Restriction Theorem for Oscillatory Integral Operator with Certain Polynomial Phase

TL;DR

This paper establishes a restriction theorem for a class of oscillatory integral operators with polynomial phases, providing conditions under which the theorem holds, advancing understanding in harmonic analysis.

Contribution

It introduces a restriction theorem for a specific oscillatory integral operator with polynomial phase and identifies necessary conditions for its validity.

Findings

Established a restriction theorem for the operator on 1 sphere

Derived necessary conditions for the restriction theorem to hold

Extended harmonic analysis techniques to polynomial phase oscillatory integrals

Abstract

We consider the following oscillatory integral operator \begin{equation}\label{opera-defi-1} T_{\alpha,m}f(x)=\int_{\mathbb R^n}e^{i(x_1^{\alpha_1} y_1^m+\cdots+x_n^{\alpha_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.