The best constant for centered Hardy-Littlewood maximal inequality

Abstract

We find the exact value of the best possible constant $C$ for the weak type $(1,1)$ inequality for the one dimensional centered Hardy-Littlewood maximal operator. We prove that $C$ is the largest root of the quadratic equation $12C^2-22C+5=0$ thus obtaining $C=\mathrm{1.5675208...}$. This is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated.

42 pages, published version