A Sharp Bound for the Stein-wainger Oscillatory Integral

Let P d denote the space of all real polynomials of degree at most d. It is an old result of Stein and Wainger [4] that sup P ∈P d ˛ ˛ ˛ ˛ p.v. Z 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 in [2] claim that the true order of magnitude of the above principal value integral is log d. We prove that sup P ∈P d ˛ ˛ ˛ ˛ p.v. Z R e iP (t) dt t ˛ ˛ ˛ ˛ ∼ log d.