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2017 Nonnegative kernels and 1-rectifiability in the Heisenberg group
Vasileios Chousionis, Sean Li
Anal. PDE 10(6): 1407-1428 (2017). DOI: 10.2140/apde.2017.10.1407


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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Vasileios Chousionis. Sean Li. "Nonnegative kernels and 1-rectifiability in the Heisenberg group." Anal. PDE 10 (6) 1407 - 1428, 2017.


Received: 21 October 2016; Revised: 7 April 2017; Accepted: 9 May 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1369.28004
MathSciNet: MR3678492
Digital Object Identifier: 10.2140/apde.2017.10.1407

Primary: 28A75
Secondary: 28C10 , 35R03

Keywords: Heisenberg group , rectifiability , singular integrals

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.10 • No. 6 • 2017
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