The norm of the centered Hardy--Littlewood maximal operator

We determine the exact $L^p(\mathbb{R})$ norm of the centered Hardy–Littlewood maximal operator for every $1<p<\mathrm{\infty }$, proving that it equals $p\mathrm{/}(p-1)$. This settles the sharp strong-type problem for the centered maximal operator on the real line. For every $p$ in this range, the norm is strictly larger than the constant associated with the critical power $\mathrm{\mid }x{\mathrm{\mid }}^{-1\mathrm{/}p}$, thereby disproving the conjecture of Dror, Ganguli, and Strichartz that this profile determines the unrestricted norm.