In this paper, the author studies a class of non-standard commutators with higher order remainders for oscillatory singular integral operators with phases more general than polynomials. For 1 〈 p 〈 ∞, the L^p-boundedness of such operators are obtained provided that their kernels belong to the spaces L^q（s^n-1） for some q 〉 1.
Let Q 2 = [0, 1]2 be the unit square in two dimension Euclidean space ?2. We study the L p boundedness properties of the oscillatory integral operators T α,β defined on the set S(?3) of Schwartz test functions f by $$ \mathcal{T}_{\alpha ,\beta } f(x,y,z) = \int_{Q^2 } {f(x - t,y - s,z - t^k s^j )e^{ - it^{ - \beta _1 } s^{ - \beta 2} } t^{ - 1 - \alpha _1 } s^{ - 1 - \alpha _2 } dtds} , $$ where β1 > α1 ? 0, β2 > α2 ? 0 and (k, j) ∈ ?2. As applications, we obtain some L p boundedness results of rough singular integral operators on the product spaces.
