Endpoint estimates for one-dimensional oscillatory integral operator

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

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