Maximal growth of the Stein-Wainger oscillatory integral

TL;DR

This paper characterizes how the Stein-Wainger oscillatory integral's growth depends on phase regularity within Denjoy-Carleman classes, resolving a prior open problem and offering new insights into related harmonic analysis estimates.

Contribution

It establishes a detailed hierarchy for the integral's growth based on phase regularity and provides a new proof of a classical theorem, advancing understanding in oscillatory integral analysis.

Findings

Resolved a problem posed by Wang--Zhang.

Provided a new proof of Nagel--Wainger's theorem.

Established sharp growth estimates near flat points.

Abstract

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.