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

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