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

TL;DR

This paper investigates Kakeya maximal operators in finite Heisenberg groups, determining exact growth exponents and establishing sharp estimates, with implications for Kakeya sets and potential new approaches to affine Kakeya problems.

Contribution

It provides the first exact exponents for horizontal Kakeya maximal operators in finite Heisenberg groups and introduces a refined-direction operator with sharp estimates.

Findings

Exact $oldsymbol{ ext{ell}^u o ext{ell}^v}$ growth exponents determined for all $n$ and directions.

Sharp $oldsymbol{ ext{ell}^2 o ext{ell}^2}$ estimate in $oldsymbol{ ext{H}_1( extbf{F}_q)}$.

Lower bounds for horizontal Heisenberg Kakeya sets with refined directions.

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 \[ \|M_{\mathbb H_1}^{\mathrm{rd}}F\|_{\ell^2(D_1)} \lesssim q^{1/2}\|F\|_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] 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.