A sharp bound for the Stein-Wainger oscillatory integral

Abstract

Let P d \mathcal {P}_d denote the space of all real polynomials of degree at most d d . It is an old result of Stein and Wainger that \[ sup P ∈ P d | p . v . ∫ R e i P ( t ) d t t | ≤ C d \sup _ {P\in \mathcal {P}_d} \bigg |p.v.\int _{\mathbb {R}} {e^{iP(t)}\frac {dt}{t}} \bigg |\leq C_d \] for some constant C d C_d depending only on d 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 \log d . We prove that \[ sup P ∈ P d | p . v . ∫ R e i P ( t ) d t t | ∼ log ⁡ d . \sup _ {P\in \mathcal {P}_d}\bigg |p.v. \int _{\mathbb {R}}{e^{iP(t)}\frac {dt}{t}}\bigg |\sim \log {d}. \]