A Search for Best Constants in the Hardy-Littlewood Maximal Theorem

Abstract

Let $Mf(x) = \sup(1/2r)\int^{x+r}_{x-r} |f(t)|\;dt$ be the centered maximal operator on the line. Through a numerical search procedure, we have conjectural best constants for the weak-type 1-1 estimate (3/2) and the L^p estimate (the constant B(p,1) such that $M(|x|^{-1/p}) = B(p,1)|x|^{-1/p}).$ We prove that these constants are lower bounds for the best constants and discuss the numerical evidence for the conjectures.