Quantitative BMO-BLO Estimates for the Hardy-Littlewood Maximal Function

Abstract: 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.