Maximal growth of the Stein-Wainger oscillatory integral

We establish a precise hierarchy for the maximal growth of the Stein-Wainger oscillatory integral as the regularity of the phase varies over Denjoy-Carleman classes, such as the Gevrey classes and their generalizations. In particular, we resolve a problem posed by Wang–Zhang, motivated by eigenfunction restriction estimates on curves, and also provide a new proof of a theorem of Nagel–Wainger on the Hilbert transform along curves. A key ingredient is the sharp estimate on the growth of a phase near a flat point.