Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00090-x ORIGINAL PAPER
Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting Chiara Boiti1 · David Jornet2 · Alessandro Oliaro3 · Gerhard Schindl4 Received: 17 April 2020 / Accepted: 7 July 2020 © The Author(s) 2020
Abstract We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending the previous work by Langenbruch. As a consequence, we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions. Keywords Weight matrices · Ultradifferentiable functions · Sequence spaces · Nuclear spaces Mathematics Subject Classification 46A04 · 46A45 · 26E10
Communicated by Jose Bonet. * Chiara Boiti [email protected] David Jornet [email protected] Alessandro Oliaro [email protected] Gerhard Schindl [email protected] 1
Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli n. 30, 44121 Ferrara, Italy
2
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, Camino de Vera, s/n, 46071 Valencia, Spain
3
Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto n. 10, 10123 Torino, Italy
4
Fakultät für Mathematik, Universität Wien, Oskar‑Morgenstern‑Platz n. 1, 1090 Wien, Austria
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C. Boiti et al.
1 Introduction The systematic study of nuclear locally convex spaces began in 1951 with the fun‑ damental dissertation of Grothendieck [20] to classify those infinite dimensional locally convex spaces which are not normed, suitable for mathematical analysis. Among the properties of a nuclear space, the existence of a Schwartz kernel for a continuous linear operator on the space is of crucial importance for the theory of linear partial differential operators. In our setting of ultradifferentiable functions, this fact helps, for instance, to study the behaviour (propagation of singularities or wave front sets) of a differential or pseudodifferential operator when acting on a distribution. See, for example, [1, 7, 16, 17, 33, 38] and the references therein. Since the middle of the last century, several authors have studied the topo‑ logical structure of global spaces of ultradifferentiable functions and, in particu‑ lar, when the spaces are nuclear. See [31], or the book [19]. More recently, the first three authors in [9] used the isomorphism established by Langenbruch [28] between global spaces of ultradifferentiable functions in the sense of Gel’fand and Shilov [18] and some sequence spaces to see that under the condition that appears in [11, Corollary 16(3)] on the weight function 𝜔 (as in [12]) the space S(𝜔) (ℝd ) of rapidly decreasing ultradifferentiable functions of Beurling type in the sense of Björck [3] is nuclea
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