Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): Lechao Xiao
The one-dimensional oscillatory integral operator associated to a real analytic phase S is given by
$T_{\lambda }f(x)=\underset{-\infty }{\overset{\infty }{\int }}{e}^{i\lambda S(x,y)}\chi (x,y)f(y)dy.$ In their fundamental work, Phong and Stein established sharp $L^2$ estimates for $T_{\lambda }$. The goal of this paper is to extend their results to all endpoints. In particular, we obtain a complete characterization for the mapping properties for $T_{\lambda }$ on $L^p(\mathbb{R})$. More precisely, we show that ${\Vert T_{\lambda }f\Vert }_p\lesssim |\lambda {|}^{-\alpha }{\Vert f\Vert }_p$ holds for some $\alpha >0$ if and only if $(\frac{1}{\alpha p},\frac{1}{\alpha p^{\prime }})$ lies in the reduced Newton polygon of S.