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Published online by Cambridge University Press: 12 September 2025
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.