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

TL;DR

The paper advances the understanding of the restricted reverse weak-type behavior of the Hardy--Littlewood maximal operator by establishing an improved lower bound for the restricted constant $C_n(\lambda)$ that holds for all $\lambda\in(0,1)$. This bound is then leveraged to analyze the local $\lambda$-median maximal operator $m_{\lambda}$ on Banach function spaces, demonstrating that boundedness of $m_{\lambda}$ for some $\lambda$ implies boundedness of the classical maximal operator $M$ on a dilated space $X^{1/p}$ for some $p>1$. An iteration-based framework together with a halo-type enlargement is developed to transfer boundedness from $m_{\lambda}$ to $M$, and a duality/regularity criterion is provided: if $m_{\lambda}$ is bounded on $X$ and on $X'$ for some $\lambda$, then under suitable assumptions $M$ is bounded on $X$. The results intersect with $A_p$-regularity conjectures and offer a strategy to deduce $M$-boundedness from $m_{\lambda}$-boundedness across a broad class of 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 $λ$-median maximal operator $m_λ$ acting on a Banach function space $X$. We show that under certain assumptions on $X$, the boundedness properties of $m_λ$ and $M$ are equivalent.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4: N26
• Corollary 1.5
• Theorem 2.1: R15
• Theorem 2.2: R14
• Lemma 3.1
• proof : Proof of Theorem \ref{['mr']}
• Lemma 4.1