2212.01845Dec 4, 202217 pages

Kakeya maximal inequality in the Heisenberg group

Katrin Fässler, Andrea Pinamonti, Pietro Wald

Classical Analysis and ODEs Metric Geometry

Abstract

We define the Heisenberg Kakeya maximal functions $M_δf$ , $0<δ<1$ , by averaging over $δ$ -neighborhoods of horizontal unit line segments in the Heisenberg group $\mathbb{H}^1$

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equipped with the Korányi distance

d_{\mathbb{H}}

. We show that

\|M_δf\|_{L^3(S^1)}\leq C(\varepsilon)δ^{-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