On sharp variational inequalities for oscillatory integral operators of Stein-Wainger type

Abstract: For any integer $n \geq 2$, we establish $L^p(\R^n)$ inequalities for the $r$-variations of Stein-Wainger type oscillatory integral operators with general radial 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). In pursuit of the objective, we use an approach incorporating the method of stationary phase, a square function estimate derived from Seeger's arguments, Stein-Tomas restriction estimates, and a crucial observation enabling the derivation of useful decay estimates near endpoints. Furthermore, we obtain the restricted weak type estimates for endpoints in the specific case of homogeneous phase functions. As an application of these outcomes, we further extend analogous results to a class of ergodic integral operators.