有限Heisenberg群中的水平Kakeya极大算子：精确指数与应用
Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications
AI总结：
研究有限Heisenberg群中与水平线相关的Kakeya极大算子，通过傅里叶分析得到秩一时精确的混合范数增长指数，并应用于组合几何中的点线配置问题。
摘要：
设 $q$ 为奇素数幂。我们研究有限Heisenberg群 $\mathbb H_n(\mathbb F_q)$ 中与水平线相关的Kakeya极大算子。我们的主要对象是细化方向极大算子，其参数记录了射影水平方向以及由水平性确定的中心齐次坐标。在秩一情形，我们证明 \[ \|M_{\mathbb H_1}^{\mathrm{rd}}F\|_{\ell^2(\mathcal D_1)} \lesssim q^{\frac{1}{2}} \|F\|_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] 其中指数 $\frac{1}{2}$ 是紧的。结合该估计与端点界和插值，我们确定精确的混合范数增长指数：\[ A^{\mathrm{rd}}_1(u,v) = \max\left\{ \frac1v,\, 1-\frac1u,\, \frac2v-\frac1u,\, 1+\frac2v-\frac3u \right\}, \qquad 1\le u,v\le\infty. \] 作为推论，若 $E\subset\mathbb H_1(\mathbb F_q)$ 在来自 $\Omega\subset D_1$ 的每个细化方向上与至少 $m$ 个点的水平线相交，则 \[ |E|\gtrsim \frac{m^2|\Omega|}{q}. \] 作为基准，我们还分析了仅由射影水平方向参数化的较粗算子，并确定了其在每个秩上的精确 $\ell^u\to\ell^v$ 增长指数。在秩一时，该基准通过自包含的 $TT^*$ 论证而非多项式消去建立，且相同的平面估计作为细化方向证明中零中心频率分量重新出现。非零中心频率由Plancherel、特征正交性和一个显式二次映射的有界纤维性质控制。因此，尖锐的细化方向估计完全通过傅里叶分析方法获得。
英文摘要：
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 $Ω\subset D_1$, 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 $TT^*$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
