R.O. Dror, S. Ganguli, and R.S. Strichartz. A search for best constants in the Hardy-Littlewood maximal theorem. Journal of Fourier Analysis and Applications 2:473-86, 1996.
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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.