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.

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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