Oscillatory integrals on unit square along surfaces

Let Q2 = [0, 1]2 be the unit square in two-dimensional Euclidean space ℝ2. We study the Lp boundedness of the oscillatory integral operator Tα,β defined on the set ℒ(ℝ2+n) of Schwartz test functions by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_{\alpha ,\beta } f(u,v,x) = \int_{Q^2 } {\frac{{f(u - t,v - s,x - \gamma (t,s))}} {{t^{1 + \alpha _1 } s^{1 + \alpha _2 } }}} e^{it - \beta _{1_s } - \beta _2 } dtds, $$\end{document} where x ∈ ℝn, (u, v) ∈ ℝ2, (t, s, γ(t, s)) = (t, s, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t^{p_1 } s^{q_1 } ,t^{p_2 } s^{q_2 } ,...,t^{p_n } s^{q_n } $$\end{document}) is a surface on ℝn+2, and β1 > α1, β2 > α2. Our results extend some known results on ℝ3.