Endpoint estimates for one-dimensional oscillatory integral operator

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.