Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces
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| Format: | Preprint |
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2024
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| _version_ | 1866912820609679360 |
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| author | Hagelstein, Paul Kosz, Dariusz Stempak, Krzysztof |
| author_facet | Hagelstein, Paul Kosz, Dariusz Stempak, Krzysztof |
| contents | 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 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 $\pm\infty$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06734 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces Hagelstein, Paul Kosz, Dariusz Stempak, Krzysztof Classical Analysis and ODEs 42B25, 26A45 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 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 $\pm\infty$. 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. |
| title | Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces |
| topic | Classical Analysis and ODEs 42B25, 26A45 |
| url | https://arxiv.org/abs/2407.06734 |