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

Let E be a 1-regular subset of the Heisenberg group . We prove that there exists a 1-homogeneous kernel K1 such that if E is contained in a 1-regular curve, the corresponding singular integral is bounded in L2(E). Conversely, we prove that there exists another 1-homogeneous kernel K2 such that the L2(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 K1 and K2 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 K1 and K2 are even and nonnegative.

Heisenberg group, rectifiability, singular integrals
Mathematical Subject Classification 2010
Primary: 28A75
Secondary: 28C10, 35R03
Received: 21 October 2016
Revised: 7 April 2017
Accepted: 9 May 2017
Published: 14 July 2017
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