Estimating Oscillatory Integrals of Convolution Type in ${\mathbb{R}}^d$

Abstract

In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in ${\mathbb{R}}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the $d=2$ case. Our main hypothesis has close connections to the property of simple nondegeneracy studied by Christ, Li, Tao and Thiele (2005).

Application of Theorem 2.1 in the proof of Theorem 1.2 assumed uniformity of the constant C which does not hold in general; in reality, there may be dependence on the phase