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

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 ${\text{dim}}_H(F)\leq 2$ , then $\mu$ must have $(2n+1)$ -polynomial growth. This is the Heisenberg counterpart of a result of Guy David from [ Dav91 ]. [ABSTRACT FROM AUTHOR]