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

Let $M$ denote the centered Hardy–Littlewood operator on $\mathbb{R}$. We prove that $$\mathrm{V}\mathrm{a}\mathrm{r}(Mf)\le \mathrm{V}\mathrm{a}\mathrm{r}(f)-\frac{1}{2}\mid \mathrm{\mid }f(\mathrm{\infty })\mathrm{\mid }-\mathrm{\mid }f(-\mathrm{\infty })\mathrm{\mid }\mid$$ for piecewise constant functions $f$ with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt \cite{BW} proved for indicator functions of bounded variation vanishing at $\pm \mathrm{\infty }$. We conjecture that the inequality holds for all functions of bounded variation, representing a stronger version of the existing conjecture $\mathrm{V}\mathrm{a}\mathrm{r}(Mf)\le \mathrm{V}\mathrm{a}\mathrm{r}(f)$. We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.