The best constant for centered Hardy-Littlewood maximal inequality


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 $C=1.5675208\ldots\,$ . This is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated.


Antonios D. Melas

Department of Mathematics, University of Athens, Athens, Greece