A study guide for "Trilinear smoothing inequalities and a variant of the triangular Hilbert transform"

Abstract

This article is a study guide for "Trilinear smoothing inequalities and a variant of the triangular Hilbert transform" by Christ, Durcik, and Roos. We first present the standard techniques in the study of oscillatory integrals with the simpler toy model of a Hilbert transform along a parabola. These standard techniques prove to be insufficient in the study of the triangular Hilbert transform with curvature. The central and novel idea in their proof of the $L^p$-boundedness of the triangular Hilbert transform with curvature is a trilinear smoothing inequality which we also examine in this article.

55 pages, 2 figures, Study guide writing workshop in UPenn(https://sites.google.com/view/studyguideworkshop2023/home), resubmit to add info on acknowledgement