On a Discrete Version of Tanaka’s Theorem for Maximal Functions

In this paper we prove a discrete version of Tanaka’s Theorem [19] for the Hardy-Littlewood maximal operator in dimension n = 1, both in the non-centered and centered cases. For the non-centered maximal operator f M we prove that, given a function f : Z→ R of bounded variation, Var(f Mf) ≤ 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 ∈ `1(Z), Var(Mf) ≤ C‖f‖`1(Z). This provides a positive solution to a question of Haj lasz and Onninen [6] in the discrete one-dimensional case.