{
  "id": "1405.2141",
  "version": "v2",
  "published": "2014-05-09T05:01:16.000Z",
  "updated": "2014-09-18T11:18:33.000Z",
  "title": "Tangential limits for harmonic functions with respect to $φ(Δ)$ : stable and beyond",
  "authors": [
    "Jaehoon Kang",
    "Panki Kim"
  ],
  "comment": "17pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we discuss tangential limits for regular harmonic functions with respect to $\\phi(\\Delta):=-\\phi(-\\Delta)$ in the $C^{1,1}$ open set $D$ in $\\mathbb{R}^d$, where $\\phi$ is the complete Bernstein function and $d \\ge 2$. When the exterior function $f$ is local $L^p$-H\\\"older continuous of order $\\beta$ on $D^c$ with $ p\\in(1,\\infty]$ and $\\beta>1/p$, for a large class of Bernstein function $\\phi$, we show that the regular harmonic function $u_f$ with respect to $\\phi(\\Delta)$, whose value is $f$ on $D^c$, converges a.e. through a certain parabola that depends on $\\phi$ and $\\phi'$. Our result includes the case $\\phi(\\lambda)=\\log(1+\\lambda^{\\alpha/2})$. Our proofs use both the probabilistic and analytic methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-09T05:01:16.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-18T11:18:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B25",
      "60J75"
    ],
    "keywords": [
      "tangential limits",
      "regular harmonic function",
      "complete bernstein function",
      "analytic methods",
      "large class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2141K"
    }
  }
}