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

TL;DR

This paper investigates lower bounds for the centered Hardy-Littlewood maximal operator on the real line, establishing new inequalities for certain ranges of p and specific classes of functions.

Contribution

It provides new lower bounds for the maximal operator on the real line, especially for 1<p<2 and for indicator and unimodal functions when p≥2.

Findings

Established a positive lower bound for 1<p<2.

Proved inequalities for indicator functions when p≥2.

Extended results to unimodal functions for p≥2.

Abstract

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 for unimodal functions.