Marstrand's density theorem in the Heisenberg group

Abstract

We prove that if $\mu$ is a Radon measure on the Heisenberg group ${\mathbb{H}}^n$ such that the density ${\Theta }^s(\mu ,\cdot )$, computed with respect to the Korányi metric $d_H$, exists and is positive and finite on a set of positive $\mu$ 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.

Cited by in corpus (2)

• Marstrand-Mattila rectifiability criterion for $1$-codimensional measures in Carnot Groups
• Geometry of $1$-codimensional measures in Heisenberg groups