Certain oscillatory integrals on unit square and their applications

Let Q2 = [0, 1]2 be the unit square in two dimension Euclidean space ℝ2. We study the Lp boundedness properties of the oscillatory integral operators Tα,β defined on the set S(ℝ3) of Schwartz test functions f by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{T}_{\alpha ,\beta } f(x,y,z) = \int_{Q^2 } {f(x - t,y - s,z - t^k s^j )e^{ - it^{ - \beta _1 } s^{ - \beta 2} } t^{ - 1 - \alpha _1 } s^{ - 1 - \alpha _2 } dtds} , $$\end{document} where β1 > α1 ⩾ 0, β2 > α2 ⩾ 0 and (k, j) ∈ ℝ2. As applications, we obtain some Lp boundedness results of rough singular integral operators on the product spaces.