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

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 result that the space OM\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {O}}_M$$\end{document} 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte für Mathematik Springer Journals

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

, Volume 198 (1) – May 1, 2022
30 pages

Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2022
ISSN
0026-9255
eISSN
1436-5081
DOI
10.1007/s00605-021-01664-z
Publisher site
See Article on Publisher Site

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 OM\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {O}}_M$$\end{document} 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.

Journal

Monatshefte für MathematikSpringer Journals

Published: May 1, 2022

Keywords: Ultrabornological (PLB)-spaces; Gelfand-Shilov spaces; Multiplier spaces; Short-time Fourier transform; Gabor frames; 46E10; 46A08; 46A13; 46A63

References

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