The best constant for the 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 12C2 - 22C + 5 = 0 thus obtaining C = 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.