Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications

Abstract

Let $q$ be an odd prime power. We study Kakeya maximal operators associated with horizontal lines in the finite Heisenberg groups ${\mathbb{H}}_n({\mathbb{F}}_q)$. Our principal object is the refined-direction maximal operator, whose parameter records the projective horizontal direction together with the central homogeneous coordinate determined by horizontality. In rank one, we prove \[ \|M_{\mathbb H_1}^{\mathrm{rd}}F\|_{\ell^2(\mathcal D_1)} \lesssim q^{\frac{1}{2}} \|F\|_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] where the exponent $\frac{1}{2}$ is sharp. Combining this estimate with endpoint bounds and interpolation, we determine the exact mixed-norm growth exponent: \[ A^{\mathrm{rd}}_1(u,v) = \max\left\{ \frac1v,\, 1-\frac1u,\, \frac2v-\frac1u,\, 1+\frac2v-\frac3u \right\}, \qquad 1\le u,v\le\infty. \] As a consequence, if $E\subset {\mathbb{H}}_1({\mathbb{F}}_q)$ meets, in at least $m$ points, a horizontal line in each refined direction from $\Omega \subset D_1$, then \[ |E|\gtrsim \frac{m^2|Ω|}{q}. \] As a benchmark, we also analyze the coarser operator parameterized only by projective horizontal directions and determine its exact ${\ell }^u\to {\ell }^v$ growth exponent in every rank. In rank one, this benchmark is established by a self-contained $T{T}^{\ast }$ argument rather than polynomial vanishing, and the same planar estimate reappears as the zero-central-frequency component of the refined-direction proof. The nonzero central frequencies are controlled by Plancherel, character orthogonality, and a bounded-fiber property of an explicit quadratic map. Thus, the sharp refined-direction estimate is obtained by purely Fourier-analytic methods.

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