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arXiv · 1407.6636

Marstrand's density theorem in the Heisenberg group

Vasilis Chousionis·Jeremy T. Tyson
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Abstract

We prove that if $μ$ is a Radon measure on the Heisenberg group $\mathbb{H}^n$ such that the density $Θ^s(μ,\cdot)$, computed with respect to the Korányi metric $d_H$, exists and is positive and finite on a set of positive $μ$ measure, then $s$ is an integer. The proof relies on an analysis of uniformly distributed measures on $(\mathbb{H}^n,d_H)$. We provide a number of examples of such measures, illustrating both the similarities and the striking differences of this sub-Riemannian setting from its Euclidean counterpart.

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Vasilis Chousionis, Jeremy T. Tyson. 2014-07-24. Marstrand's density theorem in the Heisenberg group. https://arxiv.org/abs/1407.6636

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