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

Paata Ivanisvili; Samuel Zbarsky

doi:10.1016/j.crma.2019.03.003

paper

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

arXiv:1807.04399 · doi:10.1016/j.crma.2019.03.003

Abstract

For $1 < p < \infty$ and $M$ the centered Hardy-Littlewood maximal operator on $R$ , we consider whether there is some $ε = ε (p) > 0$ such that $∥ M f ∥_{p} \geq (1 + ε) ∣∣ f ∣ ∣_{p}$ . We prove this for $1 < p < 2$ . For $2 \leq p < \infty$ , we prove the inequality for indicator functions and for unimodal functions.