A note on the best constant for uncentered maximal functions with measure on <i>ℝ</i>

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.