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

We study singular integral operators induced by $3$-dimensional Calder'on-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 $C^{1,\alpha }$ functions over vertical planes. In particular, the result applies to the operator $\mathcal{R}$ induced by the kernel $$\mathcal{K}(z)={\mathrm{\nabla }}_{\mathbb{H}}\mathrm{\parallel }z{\mathrm{\parallel }}^{-2},z\in \mathbb{H}\setminus \{0\},$$ the horizontal gradient of the fundamental solution of the sub-Laplacian. The $L^2$ boundedness of $\mathcal{R}$ 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.