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 ${\dim }_H(F)\le 2$, then $\mu$ must have $(2n+1)$-polynomial growth. This is the Heisenberg counterpart of a result of Guy David from 1991.