$L^2$ estimates of trilinear oscillatory integrals of convolution type on $\mathbb{R}^2$

TL;DR

This paper establishes sharp $L^2$ decay estimates for trilinear oscillatory integrals of convolution type on $R^2$, covering both smooth and polynomial phases, advancing understanding of their boundedness properties.

Contribution

It provides the first sharp $L^2$ decay estimates for these integrals with smooth phases and uniform bounds for polynomial phases, filling a gap in harmonic analysis.

Findings

Sharp $L^2$ decay estimates for smooth phases.

Uniform $L^2$ bounds for polynomial phases.

Enhanced understanding of oscillatory integral behavior.

Abstract

This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $\mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.