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In this paper, we prove that the weighted BMO space as followsAs an application, we characterize this space by the boundedness of the bilinear commutators [b, T ] j (j = 1, 2), generated by the bilinear convolution type Calderón-Zygmund operators and the symbol b, from1/p 2 and ω ∈ A 1 . Thus we answer the open problem proposed in [2] affirmatively.

Let I α be the bilinear fractional integral operator, B α be a more singular family of bilinear fractional integral operators and b = (b, b). Bényi et al. in [1] showeda compact operator. Also, the authors characterize the compactness of the iterated commutator [Π b, I α ] of bilinear fractional integral operator. More precisely, the commutator [Π b, I α ] is a compact operator if and only if b ∈ CMO.2010 Mathematics Subject Classification. Primary 42B20, 47B07; Secondary: 42B25,47G99.

Let m ∈ N and b = (b 1 , · · · , b m ) be a collection of locally integrable functions.As an application, we show that if the linear commutator of certain multilinear Calderón-Zygmund operator [Σ b, T ] is bounded from L p1 × · · · × L pm to L p with m i=1 1/p i = 1/p and 1 < p, p 1 , · · · , p m < ∞, then b 1 , · · · , b m ∈ BM O. Therefore, the result of Chaffee [2] (or Li and Wick [11]) is extended to the general case.

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