Let P be a polynomial and TP,β be defined by

TP,β(f, g)(x) = Z 1

−1

f(x − t) g(x − P(t)) e

i|t|−β dt

|t|

.

If β > 1 and P is a homogeneous polynomial, then TP,β maps L

∞ × L

2

to L

2

. This is a

joint work with D. Fan.

This problem was motivated by Hilbert transform along curves and the bilinear Hilbert

transform. One crucial point in the proof is the stability of the critical points of the phase

function aξt + bηt2 + f(t) for some a, b ∈ R and C

∞ function. The proof is also based

on a T T∗

argument and an uniform estimate of a bilinear oscillatory integral proved by

Phong and Stein.