On a non-standard characterization of the $A_p$ condition

TL;DR

This paper presents a new characterization of the Muckenhoupt $A_p$ condition, linking it to strong weighted inequalities and providing examples of Banach spaces with specific maximal operator boundedness properties.

Contribution

It introduces an alternative characterization of the $A_p$ condition and explores its implications for weighted inequalities and Banach function spaces.

Findings

Strong weighted inequalities hold iff $w \\in A_p$

New examples of Banach spaces with specific maximal operator boundedness

Alternative $A_p$ characterization established

Abstract

The classical Muckenhoupt's $A_p$ condition is necessary and sufficient for the boundedness of the maximal operator $M$ on $L^p(w)$ spaces. In this paper we obtain another characterization of the $A_p$ condition. As a result, we show that some strong versions of the weighted $L^p(w)$ Coifman--Fefferman and Fefferman--Stein inequalities hold if and only if $w\in A_p$. We also give new examples of Banach function spaces $X$ such that $M$ is bounded on $X$ but not bounded on the associate space $X'$.