# Local Fourier spaces and weighted Beurling density

Local Fourier spaces and weighted Beurling density We consider Banach spaces of functions or distributions on Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^d$$\end{document} for which the norm is defined in terms of a weighted Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-norm of the Fourier transform of the elements and the weight w in question is assumed to be tempered and moderate. We study in particular subspaces of these spaces obtained by taking the closure in the corresponding norm of the test functions with compact support in a fixed open subset U of Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^d$$\end{document}, usually assumed to be bounded. We consider weighted inequalities involving the Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-norm of the Fourier transform of the elements of the subspace with respect to a positive Borel measure μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu$$\end{document} on Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^d$$\end{document} and the original norm defined on the subspace. We obtain, in particular, an exact characterization for these inequalities to hold in the case where U is a ball with a small enough radius using a suitable weighted version of the Beurling density. Exploiting duality, we then use these results to characterize the positive Borel measures μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu$$\end{document} having the property that the inverse Fourier transform of any measure Fdμ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F\,d\mu$$\end{document}, where F∈Lq(μ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F\in L^q(\mu )$$\end{document}, agrees on any open ball B of sufficiently small radius with the inverse Fourier transform of a tempered function g, where g∈Lq(w~)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g\in L^q({\tilde{w}})$$\end{document}, for a weight w~\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\tilde{w}}$$\end{document} related to w and, if it is the case, we also obtain a necessary and sufficient condition for the associated mapping F-1Lq(μ)|B→F-1Lq(w~)|B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {F}}^{-1} L^q(\mu )|_B \rightarrow {\mathcal {F}}^{-1} L^q({\tilde{w}})|_B$$\end{document} to be surjective. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Operator Theory Springer Journals

# Local Fourier spaces and weighted Beurling density

, Volume 5 (3) – Jul 5, 2020
32 pages

/lp/springer-journals/local-fourier-spaces-and-weighted-beurling-density-VCV9slAtKo
Publisher
Springer Journals
ISSN
2662-2009
eISSN
2538-225X
DOI
10.1007/s43036-020-00086-2
Publisher site
See Article on Publisher Site

### Abstract

We consider Banach spaces of functions or distributions on Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^d$$\end{document} for which the norm is defined in terms of a weighted Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-norm of the Fourier transform of the elements and the weight w in question is assumed to be tempered and moderate. We study in particular subspaces of these spaces obtained by taking the closure in the corresponding norm of the test functions with compact support in a fixed open subset U of Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^d$$\end{document}, usually assumed to be bounded. We consider weighted inequalities involving the Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-norm of the Fourier transform of the elements of the subspace with respect to a positive Borel measure μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu$$\end{document} on Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^d$$\end{document} and the original norm defined on the subspace. We obtain, in particular, an exact characterization for these inequalities to hold in the case where U is a ball with a small enough radius using a suitable weighted version of the Beurling density. Exploiting duality, we then use these results to characterize the positive Borel measures μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu$$\end{document} having the property that the inverse Fourier transform of any measure Fdμ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F\,d\mu$$\end{document}, where F∈Lq(μ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F\in L^q(\mu )$$\end{document}, agrees on any open ball B of sufficiently small radius with the inverse Fourier transform of a tempered function g, where g∈Lq(w~)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g\in L^q({\tilde{w}})$$\end{document}, for a weight w~\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\tilde{w}}$$\end{document} related to w and, if it is the case, we also obtain a necessary and sufficient condition for the associated mapping F-1Lq(μ)|B→F-1Lq(w~)|B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {F}}^{-1} L^q(\mu )|_B \rightarrow {\mathcal {F}}^{-1} L^q({\tilde{w}})|_B$$\end{document} to be surjective.