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In this paper, we study the behaviors of the commutators ~b, T_ generated by multilinear fractional Calderón-Zygmund operators T_γ with vec b= (b₁, …, bₘ) ∈ (Lₗoc¹) ᵐ on weighted Hardy spaces. Generally, for vec b∈ (BMO) ᵐ, ~b, ~T_ may be not bounded from the product Hardy spaces to Lebesgue spaces. We show that for some pᵢ∈ (0, 1 with 1/q=1/p₁+·s+1/pₘ-γ/n, ωi∈ RHqᵢ/pᵢand bᵢ∈ mathcal BMOωi, pᵢ which are a class of non-trivial subspaces of rm BMO (bounded mean oscillation), i=1, …, m, the commutators ~b, T_ are bounded from Hᵖ₁ω1
Chen et al. (Fri,) studied this question.