Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces

Paul Hagelstein, Dariusz Kosz, Krzysztof Stempak

Let $M$ denote the centered Hardy--Littlewood operator on $\mathbb{R}$. We prove that \[ {\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big| \] for piecewi…