Kakeya maximal inequality in 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 Korányi distance $d_{\mathbb{H}}$. We show that $$\Vert M_{\delta }f{\Vert }_{L^3(S^1)}\le C(\epsilon ){\delta }^{-1/3-\epsilon }\Vert f{\Vert }_{L^3({\mathbb{H}}^1)},f\in L^3({\mathbb{H}}^1),$$ for all $\epsilon >0$. 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.

17 pages

References in corpus (5)

• A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in ${\mathbb{R}}^3$
• On the Dimension of Kakeya Sets in the First Heisenberg Group
• Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements
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Cited by in corpus (1)

• An $L^{4/3}$ $S{L}_2$ Kakeya maximal inequality