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

Abstract

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

accepted manuscript