Necessary condition for the $L^2$ boundedness of the Riesz transform on Heisenberg groups

Abstract: Let $\mu$ be a Radon measure on the $n$-th Heisenberg group $\mathbb{H}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on $\mathbb{H}^n$ is $L^{2(\mu)$-bounded,} and if $\mu(F)=0$ for all Borel sets with $\dim_H(F)\leq 2$, then $\mu$ must have $(2n+1)$-polynomial growth. This is the Heisenberg counterpart of a result of Guy David from 1991.