Non-Transversal Multilinear Duality and Joints

TL;DR

This paper develops a duality framework for multilinear operators, extending previous theories to non-transversal cases, and applies it to joints problems, resulting in a discrete analogue of Euclidean geometric results.

Contribution

It introduces a duality theory for non-transversal multilinear operators and applies it to joints problems, providing a new factorisation theorem in the discrete setting.

Findings

Established a duality framework for non-transversal multilinear operators.

Derived a factorisation theorem for joints and multijoints.

Provided a discrete analogue of Euclidean geometric results by Bourgain and Guth.

Abstract

We develop a framework for a duality theory for general multilinear operators which extends that for transversal multilinear operators which has been established in arXiv:1809.02449. We apply it to the setting of joints and multijoints, and obtain a "factorisation" theorem which provides an analogue in the discrete setting of results of Bourgain and Guth (arXiv:0811.2251 and arXiv:1012.3760) from the Euclidean setting.