Singular integrals on Ahlfors-David regular subsets of the Heisenberg group (Q2430508)

A Borel measure \(\mu\) on a metric space \(X\) is Ahlfors-David regular if for some positive numbers \(s\) and \(C\), \[ r^s/C\leq \mu (B(x,r))\leq C r^s\quad\text{for all}\quad x\in \mathrm{spt} \mu, 0<r<\mathrm{diam}(\mathrm{spt} \mu), \] where \(\mathrm{spt} \mu\) stands for the support of \(\mu\). The authors investigate certain singular integral operators with Riesz-type kernels on \(s\)-dimensional Ahlfors-David regular subsets of Heisenberg groups. The authors show that \(L^2\)-boundedness, and even a little less, implies that \(s\) must be an integer and that the set can be approximated at some arbitrarily small scales by homogeneous subgroups. It follows that the operators cannot be bounded on many self-similar fractal subsets of Heisenberg groups.