A sharp bound for the Stein-Wainger oscillatory integral

Abstract:

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