On a discrete version of Tanaka’s theorem for maximal functions

Source abstract

In this paper we prove a discrete version of Tanaka’s theorem for the Hardy-Littlewood maximal operator in dimension n = 1 n=1 , both in the non-centered and centered cases. For the non-centered maximal operator M ~ \widetilde {M} we prove that, given a function f : Z → R f: \mathbb {Z} \to \mathbb {R} of bounded variation, $$Var ⁡ ( M ~ f ) ≤ Var ⁡ ( f ) , \operatorname {Var}(\widetilde {M} f) \leq \operatorname {Var}(f),$$ where Var ⁡ ( f ) \operatorname {Var}(f) represents the total variation of f f . For the centered maximal operator M M we prove that, given a function f : Z → R f: \mathbb {Z} \to \mathbb {R} such that f ∈ ℓ 1 ( Z ) f \in \ell ^1(\mathbb {Z}) , $$Var ⁡ ( M f ) ≤ C ‖ f ‖ ℓ 1 ( Z ) . \operatorname {Var}(Mf) \leq C \|f\|_{\ell ^1(\mathbb {Z})}.$$ This provides a positive solution to a question of Hajłasz and Onninen in the discrete one-dimensional case.