Preprint

Inserted: 6 mar 2026
Last Updated: 6 mar 2026

Year: 2026

Abstract:

We study Kakeya maximal operators associated with horizontal lines in finite Heisenberg groups $\mathbb H_n(\mathbb F_q)$. For the operator parameterized only by projective horizontal directions, we show that projection to $\mathbb F_q^{2n}$ reduces the problem to the affine finite field Kakeya maximal operator, and we determine the exact $\ell^u \to \ell^v$ growth exponent for all $n$ and all $1 \le u,v \le \infty$. We then introduce a refined-direction operator that also records the central slope of a horizontal line. In $\mathbb H_1(\mathbb F_q)$, we prove the sharp $\ell^2 \to \ell^2$ estimate \[ \
\mathcal M_{\mathbb H_1}^{\mathrm{rd}}F\
_{\ell^2(\mathcal D_1)} \lesssim q^{1/2}\
F\
_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] deduce the exact mixed-norm exponent formula, and obtain lower bounds for horizontal Heisenberg Kakeya sets with prescribed refined directions. The argument is purely Fourier-analytic and does not use the polynomial method. An outlook toward a new approach to the affine Kakeya problem in $\mathbb{F}_q^3$ will be discussed in this paper.

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• Kakeya_Maximal_Operator_Estimates_in_the_Heisenberg_Group_over_Finite_Fields-10.pdf