{
  "id": "2310.18914",
  "version": "v1",
  "published": "2023-10-29T05:59:13.000Z",
  "updated": "2023-10-29T05:59:13.000Z",
  "title": "Zygmund regularity of even singular integral operators on domains",
  "authors": [
    "Andrei Vasin"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Given a bounded Lipschitz domain $D\\subset \\mathbb{R}^d,$ a convolution Calder\\'{o}n-Zygmund operator $T$ and a growth function $\\omega(x)$ of type $n$, we study what conditions on the boundary of the domain are sufficient for boundedness of the restricted even operator $T_D$ on the generalized Zygmund space $C^{\\omega}_*(D)$. Based on a recent T(P) theorem, we prove that this holds if the smoothness of the boundary of a domain $D$ is by one point, in a sense, greater than the smoothness of the corresponding Zygmund space $C^{\\omega}_*(D)$. The main argument of the proof are the higher order gradient estimates of the transform $T_D\\chi_D$ of the characteristic function of a domain with the polynomial boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-29T05:59:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular integral operators",
      "zygmund regularity",
      "higher order gradient estimates",
      "growth function",
      "lipschitz domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}