ON A DISCRETE VERSION OF TANAKA'S THEOREM FOR MAXIMAL FUNCTIONS

In this paper we prove a discrete version of Tanaka's theorem for the Hardy-Littlewood maximal operator in dimension n = 1, both in the non-centered and centered cases. For the non-centered maximal operator (M) over tilde we prove that, given a function f : Z -> R of bounded variation, Var((M) over tilde (f)) <= Var(f), where Var(f) represents the total variation of f. For the centered maximal Operator M we prove that, given a function f : Z -> R such that f is an element of l(1)(Z), Var(Mf) <= C parallel to f parallel to(l1(z)). This provides a positive solution to a question of Hajlasz and Onninen in the discrete one-dimensional case.