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

Abstract Let $M$ denote the centered Hardy–Littlewood operator on $\mathbb{R}$ . We prove that \begin{equation*}{\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big|,\end{equation*} for piecewise constant functions $f$ with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt [3] proved for indicator functions of bounded variation vanishing at $\pm\infty$ . We conjecture that the inequality holds for all functions of bounded variation, representing a stronger version of the existing conjecture ${\rm Var} (Mf)\le {\rm Var} (f)$ . We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.