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

TL;DR

This paper proves that certain singular integral operators are bounded on a broad class of intrinsic graphs in the Heisenberg group, linking boundedness to geometric properties and non-removability of Lipschitz harmonic functions.

Contribution

It establishes that $L^{2}$ boundedness on vertical planes implies boundedness on all $C^{1,eta}$ intrinsic graphs, extending understanding of singular integrals in the Heisenberg group.

Findings

Boundedness of singular integrals on vertical planes implies boundedness on intrinsic graphs.

The operator $ abla_{ ext{Heisenberg}} orm{z}^{-2}$ is $L^{2}$ bounded on these graphs.

Intrinsic graphs are shown to be non-removable for Lipschitz harmonic functions.

Abstract

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 $$\mathcal{K}(z) = \nabla_{\mathbb{H}} \| z \|^{-2}, \quad z \in \mathbb{H} \setminus \{0\},$$ 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.