Nonnegative kernels and 1-rectifiability in the Heisenberg group

Chousionis, Vasileios; Li, Sean

doi:10.2140/apde.2017.10.1407

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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

DOI: 10.2140/apde.2017.10.1407

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 $L^{2} (E)$ . 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 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

Milestones

Received: 21 October 2016

Revised: 7 April 2017

Accepted: 9 May 2017

Published: 14 July 2017

Authors

Vasileios Chousionis
Department of Mathematics University of Connecticut Storrs, CT 06269 United States

Sean Li
Department of Mathematics University of Chicago Chicago, IL 60637 United States

Vol. 10, No. 6, 2017