Sharp variational estimates of Stein-Wainger type operators

Abstract

For any integer $n\ge 2$, we establish $L^p({\mathbb{R}}^n)$ inequalities for the $r$-variations of Stein-Wainger type oscillatory integral operators with general phase functions. These inequalities closely related to Carleson's theorem are sharp, up to endpoints. In particular, when the phase function is chosen as $|t|^\A$ with $\A\in (0,1)$, our results provide an affirmative answer to a question posed in Guo-Roos-Yung (Anal. PDE, 2020). Furthermore, we obtain the restricted weak type estimates for endpoints in the specific case of homogeneous phase functions.

29 pages; a detailed comparison with the previous Version 2: remove the application, modify notation and proof unchanged