A restriction theorem for oscillatory integral operator with certain polynomial phase

We consider the oscillatory integral operator Tα,mf(x)=∫ℝnei(x1α1y1m+⋯x1αnynm)f(y)dy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T_{\alpha ,m}}f\left( x \right) = {\smallint _{{\mathbb{R}^n}}}{e^i}\left( {x_1^{{\alpha _1}}y_1^m + \cdots x_1^{{\alpha _n}}y_n^m} \right)f\left( y \right)dy$$\end{document} where the function f is a Schwartz function. In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.