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A note on the best constant for uncentered maximal functions with measure on ℝ

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  12 September 2025

Zhipeng Wu  and
Qingdong Guo
Zhipeng Wu
Affiliation:
School of Mathematical Sciences, https://ror.org/00mcjh785 Xiamen University , Xiamen 361005, People’s Republic of China e-mail: wuzhp@stu.xmu.edu.cn
Qingdong Guo*
Affiliation:
School of Mathematics and Statistics, https://ror.org/00mcjh785 Xiamen University of Technology , Xiamen 361024, People’s Republic of China
*
e-mail: qingdongmath@stu.xmu.edu.cn
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Abstract

Let $M_\mu $ be the uncentered Hardy–Littlewood maximal operator with a Borel measure $\mu $ on $\mathbb {R}$. In this note, we verify that the norm of $M_\mu $ on $L^p(\mathbb {R},\mu )$ with $p\in (1,\infty )$ is just the upper bound $\theta _p$ obtained by Grafakos and Kinnunen and reobtain the norm of $M_\mu $ from $L^1(\mathbb {R},\mu )$ to $L^{1,\infty }(\mathbb {R},\mu )$. Moreover, the norm of the “strong” maximal operator $N_{\vec {\mu }}^{n}$ on $L^p(\mathbb {R}^n, \vec {\mu })$ is also given.

Keywords

Uncentered Hardy–Littlewood maximal functionbest constantBorel measure

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory 42B99: None of the above, but in this section

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 9
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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