Refine search
Actions for selected content:
1 results
Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 03 November 2025, pp. 1-13
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
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.
