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

Buckley (1993) [3] proved the linear dependence parallel to M parallel to L2(Rn,w) <= c(n)A of the L-2(R-n, w)-norm for the Hardy-Littlewood maximal operator M on the classical A(2)-constant A =A(2)(w) = supQfQwfQw(-1) where the supremum is taken over all cubes with sides parallel to the axes. We prove in the case n = 1 that, for P-0 = 1 + root A-1/A < p <= 2, the dependence on the constant A is precisely preserved parallel to M parallel to(p)(L)((R,w)) <= c(p) [a/1 - p(2 - p)A](1/p-1) and it is impossible to decrease the value of p(0). Similar exact continuation theorems hold for the L-2-norm inequalities of weighted maximal operators. (C) 2012 Elsevier Inc. All rights reserved.