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

TL;DR

This paper studies singular integral operators on regular subsets of the Heisenberg group, showing that boundedness implies the set's structure closely resembles homogeneous subgroups, especially for integer dimensions.

Contribution

It establishes a link between the boundedness of Riesz-type singular integrals and the geometric structure of subsets in the Heisenberg group, revealing restrictions on fractal sets.

Findings

Boundedness implies the set's dimension is an integer.

Sets can be approximated by homogeneous subgroups at small scales.

Operators are not bounded on many fractal subsets.

Abstract

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 arbitrary small scales by homogeneous subgroups. It follows that the operators cannot be bounded on many self similar fractal subsets of Heisenberg groups.