  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.
