Let be a -regular subset of the Heisenberg group . We prove that there exists a -homogeneous kernel such that if is contained in a -regular curve, the corresponding singular integral is bounded in . Conversely, we prove that there exists another -homogeneous kernel such that the -boundedness of its corresponding singular integral implies that is contained in a -regular curve. These are the first non-Euclidean examples of kernels with such properties. Both and are weighted versions of the Riesz kernel corresponding to the vertical component of . Unlike the Euclidean case, where all known kernels related to rectifiability are antisymmetric, the kernels and are even and nonnegative.
"Nonnegative kernels and 1-rectifiability in the Heisenberg group." Anal. PDE 10 (6) 1407 - 1428, 2017. https://doi.org/10.2140/apde.2017.10.1407