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

Abstract

Let Mf (x) = sup(1/2r) 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.[8] Stein, E. M., and Strömberg, J.-O. (1983). Behavior of maximal functions in R n for large n. Ark. Mat. 21, 259-269.

Random martingales and localization of maximal inequalities

“…This proves (20); the proof of ( 21) is identical. To prove ( 22) take an non-negative h ∈ L p (G), and observe the pointwise bound…”

“…(the case of the strong (p, p) norm of M, p > 1, when X = R, remains open, but we refer to [20,25] for some partial results).…”

“…Inequalities (121), (122), (123) show that the metric measure space (G, d G , µ G ) is Ahlfors-David n-regular. It remains to prove the estimates (20), (21), (22). By assertion (3) of Lemma 6.1 we can find f : X → R + with f L 1 (X) = 1, and λ > 0, such that…”