A sharp bound for the Stein-Wainger oscillatory integral

TL;DR

This paper proves that the principal value oscillatory integral involving polynomials of degree d has an order of magnitude proportional to log d, confirming a conjecture by Carbery, Wainger, and Wright.

Contribution

The paper establishes the precise order of the oscillatory integral as log d, resolving a longstanding conjecture in harmonic analysis.

Findings

The integral's magnitude is proportional to log d.

The conjecture by Carbery, Wainger, and Wright is confirmed.

Provides a sharp bound for oscillatory integrals with polynomial phases.

Abstract

Let Pd denote the space of all real polynomials of degree at most d. It is an old result of Stein and Wainger that for every polynomial P in Pd: |p.v.\int_R {e^{iP(t)} dt/t} | < C(d) for some constant C(d) depending only on d. On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is log d. We prove this conjecture.