Lower bounds for the centered Hardy-Littlewood maximal operator on the real line

TL;DR

This paper establishes a quantitative lower bound for the centered Hardy-Littlewood maximal operator on the real line, showing it amplifies the L^p norm of functions by at least a fixed factor greater than one.

Contribution

It proves the existence of a positive epsilon depending on p, providing a new lower bound for the operator's norm on L^p spaces, which was previously unknown.

Findings

Existence of epsilon_p > 0 for all 1 < p < ∞

Lower bound for the L^p norm of the maximal operator

Quantitative improvement over the trivial bound

Abstract

Let $1<p<\infty$. We prove that there exists an $\varepsilon_p>0$ such that for each $f\in L^p(\mathbb{R})$, the centered Hardy-Littlewood maximal operator $M$ on $\mathbb{R}$ satisfies the lower bound $\|Mf\|_{L^p(\mathbb{R})}\ge (1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$.