Weighted (PLB)-spaces of ultradifferentiable functions and multiplier spaces

We study weighted ( PLB )-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck’s classical resu...

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Bibliographic Details
Published inMonatshefte für Mathematik Vol. 198; no. 1; pp. 31 - 60
Main Authors Debrouwere, Andreas, Neyt, Lenny
Format Journal Article
LanguageEnglish
Published Vienna Springer Vienna 01.05.2022
Springer Nature B.V
Subjects
Mathematics
Mathematics and Statistics
Weighting functions
46A08
46E10
Gelfand-Shilov spaces
46A13
Ultrabornological (PLB)-spaces
Multiplier spaces
46A63
Gabor frames
Short-time Fourier transform
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Summary:We study weighted ( PLB )-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck’s classical result that the space O M of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-021-01664-z