Commutators of singular integral operators satisfying a variant of a Lipschitz condition

Let T be a singular integral operator with its kernel satisfying |K(x - y) - ∑(k=1)(ℓ) ‍Bk (x)ϕk(y)| ≤ C| y| (γ)/|x - y| (n+γ), |x | > 2| y| > 0, where B(k) and ϕk (k = 1,…, ℓ) are appropriate functions and γ and C are positive constants. For b = b1'...,bm) with bj ∈ BMO(ℝ(n)), the multilinear commutator Tb generated by T and b is formally defined by T(b)ƒ(x) = ∫(ℝn)[∏(j=1)(m) (b(j)(x) - b(j)(y))] K(x, y)ƒ(y)dy. In this paper, the weighted L(p)-boundedness and the weighted weak type L log estimate for the multilinear commutator T(b) are established.