On the space of Laplace transformable distributions
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On the space of Laplace transformable distributions Andreas Debrouwere1 · Eduard A. Nigsch2 Received: 19 February 2020 / Accepted: 13 July 2020 © The Author(s) 2020
Abstract We show that the space S () of Laplace transformable distributions, where ⊆ Rd is a non-empty convex open set, is an ultrabornological (PLS)-space. Moreover, we determine an explicit topological predual of S (). Keywords Laplace transform · Distributions · Ultrabornological (PLS)-spaces · Short-time Fourier transform Mathematics Subject Classification Primary 46F05 · 46A13 · Secondary 81S30
1 Introduction Schwartz introduced the space S () of Laplace transformable distributions as S () = { f ∈ D (Rd ) | e−ξ ·x f (x) ∈ S (Rdx ) ∀ξ ∈ },
where ⊆ Rd 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 (Rd ), 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,
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Eduard A. Nigsch [email protected] Andreas Debrouwere [email protected]
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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
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Institute for Analysis and Scientific Computing, TU Vienna, Wiedner Hauptstraße 8–10, 1040 Vienna, Austria 0123456789().: V,-vol
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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 Doma´nski 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 OC is ultrabornological. He proved that OC is isomorphic to a coms and verified directly that the latter space is plemented subspace of the sequence space s ⊗ ultrabornological. Much later, a different proof was given by Larcher and Wengenroth using homological methods [8]. The first author and Vindas [9] extended t
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