Singular Integrals on Ahlfors-David Regular Subsets of the Heisenberg Group

We investigate certain singular integral operators with Riesz-type kernels on s-dimensional Ahlfors-David regular subsets of Heisenberg groups. We show that L-2-boundedness, and even a little less, implies that s must be an integer and 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.