Nonnegative kernels and 1-rectifiability in the Heisenberg group

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.