(x)\) for all \(x \in {\mathbb {R}}^d\).
- (b):
-
\(h_{K_1+K_2}(x) = h_{K_1}(x) + h_{K_2}(x)\) for all \(x \in {\mathbb {R}}^d\).
Example 5.3
For \(r > 0\) we have \(h_{{\overline{B}}(0,r)}(x) = r \left| x\right| \) for all \(x \in {\mathbb {R}}^d\), where \({\overline{B}}(0,r)\) denotes the closed ball in \({\mathbb {R}}^d\) centered at the origin with radius r. Next, let K be a non-empty compact convex subset of \({\mathbb {R}}^d\) and \(\varepsilon > 0\). We set \(K_\varepsilon = K + {\overline{B}}(0,\varepsilon )\). Lemma 5.2 and the above yield that \(h_{K_\varepsilon }(x) = h_K(x) + \varepsilon \left| x\right| \) for all \(x \in {\mathbb {R}}^d\).
Let \(\emptyset \ne \Gamma \subseteq {\mathbb {R}}^d\) be open and convex and let \((K_N)_{N \in {\mathbb {N}}} \subset {\text {CCS}}(\Gamma )\) be such that \(K_N \subseteq K_{N+1}\) for all \(N \in {\mathbb {N}}\) and \(\Gamma = \bigcup _{N} K_N\). Lemma 5.2 yields that \({\mathcal {W}} = (e^{h_{-K_N}})_{N \in {\mathbb {N}}}\) is an increasing weight system. We set \(C_\Gamma ({\mathbb {R}}^d) \mathrel {\mathop :}={\mathcal {W}}C({\mathbb {R}}^d)\). Clearly, the definition of \(C_\Gamma ({\mathbb {R}}^d)\) is independent of the chosen sequence \((K_N)_{N \in {\mathbb {N}}}\). The next result is the key observation of this article.
Proposition 5.4
Let \(\emptyset \ne \Gamma \subseteq {\mathbb {R}}^d\) be open and convex and let \(\psi \in {\mathcal {D}}({\mathbb {R}}^d)\). Then, the mappings
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 \(C_\Gamma ({\mathbb {R}}^d_x) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}_{{\text {pol}}}C({\mathbb {R}}^d_\xi )\) is generated by the system of seminorms
For \(k,n \in {\mathbb {N}}\) we write
The topology of \( {\mathcal {S}}({\mathbb {R}}^d)\) is generated by the system of seminorms \(\{ \Vert \,\cdot \, \Vert _{{\mathcal {S}}^n_k} \, | \, k,n \in {\mathbb {N}}\}\). We now give two technical lemmas.
Lemma 5.5
Let \(\psi \in {\mathcal {D}}({\mathbb {R}}^d)\), \(K \subset {\mathbb {R}}^d\) be compact, \(v \in {\overline{V}}({\mathcal {V}}_{{\text {pol}}})\) and \(\varepsilon > 0\). Then,
Proof
Choose \(r > 0\) such that \({\text {supp}} \psi \subseteq {\overline{B}}(0,r)\) and \(R \ge 1\) such that \(K \subseteq {\overline{B}}(0,R)\). For all \(k,n \in {\mathbb {N}}\) we have that
\(\square \)
Lemma 5.6
Let \(\psi \in {\mathcal {D}}({\mathbb {R}}^d)\) and \(\eta \in {\mathbb {R}}^d\). Then, for all \(k,n \in {\mathbb {N}}\) and \(\varphi \in {\mathcal {S}}({\mathbb {R}}^d)\),
where
In particular, \(\sup _{\eta \in K} C_{\eta ,k,n,\psi } < \infty \) for all \(K \subset {\mathbb {R}}^d\) compact.
Proof
We have that
\(\square \)
Proof of Proposition 5.4
(i) \(V_\psi :{\mathcal {S}}'(\Gamma ) \rightarrow C_\Gamma ({\mathbb {R}}^d_x) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}_{{\text {pol}}}C({\mathbb {R}}^d_\xi )\) is well-defined and continuous: Let \(K \in {\text {CCS}}(\Gamma )\) and \(v \in {\overline{V}}({\mathcal {V}}_{{\text {pol}}})\) be arbitrary. Choose \(\varepsilon > 0\) so small that \(K_\varepsilon \in {\text {CCS}}(\Gamma )\) and pick, for \(x \in {\mathbb {R}}^d\) fixed, \(\eta _x \in K\) such that \(h_{-K}(x) \le (-\eta _x \cdot x) + 1\). Example 5.3 implies that, for all \(f \in {\mathcal {S}}'(\Gamma )\) and \((x,\xi ) \in {\mathbb {R}}^{2d}\),
where
by Lemma 5.5.
