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For any integer $n \geq 2$, we establish $L^p(\R^n)$ inequalities for the$r$-variations of Stein-Wainger type oscillatory integral operators withgeneral radial phase functions. These inequalities closely related toCarleson's theorem are sharp, up to endpoints. In particular, when the phasefunction is chosen as $|t|^\A$ with $\A\in (0,1)$, our results provide anaffirmative answer to a question posed in Guo-Roos-Yung (Anal. PDE, 2020). Inpursuit of the objective, we use an approach incorporating the method ofstationary phase, a square function estimate derived from Seeger's arguments,Stein-Tomas restriction estimates, and a crucial observation enabling thederivation of useful decay estimates near endpoints. Furthermore, we obtain therestricted weak type estimates for endpoints in the specific case ofhomogeneous phase functions. As an application of these outcomes, we furtherextend analogous results to a class of ergodic integral operators.

Sharp variational inequalities for oscillatory integral operators associated with Carleson's theorem