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We study the compactness of commutator of a locally integrable function and an oscillatory singular integral operator defined by a real-valued polynomial and a kernel function satisfying the Hölder condition. We show that such a commutator is compact from L p {L^{p}} to L q {L^{q}} if and only if the locally integrable function is the sum of a constant and a function in certain Lebesgue space. [ABSTRACT FROM AUTHOR]

Let T be a bilinear Calderón–Zygmund singular integral operator and let T * {T^{*}} be its corresponding truncated maximal operator. For any b ∈ BMO ⁡ (ℝ n) {b\in\operatorname{BMO}({\mathbb{R}^{n}})} and b → = (b 1 , b 2) ∈ BMO ⁡ (ℝ n) × BMO ⁡ (ℝ n) {\vec{b}=(b_{1},b_{2})\in\operatorname{BMO}({\mathbb{R}^{n}})\times% \operatorname{BMO}({\mathbb{R}^{n}})} , let T b , j * {T^{*}_{b,j}} ( j = 1 , 2 {j=1,2}) and T b → * {T^{*}_{\vec{b}}} be the commutators in the j-th entry and the iterated commutators of T * {T^{*}} , respectively. In this paper, for all 1 < p 1 , p 2 < ∞ {1