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

TL;DR

This paper establishes decay estimates for a class of oscillatory integrals of convolution type in multi-dimensional space, extending previous one-dimensional results and analyzing their sharpness in specific cases.

Contribution

It provides a new $L^2-L^2-L^2$ decay estimate for oscillatory integrals in $R^d$, generalizing earlier 1D results and exploring sharpness in 2D.

Findings

Proves an $L^2-L^2-L^2$ decay estimate for convolution-type oscillatory integrals in $R^d$

Recovers Li's 2013 result in the case $d=1$

Discusses the sharpness of the estimate in the case $d=2$

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).