Singular integrals on self-similar sets and removability for Lipschitz harmonic functions in Heisenberg groups

TL;DR

This paper investigates singular integrals on measure-zero, lower-dimensional subsets within metric groups like Euclidean and Heisenberg groups, extending Euclidean results and applying them to harmonic functions in the Heisenberg setting.

Contribution

It generalizes singular integral results to broader kernels and applies these findings to harmonic functions in Heisenberg groups, bridging Euclidean and non-Euclidean analysis.

Findings

Extended singular integral results to general kernels.

Established removability criteria for Lipschitz harmonic functions in Heisenberg groups.

Connected Euclidean and Heisenberg harmonic analysis.

Abstract

In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces.