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

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 $[\Sigma \vec{b},\mathcal{M}]$.