Necessary and sufficient conditions for boundedness of commutators of bilinear Hardy-Littlewood maximal function

TL;DR

This paper characterizes weighted BMO spaces through various commutators of the bilinear Hardy-Littlewood maximal function, establishing conditions for their boundedness.

Contribution

It provides new characterizations of weighted BMO spaces via multiple types of commutators of the bilinear Hardy-Littlewood maximal function.

Findings

Characterizations of weighted BMO space using commutators.

Boundedness conditions for different commutators.

Connections between commutator boundedness and BMO space.

Abstract

Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of several different commutators of bilinear Hardy-Littlewood maximal function, respectively; these commutators include the maximal iterated commutator $\mathcal{M}_{\Pi \vec{b}}$, the maximal linear commutator $\mathcal{M}_{\Sigma\vec{b}}$, the iterated commutator $[\Pi \vec{b},\mathcal{M}]$ and the linear commutator $[\Sigma \vec{b},\mathcal{M}]$.