Boundedness of singular integrals on C^{1,alpha} intrinsic graphs in the Heisenberg group

We study singular integral operators induced by 3-dimensional Calderon-Zygmund kernels in the Heisenberg group. We show that if such an operator is L (2) bounded on vertical planes, with uniform constants, then it is also L-2 bounded on all intrinsic graphs of compactly supported C-1,C-alpha functions over vertical planes. In particular, the result applies to the operator R, induced by the kernel K(z) = del(H )parallel to z parallel to(-2), z is an element of H \ {0}, the horizontal gradient of the fundamental solution of the sub-Laplacian. The L-2 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. (C) 2019 Elsevier Inc. All rights reserved.Peer reviewe