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A note on the best constant for uncentered maximal functions with measure on ℝ
Published online by Cambridge University Press: 12 September 2025
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.
MSC classification
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
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