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Mellin integral transforms and generalized Lipschitz and Zygmund spaces

Mellin integral transforms and generalized Lipschitz and Zygmund spaces Let c∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ c\in {\mathbb {R}} $$\end{document} and Xc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ X_{c} $$\end{document} be the set of functions f:(0,+∞)→C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ f: (0,+\infty )\rightarrow {\mathbb {C}} $$\end{document} such that f(·)(·)c-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ f(\cdot )(\cdot )^{c-1} $$\end{document} is integrable in the Lebesgue’s sense over (0,+∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ (0,+\infty ) $$\end{document}. The Mellin integral transformation of f∈Xc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ f \in X_{c} $$\end{document} is given by M[f](s):=∫0+∞us-1f(u)du,s=c+it,t∈R.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} {\mathcal {M}}[f](s):=\int _{0}^{+\infty }u^{s-1}f(u)du, \;\; s=c+it, \; t \in {\mathbb {R}}. \end{aligned}$$\end{document}The aim of this research is to study the smoothness property of f with an absolutely convergent Mellin transform. More precisely, we study the order of magnitude of the Mellin transform for complex-valued functions belonging to the generalized Mellin Lipschitz and Zygmund classes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Pseudo-Differential Operators and Applications Springer Journals

Mellin integral transforms and generalized Lipschitz and Zygmund spaces

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References (31)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
1662-9981
eISSN
1662-999X
DOI
10.1007/s11868-023-00517-7
Publisher site
See Article on Publisher Site

Abstract

Let c∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ c\in {\mathbb {R}} $$\end{document} and Xc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ X_{c} $$\end{document} be the set of functions f:(0,+∞)→C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ f: (0,+\infty )\rightarrow {\mathbb {C}} $$\end{document} such that f(·)(·)c-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ f(\cdot )(\cdot )^{c-1} $$\end{document} is integrable in the Lebesgue’s sense over (0,+∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ (0,+\infty ) $$\end{document}. The Mellin integral transformation of f∈Xc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ f \in X_{c} $$\end{document} is given by M[f](s):=∫0+∞us-1f(u)du,s=c+it,t∈R.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} {\mathcal {M}}[f](s):=\int _{0}^{+\infty }u^{s-1}f(u)du, \;\; s=c+it, \; t \in {\mathbb {R}}. \end{aligned}$$\end{document}The aim of this research is to study the smoothness property of f with an absolutely convergent Mellin transform. More precisely, we study the order of magnitude of the Mellin transform for complex-valued functions belonging to the generalized Mellin Lipschitz and Zygmund classes.

Journal

Journal of Pseudo-Differential Operators and ApplicationsSpringer Journals

Published: Jun 1, 2023

Keywords: Mellin transforms; Mellin translation operator; Lipschitz classes; Zygmund classes; Boas’ theorems; 33D15; 33E30; 44A05

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