(ii) \(V^*_\psi :C_\Gamma ({\mathbb {R}}^d_x) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}_{{\text {pol}}}C({\mathbb {R}}^d_\xi ) \rightarrow {\mathcal {S}}'(\Gamma )\) is well-defined and continuous: We start by showing that \(V^*_\psi F \in {\mathcal {S}}'(\Gamma )\) for all \(F \in C_\Gamma ({\mathbb {R}}_x^d) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}_{{\text {pol}}}C({\mathbb {R}}_\xi ^d)\). Lemma 5.6 implies that, for all \(\eta \in \Gamma \),
is a well-defined continous linear functional on \({\mathcal {S}}({\mathbb {R}}^d)\). Since \(e^{-\eta \cdot t}V^*_\psi F(t) = {f_\eta (t)}|_{{\mathcal {D}}({\mathbb {R}}^d)}\), we obtain that \(e^{-\eta \cdot t}V^*_\psi F(t) \in {\mathcal {S}}'({\mathbb {R}}^d)\) and that
Next, we show that \(V^*_\psi \) is continuous. Let \(K \in {\text {CCS}}(\Gamma )\) and \(B \in {\mathfrak {B}}({\mathcal {S}}({\mathbb {R}}^d))\) be arbitrary. Choose \(\varepsilon > 0\) so small that \(K_\varepsilon \in {\text {CCS}}(\Gamma )\). Lemma 5.6 implies that there is \(v \in {\overline{V}}({\mathcal {V}}_{{\text {pol}}})\) such that
for all \(\eta \in K\) and \(\varphi \in B\). Set \(w(\xi ) = v(\xi ) (1+ \left| \xi \right| )^{d+1} \in {\overline{V}}({\mathcal {V}}_{{\text {pol}}})\). Example 5.3 implies that, for all \(F \in C_\Gamma ({\mathbb {R}}_x^d) {\widehat{\otimes }}_\varepsilon {\mathcal {V}}_{{\text {pol}}}C({\mathbb {R}}_\xi ^d)\),
where
\(\square \)
We now combine Theorem 4.1 with the results from Sect. 4 to study the space \({\mathcal {S}}'(\Gamma )\). Let \(\emptyset \ne \Gamma \subseteq {\mathbb {R}}^d\) be open and convex and let \((K_N)_{N \in {\mathbb {N}}} \subset {\text {CCS}}(\Gamma )\) be such that \(K_N \subseteq K_{N+1}\) for all \(N \in {\mathbb {N}}\) and \(\Gamma = \bigcup _{N} K_N\). For \({\mathcal {W}} = (e^{h_{-K_N}})_{N \in {\mathbb {N}}}\) we set \({\mathcal {B}}'_\Gamma ({\mathbb {R}}^d) \mathrel {\mathop :}={\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d)\) and \({\mathcal {O}}'_{C, \Gamma }({\mathbb {R}}^d) = {\mathcal {O}}'_{C, {\mathcal {W}}}({\mathbb {R}}^d)\). Clearly, these definitions are independent of the chosen sequence \((K_N)_{N \in {\mathbb {N}}}\). We are ready to state and prove our main theorem.
Theorem 5.7
Let \(\emptyset \ne \Gamma \subseteq {\mathbb {R}}^d\) be open and convex. Then, \({\mathcal {S}}'(\Gamma ) = {\mathcal {B}}'_\Gamma ({\mathbb {R}}^d) = {\mathcal {O}}'_{C, \Gamma }({\mathbb {R}}^d)\) as locally convex spaces and \({\mathcal {S}}'(\Gamma )\) is an ultrabornological (PLS)-space.
Proof
Let \((K_N)_{N \in {\mathbb {N}}} \subset {\text {CCS}}(\Gamma )\) be such that \(K_N \subseteq K_{N+1}\) for all \(N \in {\mathbb {N}}\) and \(\Gamma = \bigcup _{N} K_N\). Set \({\mathcal {W}} = (e^{h_{-K_N}})_{N \in {\mathbb {N}}}\). Lemma 5.2 and Example 5.3 imply that \({\mathcal {W}}\) satisfies (2.1), (2.2) and (2.3). Hence, in view of the reconstruction formula (3.1), the topological identity \({\mathcal {S}}'(\Gamma ) = {\mathcal {O}}'_{C, \Gamma }({\mathbb {R}}^d)\) follows from Proposition 4.1 and Proposition 5.4. Since \({\mathcal {W}}\) also satisfies (4.1) (again by Lemma 5.2 and Example 5.3), the other statements are a direct consequence of Theorem 4.2. \(\square \)
Notes
To be precise, the spaces considered in [9], denoted there by \((\dot{{\mathcal {B}}}_{{\mathcal {W}}^\circ } ({\mathbb {R}}^d))'\) and \({\mathcal {O}}'_C({\mathcal {D}}, L^1_{{\mathcal {W}}})\), differ from \({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d)\) and \( {\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\) defined above. However, if \({\mathcal {W}}\) satisfies (2.1), (2.2) and (2.3), then \({\mathcal {B}}'_{{\mathcal {W}}}({\mathbb {R}}^d) = (\dot{{\mathcal {B}}}_{{\mathcal {W}}^\circ } ({\mathbb {R}}^d))'\) and \({\mathcal {O}}'_C({\mathcal {D}}, L^1_{{\mathcal {W}}}) = {\mathcal {O}}'_{C,{\mathcal {W}}}({\mathbb {R}}^d)\); the first equality is clear, while the second one follows from [9, Prop. 6.2]. Moreover, under these conditions, all statements and proofs from [9] remain valid if one replaces \(L^1_{{\mathcal {W}}}({\mathbb {R}}^d)\) by \({\mathcal {W}}C({\mathbb {R}}^d)\).
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Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). We thank N. Ortner for suggesting the topic of this paper and the anonymous referee for helpful comments. Nigsch acknowledges support by the Austrian Science Fund (FWF) Grants P26859 and P30233. Debrouwere was supported by FWO-Vlaanderen through the postdoctoral Grant 12T0519N.
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Debrouwere, A., Nigsch, E.A. On the space of Laplace transformable distributions. RACSAM 114, 185 (2020). https://doi.org/10.1007/s13398-020-00907-2
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DOI: https://doi.org/10.1007/s13398-020-00907-2