{
  "id": "1708.08444",
  "version": "v1",
  "published": "2017-08-28T17:59:25.000Z",
  "updated": "2017-08-28T17:59:25.000Z",
  "title": "Boundedness of singular integrals on $C^{1,α}$ intrinsic graphs in the Heisenberg group",
  "authors": [
    "Vasileios Chousionis",
    "Katrin Fässler",
    "Tuomas Orponen"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-28T17:59:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intrinsic graphs",
      "heisenberg group",
      "vertical planes",
      "boundedness",
      "study singular integral operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}