Author:
DĄBROWSKI DAMIAN,VILLA MICHELE
Abstract
AbstractLet
$\mu$
be a Radon measure on the nth 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].
Publisher
Cambridge University Press (CUP)