Fractional Maximal Operator on Musielak–Orlicz Spaces Over Unbounded Quasi-Metric Measure Spaces

被引:0
作者
Yoshihiro Sawano
Tetsu Shimomura
机构
[1] Chuo University,Department of Mathematics
[2] Hiroshima University, Department of Mathematics, Graduate School of Humanities and Social Sciences
来源
Results in Mathematics | 2021年 / 76卷
关键词
Fractional maximal operator; Musielak–Orlicz space; metric measure space; Primary 42B25; Secondary 42B35; 46E30;
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摘要
The main target of this article is the boundedness of the fractional maximal operator [inline-graphic not available: see fulltext], on Musielak–Orlicz spaces LΦ(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\Phi }(X)$$\end{document} over unbounded quasi-metric measure spaces as an extension of recent results by Cruz-Uribe and Shukla (Studia Math 242(2):109–139, 2018) and the authors (2019), where η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} is the order of the fractional maximal operator and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is its modification rate. Our results are new even for the Hardy–Littlewood maximal operator Mλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\lambda }$$\end{document} or for the Orlicz spaces Lp(·)(logL)q(·)(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p(\cdot )}(\log L)^{q(\cdot )}(X)$$\end{document}. Usually, for the proof of the boundedness, the three-line theorem is employed. This new technique of using the three-line theorem enables us to extend the function spaces with ease. An example explains why we can not remove the modification parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}.
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