A Bilinear Kakeya Inequality in the Heisenberg Group
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913001546711040 |
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| author | Galanos, Yannis |
| author_facet | Galanos, Yannis |
| contents | 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 Fässler, 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 Szemerédi--Trotter-type clustering phenomena. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02984 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Bilinear Kakeya Inequality in the Heisenberg Group Galanos, Yannis Classical Analysis and ODEs Metric Geometry 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 Fässler, 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 Szemerédi--Trotter-type clustering phenomena. |
| title | A Bilinear Kakeya Inequality in the Heisenberg Group |
| topic | Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2604.02984 |