The local symmetry condition in the Heisenberg group

TL;DR

This paper introduces a Heisenberg group analogue of the local symmetry condition and demonstrates its connection to singular integral boundedness and geometric properties of 3-regular sets.

Contribution

It establishes a new local symmetry condition in the Heisenberg group and links it to singular integral boundedness and geometric lemmas for 3-regular sets.

Findings

LSC in the Heisenberg group is implied by boundedness of certain singular integrals.

LSC implies the weak geometric lemma for vertical beta-numbers.

The work extends geometric measure theory concepts to the Heisenberg group.

Abstract

I propose an analogue in the first Heisenberg group $\mathbb{H}$ of David and Semmes' local symmetry condition (LSC). For closed $3$-regular sets $E \subset \mathbb{H}$, I show that the (LSC) is implied by the $L^{2}(\mathcal{H}^{3}|_{E})$ boundedness of $3$-dimensional singular integrals with horizontally antisymmetric kernels, and that the (LSC) implies the weak geometric lemma for vertical $\beta$-numbers.