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We study singular integral operators induced by $3$-dimensionalCalder\'on-Zygmund kernels in the Heisenberg group. We show that if such anoperator is $L^{2}$ bounded on vertical planes, with uniform constants, then itis also $L^{2}$ bounded on all intrinsic graphs of compactly supported$C^{1,\alpha}$ functions over vertical planes. In particular, the resultapplies to the operator $\mathcal{R}$ induced by the kernel $$\mathcal{K}(z) =\nabla_{\mathbb{H}} \| z \|^{-2}, \quad z \in \mathbb{H} \setminus \{0\},$$ thehorizontal gradient of the fundamental solution of the sub-Laplacian. The$L^{2}$ boundedness of $\mathcal{R}$ is connected with the question ofremovability for Lipschitz harmonic functions. As a corollary of our result, weinfer that the intrinsic graphs mentioned above are non-removable. Apart fromsubsets of vertical planes, these are the first known examples of non-removablesets with positive and locally finite $3$-dimensional measure.

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