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The author studies the commutators generated by a suitable function a(x) on ℝⁿ and the oscillatory singular integral with rough kernel Ω(x)|x|ⁿ and polynomial phase, where Ω is homogeneous of degree zero on ℝⁿ, and a(x) is a BMO function or a Lipschitz function. Some mapping properties of these higher order commutators on $L^p\left(ℝⁿ\right)$, which are essential improvements of some well known results, are given.

This paper is devoted to investigating the properties of multilinear $A_{P⃗}$ conditions and $A_{(P⃗,q)}$ conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of $A_{P⃗}$ and $A_{(P⃗,q)}$ classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear $A_{(P⃗,q)}$ classes in terms of the linear $A_p$ classes are established. These results essentially improve and extend...