Abstract
We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space \(\mathcal {S}_{(M_p)}\) to be nuclear. As a consequence, we obtain that for a weight function ω satisfying the mild condition: 2ω(t) ≤ ω(Ht) + H for some H > 1 and for all t ≥ 0, the space \(\mathcal {S}_\omega \) in the sense of Björck is also nuclear.
Dedicated to Prof. Luigi Rodino on the occasion of his 70th birthday.
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Acknowledgements
We are grateful to Prof. Gerhard Schindl for pointing out that (M2)′ is equivalent to (3), under (M1).
The authors were partially supported by the Projects FAR 2017, FAR 2018 and FIR 2018 (University of Ferrara), FFABR 2017 (MIUR). The research of the second author was partially supported by the project MTM2016-76647-P. The first and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Boiti, C., Jornet, D., Oliaro, A. (2020). About the Nuclearity of \(\mathcal {S}_{(M_{p})}\) and \(\mathcal {S}_{\omega }\) . In: Boggiatto, P., et al. Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36138-9_6
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DOI: https://doi.org/10.1007/978-3-030-36138-9_6
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