Kakeya maximal inequality in the Heisenberg group

TL;DR

This paper establishes a Kakeya maximal inequality in the Heisenberg group using advanced harmonic analysis techniques, leading to a sharp lower bound on the Hausdorff dimension of Heisenberg Kakeya sets.

Contribution

It introduces a new Kakeya maximal inequality in the Heisenberg group and applies recent variants of Wolff's theorem to this non-Euclidean setting.

Findings

Proves a bound for the Heisenberg Kakeya maximal function in L^3 norm.

Recovers the sharp Hausdorff dimension lower bound for Heisenberg Kakeya sets.

Extends techniques from planar harmonic analysis to the Heisenberg group.

Abstract

We define the Heisenberg Kakeya maximal functions $M_{\delta}f$, $0<\delta<1$, by averaging over $\delta$-neighborhoods of horizontal unit line segments in the Heisenberg group $\mathbb{H}^1$ equipped with the Kor\'{a}nyi distance $d_{\mathbb{H}}$. We show that $$ \|M_{\delta}f\|_{L^3(S^1)}\leq C(\varepsilon)\delta^{-1/3-\varepsilon}\|f\|_{L^3(\mathbb{H}^1)},\quad f\in L^3(\mathbb{H}^1),$$ for all $\varepsilon>0$. The proof is based on a recent variant, due to Pramanik, Yang, and Zahl, of Wolff's circular maximal function theorem for a class of planar curves related to Sogge's cinematic curvature condition. As an application of our Kakeya maximal inequality, we recover the sharp lower bound for the Hausdorff dimension of Heisenberg Kakeya sets of horizontal unit line segments in $(\mathbb{H}^1,d_{\mathbb{H}})$, first proven by Liu.