#### Vol. 10, No. 6, 2017

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Nonnegative kernels and 1-rectifiability in the Heisenberg group

### Vasileios Chousionis and Sean Li

Vol. 10 (2017), No. 6, 1407–1428
##### Abstract

Let $E$ be a $1$-regular subset of the Heisenberg group $ℍ$. 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 ${L}^{2}\left(E\right)$. Conversely, we prove that there exists another $-1$-homogeneous kernel ${K}_{2}$ such that the ${L}^{2}\left(E\right)$-boundedness of its corresponding singular integral implies that $E$ is contained in a $1$-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 $ℍ$. Unlike the Euclidean case, where all known kernels related to rectifiability are antisymmetric, the kernels ${K}_{1}$ and ${K}_{2}$ are even and nonnegative.

##### Keywords
Heisenberg group, rectifiability, singular integrals
##### Mathematical Subject Classification 2010
Primary: 28A75
Secondary: 28C10, 35R03