Restriction Theorem for Oscillatory Integral Operator with Certain Polynomial Phase

Abstract

We consider the following oscillatory integral operator \begin{equation}\label{opera-defi-1} T_{α,m}f(x)=\int_{\mathbb R^n}e^{i(x_1^{α_1} y_1^m+\cdots+x_n^{α_n} y_n^m)}f(y)dy, \end{equation} where the function $f$ is a Schwartz function. In this paper, the restriction theorem on ${\mathbb{S}}^{n-1}$ for this operator is obtained. Moreover, we obtain a necessary condition which ensures the restriction theorem hold.