Quantitative BMO-BLO Estimates for the Hardy-Littlewood Maximal Function
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918218643275776 |
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| author | Claros, Alejandro |
| author_facet | Claros, Alejandro |
| contents | In this note, we study a quantitative extension of the John-Nirenberg inequality for the Hardy-Littlewood maximal function of a $\operatorname{BMO}$ function. More precisely, for every nonconstant locally integrable function $f$ such that $Mf$ is not identically infinite, we prove the inequality \begin{equation*}
\left( \frac{1}{w(Q)}\int_Q \left( \frac{Mf(x) - \operatorname{ess\,inf}_{Q} Mf }{M^\# f(x)} \right)^p w(x)\,dx\right)^\frac{1}{p} \le c_n \, [w]_{A_\infty}\, p \end{equation*} for every cube $Q$, every $1\le p<\infty$ and every weight $w\in A_\infty$, where $[w]_{A_\infty}$ denotes the Fujii-Wilson $A_\infty$ constant. This result extends the classical boundedness $\|Mf\|_{\operatorname{BLO}}\le C_n\|f\|_{\operatorname{BMO}}$ proved by Bennett, DeVore, and Sharpley (Ann. of Math. (2) 113 (1981)) and by Bennett (Proc. Amer. Math. Soc. 85 (1982)). Furthermore, we show that the class $A_\infty$ is both necessary and sufficient for this inequality to hold, providing a new characterization of $A_\infty$ in terms of the action of the maximal operator on bounded oscillation spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative BMO-BLO Estimates for the Hardy-Littlewood Maximal Function Claros, Alejandro Classical Analysis and ODEs Functional Analysis In this note, we study a quantitative extension of the John-Nirenberg inequality for the Hardy-Littlewood maximal function of a $\operatorname{BMO}$ function. More precisely, for every nonconstant locally integrable function $f$ such that $Mf$ is not identically infinite, we prove the inequality \begin{equation*} \left( \frac{1}{w(Q)}\int_Q \left( \frac{Mf(x) - \operatorname{ess\,inf}_{Q} Mf }{M^\# f(x)} \right)^p w(x)\,dx\right)^\frac{1}{p} \le c_n \, [w]_{A_\infty}\, p \end{equation*} for every cube $Q$, every $1\le p<\infty$ and every weight $w\in A_\infty$, where $[w]_{A_\infty}$ denotes the Fujii-Wilson $A_\infty$ constant. This result extends the classical boundedness $\|Mf\|_{\operatorname{BLO}}\le C_n\|f\|_{\operatorname{BMO}}$ proved by Bennett, DeVore, and Sharpley (Ann. of Math. (2) 113 (1981)) and by Bennett (Proc. Amer. Math. Soc. 85 (1982)). Furthermore, we show that the class $A_\infty$ is both necessary and sufficient for this inequality to hold, providing a new characterization of $A_\infty$ in terms of the action of the maximal operator on bounded oscillation spaces. |
| title | Quantitative BMO-BLO Estimates for the Hardy-Littlewood Maximal Function |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2511.18949 |