Necessary condition for the L2 boundedness of the Riesz transform on Heisenberg groups

Abstract

Let μ be a Radon measure on the nth Heisenberg group Hn. In this note we prove that if the (2n+1) -dimensional (Heisenberg) Riesz transform on Hn is L2(μ) -bounded, and if μ(F)=0 for all Borel sets with dimH(F)≤2 , then μ must have (2n+1) -polynomial growth. This is the Heisenberg counterpart of a result of Guy David from [ Dav91 ].