Maximal Decay Inequalities for Trilinear oscillatory integrals of convolution type

TL;DR

This paper establishes sharp decay estimates for specific trilinear oscillatory integrals of convolution type in two dimensions, enhancing understanding of their boundedness and stability properties.

Contribution

It provides the first sharp $L^ abla$-$L^ abla$-$L^ abla$ decay results for these integrals, linking them to prior $L^2$ bounds and sublevel set estimates.

Findings

Proves sharp $L^ abla$-$L^ abla$-$L^ abla$ decay estimates.

Connects decay results to sublevel set estimates by Christ and others.

Extends multilinear oscillatory integral theory with new sharp bounds.

Abstract

In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ and Christ, Li, Tao, and Thiele. New connections to the multilinear results of Phong, Stein, and Sturm are also considered.