The exact dependence on for the maximal inequalities

Buckley (1993) [3] proved the linear dependence ‖ M ‖ L 2 ( R n , 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 ) = sup Q ⨏ Q w ⨏ Q w − 1 where the supremum is taken over all cubes with sides parallel to the axes. ve in the case n = 1 that, for p 0 = 1 + A − 1 A < p ≤ 2 , the dependence on the constant A is precisely preserved ‖ M ‖ L p ( R , w ) ≤ c ( p ) [ A 1 − p ( 2 − p ) A ] 1 p − 1 and it is impossible to decrease the value of p 0 . r exact continuation theorems hold for the L 2 -norm inequalities of weighted maximal operators.