{
  "id": "1912.11257",
  "version": "v1",
  "published": "2019-12-24T09:20:48.000Z",
  "updated": "2019-12-24T09:20:48.000Z",
  "title": "$L^2$-bounded singular integrals on a purely unrectifiable set in $\\mathbb{R}^d$",
  "authors": [
    "Joan Mateu",
    "Laura Prat"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We construct an example of a purely unrectifiable measure $\\mu$ in $\\mathbb{R}^d$ for which the singular integrals associated to the kernels $\\displaystyle{K(x)=\\frac{P_{2k+1}(x)}{|x|^{2k+d}}}$, with $k\\geq 1$ and $P_{2k+1}$ a homogeneous harmonic polynomial of degree $2k+1$, are bounded in $L^2(\\mu)$. This contrasts starkly with the results concerning the Riesz kernel $\\displaystyle{\\frac{x}{|x|^{d}}}$ in $\\mathbb{R}^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-24T09:20:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "bounded singular integrals",
      "purely unrectifiable set",
      "riesz kernel",
      "homogeneous harmonic polynomial",
      "purely unrectifiable measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}