The exact dependence on A2(w)A2(w) for the Lp(R,w)Lp(R,w) maximal inequalities

Buckley (1993) [3] proved the linear dependence ‖M‖L2(Rn,w)≤c(n)A of the L2(Rn,w)L2(Rn,w)-norm for the Hardy–Littlewood maximal operator MM on the classical A2A2-constant A=A2(w)=supQ⨏Qw⨏Qw−1 where the supremum is taken over all cubes with sides parallel to the axes.We prove in the case n=1n=1 that, for p0=1+A−1A