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| Denote by $T$ and $I_{lpha}$ the bilinear Calder'{o}n-Zygmund singular integral operator and bilinear fractional integral operator. In this paper, we give some characterizations of $mathrm{BMO}$ space via the high order commutators of the bilinear singular integral operator $T_{b}^{m}$ and the bilinear fractional integral operator $I_{lpha,b}^{m}$, respectively. More precisely, we prove that the corresponding commutators $T_{b}^{m}$ and $I_{lpha,b}^{m}$ are all bounded operators from $L^{p_{1}} imes L^{p_{2}}$ into $L^{p}$ if $binmathrm{BMO}$ for some suitable indexes $p_{1}$, $p_{2}$ and $p$. Conversely, $binmathrm{BMO}$ if the commutators $T_{b}^{m}$ and $I_{lpha,b}^{m}$ map $L^{p_{1}} imes L^{p_{2}}$ into $L^{p}$ for some suitable indexes $p_{1}$, $p_{2}$, $p$ and $m$ is an even integer. |
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| Keywords:High order commutator, Bilinear bilinear singular integral operator, Bilinear fractional integral operator, BMO space |
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