On trilinear singular Brascamp-Lieb integrals

TL;DR

This paper classifies all trilinear singular Brascamp-Lieb forms, advancing understanding of their structure and providing bounds for specific cases, which is crucial for progress on higher-dimensional bilinear Hilbert transforms.

Contribution

It completes the classification of trilinear singular Brascamp-Lieb forms in two dimensions and introduces a new method of rotations for analyzing higher-dimensional kernels.

Findings

Classified all trilinear singular Brascamp-Lieb forms in 2D.

Proved new bounds for certain forms from the classification.

Developed a rotation method to decompose kernels in higher dimensions.

Abstract

We classify all trilinear singular Brascamp-Lieb forms, completing the classification in the two dimensional case by Demeter and Thiele in arXiv:0803.1268. We use known results in the representation theory of finite dimensional algebras, namely the classification of indecomposable representations of the four subspace quiver. Our classification lays out a roadmap for achieving bounds for all degenerate higher dimensional bilinear Hilbert transforms. As another step towards this goal, we prove new bounds for a particular class of forms that arises as a natural next candidate from our classification. We further prove conditional bounds for forms associated with mutually related representations. For this purpose we develop a method of rotations that allows us to decompose any homogeneous $d$-dimensional singular integral kernel into $(d-1)$-dimensional kernels on hyperplanes.