On an improved restricted reverse weak-type bound for the maximal operator

TL;DR

This paper improves the lower bounds for the restricted reverse weak-type estimate of the Hardy-Littlewood maximal operator and explores the equivalence of boundedness properties between the median and maximal operators on Banach spaces.

Contribution

It provides an improved lower bound for the restricted reverse weak-type estimate and establishes conditions under which median and maximal operators have equivalent boundedness on Banach spaces.

Findings

Improved lower bound for the restricted reverse weak-type estimate of M.

Conditions under which m_λ and M have equivalent boundedness.

Application of results to Banach function spaces.

Abstract

We obtain an improved lower bound for the restricted reverse weak-type estimate of the Hardy-Littlewood maximal operator $M$. This result is applied to the $\lambda$-median maximal operator $m_{\lambda}$ acting on a Banach function space $X$. We show that under certain assumptions on $X$, the boundedness properties of $m_{\lambda}$ and $M$ are equivalent.