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For $1<p<\infty$ and $M$ the centered Hardy-Littlewood maximal operator on$\mathbb{R}$, we consider whether there is some $\varepsilon=\varepsilon(p)>0$such that $\|Mf\|_p\ge (1+\varepsilon)||f||_p$. We prove this for $1<p<2$. For$2\le p<\infty$, we prove the inequality for indicator functions and forunimodal functions.

Centered Hardy--Littlewood maximal operator on the real line: lower bounds