A Bilinear Kakeya Inequality in the Heisenberg Group

TL;DR

This paper establishes a new bilinear Kakeya inequality in the Heisenberg group and Euclidean space, combining geometric and harmonic analysis techniques with novel hypotheses to control complex configurations.

Contribution

It introduces a bilinear Kakeya inequality in the Heisenberg group and sharp estimates for curved tubes, utilizing a novel broadness hypothesis and advanced projection techniques.

Findings

Proved a bilinear Kakeya inequality in the Heisenberg group.

Established sharp bilinear estimates for Euclidean curved tubes.

Developed a broadness hypothesis to exclude problematic configurations.

Abstract

We prove a bilinear Kakeya inequality in the first Heisenberg group and a sharp bilinear Kakeya estimate for Euclidean curved tubes in $\R^2$. By adapting an argument of F\"assler, Pinamonti and Wald involving Heisenberg projections, we show that the latter implies the former. We prove the estimate for curved tubes using a combination of techniques developed by Pramanik, Yang and Zahl, Wolff and Schlag. We introduce a novel broadness hypothesis inspired by works of Zahl, which rules out bush-type configurations that break transversal structure. We argue that such a hypothesis is needed for proving the bilinear estimates we present. We also introduce necessary additional linear terms to the estimate to counteract Szemer\'edi--Trotter-type clustering phenomena.