Nonnegative kernels and $1$-rectifiability in the Heisenberg group

TL;DR

This paper constructs specific nonnegative kernels in the Heisenberg group that characterize 1-rectifiability of Ahlfors regular sets via bounded singular integrals, extending Euclidean rectifiability theory to a non-Euclidean setting.

Contribution

It introduces the first non-Euclidean kernels with properties linking singular integral boundedness to 1-rectifiability in the Heisenberg group.

Findings

Existence of a kernel $K_1$ with bounded singular integral on sets contained in 1-regular curves.

Existence of a kernel $K_2$ whose boundedness implies the set lies in a 1-regular curve.

Both kernels are even, nonnegative, and related to the vertical component of $bH$.

Abstract

Let $E$ be an $1$-Ahlfors regular subset of the Heisenberg group $\mathbb{H}$. We prove that there exists a $-1$-homogeneous kernel $K_1$ such that if $E$ is contained in a $1$-regular curve the corresponding singular integral is bounded in $L^2(E)$. Conversely, we prove that there exists another $-1$-homogeneous kernel $K_2$, such that the $L^2(E)$-boundedness of its corresponding singular integral implies that $E$ is contained in an $1$-regular curve. These are the first non-Euclidean examples of kernels with such properties. Both $K_1$ and $K_2$ are weighted versions of the Riesz kernel corresponding to the vertical component of $\mathbb{H}$. Unlike the Euclidean case, where all known kernels related to rectifiability are antisymmetric, the kernels $K_1$ and $K_2$ are even and nonnegative.