Norm of the Hausdorff Operator on the Real Hardy Space H1(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1({{\mathbb {R}}})$$\end{document}

被引：0

作者：

Ha Duy Hung

Luong Dang Ky

Thai Thuan Quang

机构：

[1] Ton Duc Thang University,Applied Analysis Research Group, Faculty of Mathematics and Statistics

[2] Quy Nhon University,Department of Mathematics

来源：

Complex Analysis and Operator Theory | 2018年 / 12卷 / 1期

关键词：

Hausdorff operator; Hardy space; Hilbert transform; Maximal function; Holomorphic function; 47B38; 42B30;

D O I：

10.1007/s11785-017-0651-y

中图分类号：

学科分类号：

摘要：

Let φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} be a nonnegative integrable function on (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}. It is well-known that the Hausdorff operator Hφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {H}}}_\varphi $$\end{document} generated by φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} is bounded on the real Hardy space H1(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1({{\mathbb {R}}})$$\end{document}. The aim of this paper is to give the exact norm of Hφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {H}}}_\varphi $$\end{document}. More precisely, we prove that ‖Hφ‖H1(R)→H1(R)=∫0∞φ(t)dt.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {H}}}_\varphi \Vert _{H^1({{\mathbb {R}}})\rightarrow H^1({{\mathbb {R}}})}= {\mathop {\int }\limits _{0}^{\infty }} \varphi (t)dt. \end{aligned}$$\end{document}

引用

页码：235 / 245

页数：10

共 18 条

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