Endpoint estimates for one-dimensional oscillatory integral operator

TL;DR

This paper characterizes the boundedness of one-dimensional oscillatory integral operators on L^p spaces using the geometry of the phase function's Newton polygon, providing sharp endpoint estimates.

Contribution

It provides a complete characterization of L^p mapping properties of oscillatory integral operators based on the Newton polygon of the phase function.

Findings

Boundedness characterized by Newton polygon geometry

Sharp estimates occur on the Newton diagram

Complete endpoint mapping properties established

Abstract

The one-dimensional oscillatory integral operator associated to a real analytic phase $S$ is given by $$ T_\lambda f(x) =\int_{-\infty}^\infty e^{i\lambda S(x,y)} \chi(x,y) f(y) dy. $$ In this paper, we obtain a complete characterization for the mapping properties of $T_\lambda $ on $L^p(\mathbb R)$ spaces, namely we prove that $\|T_\lambda\|_p \lesssim |\lambda|^{-\alpha}\|f\|_p$ for some $\alpha>0$ if and only if the point $(\frac 1 {\alpha p} , \frac 1 {\alpha p'})$ lies in the reduced Newton polygon of $S$, and this estimate is sharp if and only if it lies on the reduced Newton diagram.