Boundedness of singular integrals on C1,α intrinsic graphs in the Heisenberg group

We study singular integral operators induced by 3-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is L2 bounded on vertical planes, with uniform constants, then it is also L2 bounded on all intrinsic graphs of compactly supported C1,α functions over ver...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2019-10, Vol.354, Article 106745
Main Authors: Chousionis, Vasileios, Fässler, Katrin, Orponen, Tuomas
Format: Article
Language:English
Subjects:
Heisenberg group
Removable sets for harmonic functions
Singular integrals
Online Access:Get full text
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Summary:We study singular integral operators induced by 3-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is L2 bounded on vertical planes, with uniform constants, then it is also L2 bounded on all intrinsic graphs of compactly supported C1,α functions over vertical planes. In particular, the result applies to the operator R induced by the kernelK(z)=∇H‖z‖−2,z∈H∖{0}, the horizontal gradient of the fundamental solution of the sub-Laplacian. The L2 boundedness of R is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that the intrinsic graphs mentioned above are non-removable. Apart from subsets of vertical planes, these are the first known examples of non-removable sets with positive and locally finite 3-dimensional measure.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2019.106745