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

F. J. Pérez Lázaro

Published 2019-08-22Version 1

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 $\|Mf\|_{L^p(\mathbb{R})}\ge (1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$.