Abstract
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 \({\mathcal {O}}_M\) 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.
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Communicated by Karlheinz Gröchenig.
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A. Debrouwere was supported by FWO-Vlaanderen through the postdoctoral Grant 12T0519N.
L. Neyt gratefully acknowledges support by FWO-Vlaanderen through the postdoctoral Grant 12ZG921N.
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Debrouwere, A., Neyt, L. Weighted (PLB)-spaces of ultradifferentiable functions and multiplier spaces. Monatsh Math 198, 31–60 (2022). https://doi.org/10.1007/s00605-021-01664-z
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DOI: https://doi.org/10.1007/s00605-021-01664-z
Keywords
- Ultrabornological (PLB)-spaces
- Gelfand-Shilov spaces
- Multiplier spaces
- Short-time Fourier transform
- Gabor frames