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

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 $$\mathrm{\parallel }{M}_{{\mathbb{H}}_1}^{\mathrm{r}\mathrm{d}}F{\mathrm{\parallel }}_{{\mathrm{\ell }}^2({\mathcal{D}}_1)}\lesssim q^{\frac{1}{2}}\mathrm{\parallel }F{\mathrm{\parallel }}_{{\mathrm{\ell }}^2({\mathbb{H}}_1({\mathbb{F}}_q))},$$ where the exponent $\frac{1}{2}$ is sharp. Combining this estimate with endpoint bounds and interpolation, we determine the exact mixed-norm growth exponent: $$A_1^{\mathrm{r}\mathrm{d}}(u,v)=\max \{\frac{1}{v},1-\frac{1}{u},\frac{2}{v}-\frac{1}{u},1+\frac{2}{v}-\frac{3}{u}\},1\le u,v\le \mathrm{\infty }\mathrm{.}$$ 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 $\mathrm{\Omega }\subset D_1$, then $$\mathrm{\mid }E\mathrm{\mid }\gtrsim \frac{m^2\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{q}\mathrm{.}$$ As a benchmark, we also analyze the coarser operator parameterized only by projective horizontal directions and determine its exact ${\mathrm{\ell }}^u\to {\mathrm{\ell }}^v$ 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.