{
  "id": "0908.1559",
  "version": "v2",
  "published": "2009-08-11T19:59:50.000Z",
  "updated": "2009-11-10T19:05:17.000Z",
  "title": "Boundary Harnack principle for $Δ+ Δ^{α/2}$",
  "authors": [
    "Zhen-Qing Chen",
    "Panki Kim",
    "Renming Song",
    "Zoran Vondraček"
  ],
  "comment": "36 pages, no figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "For $d\\geq 1$ and $\\alpha \\in (0, 2)$, consider the family of pseudo differential operators $\\{\\Delta+ b \\Delta^{\\alpha/2}; b\\in [0, 1]\\}$ on $\\R^d$ that evolves continuously from $\\Delta$ to $\\Delta + \\Delta^{\\alpha/2}$. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $\\Delta +b \\Delta^{\\alpha/2}$ (or equivalently, the sum of a Brownian motion and an independent symmetric $\\alpha$-stable process with constant multiple $b^{1/\\alpha}$) in $C^{1, 1}$ open sets. Here a \"uniform\" BHP means that the comparing constant in the BHP is independent of $b\\in [0, 1]$. Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to $\\Delta + b \\Delta^{\\alpha/2}$ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-10T19:05:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B25",
      "60J45"
    ],
    "keywords": [
      "explicit boundary decay rate",
      "uniform boundary harnack principle",
      "uniform carleson type estimate",
      "nonnegative functions",
      "pseudo differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.1559C"
    }
  }
}