Boundedness of singular integrals on C1,α intrinsic graphs in the Heisenberg group
Published in:
 Advances in Mathematics.  2019, vol. 354, p. 106745
English
We study singular integral operators induced by 3dimensional CalderónZygmund 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 subLaplacian. 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 nonremovable. Apart from subsets of vertical planes, these are the first known examples of nonremovable sets with positive and locally finite 3 dimensional measure.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/308182
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