A Bilinear Kakeya Inequality in the Heisenberg Group

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 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.

• The paper proves a sharp bilinear Kakeya inequality in ℍ¹ for δ-tubes under (δ, α)-broad conditions.
• It adapts Euclidean parabolic incidence techniques to overcome noncommutative and horizontal geometric challenges.
• The method yields new insights into Kakeya-type problems and restriction phenomena in sub-Riemannian settings.

A Bilinear Kakeya Inequality in the Heisenberg Group

Introduction and Problem Context

The Kakeya problem remains a central object of study in geometric measure theory and harmonic analysis, with deep connections to restriction phenomena, maximal function estimates, and discrete incidence geometry. While much of the established theory concerns Euclidean spaces, the adaptation to non-commutative groups, particularly the Heisenberg group $\mathbb{H}$, has garnered increasing attention. The non-Euclidean geometry, horizontal structure, and polynomial submanifolds of $\mathbb{H}$ demand new techniques and lead to distinct phenomena not encounted in $\mathbb{R}^n$.

This paper investigates bilinear and multilinear Kakeya-type estimates in $\mathbb{H}^1$ (the first Heisenberg group). Building on previous works that establish sharp linear maximal inequalities via reductions to planar geometric incidence theorems (including "A Kakeya maximal inequality in the Heisenberg group" [FPW, 2025]), this work proposes and proves a bilinear Kakeya inequality in $\mathbb{H}^1$, together with corresponding estimates for Euclidean curved tubes in the plane. The results rely on a fusion of analytic techniques developed in the study of the restriction problem, geometric combinatorics in the spirit of Szemerédi-Trotter, and incidence theory for families of parabolas.

Main Results and Methods

The central contribution is the establishment of a sharp bilinear Kakeya inequality for Heisenberg $\delta$-tubes with $(\delta, \alpha)$-broadness hypotheses on their horizontal directions. The key technical theorem (Theorem 1 in the paper) asserts that if $\mathcal{T}_1$ and $\mathcal{T}_2$ are collections of Heisenberg $\delta$-tubes whose core lines are $\mathbb{H}$0-broad and sufficiently transverse, then: $\mathbb{H}$1 where the implied constant depends only on $\mathbb{H}$2.

The proof strategy structurally mirrors the Euclidean case but diverges in crucial geometric and combinatorial aspects. Using projections from $\mathbb{H}$3 to $\mathbb{H}$4 (the so-called Heisenberg projection), each horizontal tube can be associated to a $\mathbb{H}$5-neighbourhood of a planar parabola. Thus, the Heisenberg bilinear Kakeya problem reduces to a bilinear Kakeya-type estimate for $\mathbb{H}$6-neighbourhoods of parabolic arcs in $\mathbb{H}$7.

The core planar theorem (Theorem 2 in the paper) controls the mixed $\mathbb{H}$8 norms over overlapping collections of thickened curves, under broadness and bipartiteness constraints. Specifically, for $\mathbb{H}$9-bipartite pairs of $\mathbb{R}^n$0-separated quadratics, $\mathbb{R}^n$1, and for $\mathbb{R}^n$2-broad subcollections, one obtains: $\mathbb{R}^n$3 This estimate is proven to be sharp with respect to all main parameters except the possible presence of an $\mathbb{R}^n$4-loss.

A pivotal structural insight of the paper is the introduction of a novel broadness hypothesis tailored to exclude configurations of tubes ("bushes" and their analogues) that collapse the bilinear structure and undermine transversal geometry. The necessity of such broadness conditions is rigorously justified by constructing counterexamples inspired by bush-type and Szemerédi-Trotter-type clustering, and the sharpness of all exponents—most notably the critical $\mathbb{R}^n$5—is demonstrated.

The combinatorial component exploits parabolic rectangle-counting lemmas—generalizations of Wolff’s and Zahl’s bounds for circle incidences—to handle the nonlinearity and variable coefficient structure of parabolic arcs. The proof orchestration uses a dyadic pigeonholing reduction (to control scales and multiplicities), application of the broad/narrow dichotomy on tangency structures, and a robust bootstrapping principle akin to the "two-ends" reduction found in incidence geometry.

Implications and Connections

The results in this paper extend the multilinear Kakeya and restriction phenomena from the Euclidean to the sub-Riemannian, noncommutative regime of the Heisenberg group, while elucidating the role of transversal and broad configurations of tubes. Notably:

• The strong bilinear inequality proved here (with the critical exponent $\mathbb{R}^n$6 and optimal $\mathbb{R}^n$7 dependence, up to $\mathbb{R}^n$8) establishes the precise mixed-norm behavior of transversal tube families required for applications in harmonic analysis on $\mathbb{R}^n$9.
• The necessity and novelty of the broadness condition is highlighted; unlike in the linear case, bilinear arguments are obstructed by clusterings (bushes, pencils, clamshells), and the proof identifies the minimal broadness exponent consistent with bilinear structure as $\mathbb{H}^1$0.
• The work frames the Heisenberg bilinear Kakeya problem as an analytic-geometric bridge between Euclidean multilinear theory (e.g., Bennett-Carbery-Tao's multilinear Kakeya [BCT, 2010]) and finite field/real incidence geometry (e.g., Szemerédi-Trotter-type theorems).
• The reduction to parabolic incidence estimates connects the analysis on $\mathbb{H}^1$1 to the cutting-edge combinatorics of variable-coefficient maximal functions, as developed in works by Pramanik-Yang-Zahl [PYZ, 2022], Zahl [Zahl, 2026], and Wolff.

Practically, these inequalities have implications for questions about the dimension and structure of Kakeya and Besicovitch sets in $\mathbb{H}^1$2, extendable to restriction and oscillatory integral estimates. The new analytic-broadness framework also suggests avenues for future work on higher dimensional nilpotent groups and more general sub-Riemannian geometries.

Future Directions

While the critical bilinear exponent for the planar (projected) problem is proven to be $\mathbb{H}^1$3, the sharpness of the mixed norm exponent for the Heisenberg bilinear Kakeya problem is left partially open, with a conjectural lower bound of $\mathbb{H}^1$4 in analogy with the Szemerédi-Trotter regime. The technical obstacles are linked to possible improvements on rectangle incidence theorems for well-spaced curves, as recently considered in the Euclidean context. Future research could aim to tighten the gap and investigate analogous phenomena in higher step nilpotent groups, as well as multidimensional variants where the combinatorics of horizontal directions become even more intricate.

The necessity of the broadness hypothesis and the precise value of the broadness exponent in $\mathbb{H}^1$5 also remain of interest. Further, connections to variable coefficient analogues in oscillatory integral theory, and robust versions of the multilinear restriction problem in sub-Riemannian geometries, are promising lines of inquiry.

Conclusion

This paper achieves a rigorous formulation and proof of a bilinear Kakeya inequality in the Heisenberg group and the corresponding sharp estimate for curved (parabolic) tubes in $\mathbb{H}^1$6, under natural broadness and transversality conditions. The results clarify the geometric and combinatorial obstacles inherent in the bilinear regime, solidify the connection between Heisenberg harmonic analysis and planar incidence geometry, and lay groundwork for further explorations of multilinear phenomena in non-Euclidean settings.

Reference:

Yannis Galanos, "A Bilinear Kakeya Inequality in the Heisenberg Group," (2604.02984).