{
  "id": "1501.02023",
  "version": "v1",
  "published": "2015-01-09T03:05:32.000Z",
  "updated": "2015-01-09T03:05:32.000Z",
  "title": "Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators",
  "authors": [
    "Zhen-Qing Chen",
    "Yan-Xia Ren",
    "Ting Yang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Suppose $d\\ge 2$ and $0<\\beta<\\alpha<2$. We consider the non-local operator $\\mathcal{L}^{b}=\\Delta^{\\alpha/2}+\\mathcal{S}^{b}$, where $$\\mathcal{S}^{b}f(x):=\\lim_{\\varepsilon\\to 0}\\mathcal{A}(d,-\\beta)\\int_{|z|>\\varepsilon}\\left(f(x+z)-f(x)\\right)\\frac{b(x,z)}{|z|^{d+\\beta}}\\,dy.$$ Here $b(x,z)$ is a bounded measurable function on $\\mathbb{R}^{d}\\times\\mathbb{R}^{d}$ that is symmetric in $z$, and $\\mathcal{A}(d,-\\beta)$ is a normalizing constant so that when $b(x, z)\\equiv 1$, $\\mathcal{S}^{b}$ becomes the fractional Laplacian $\\Delta^{\\beta/2}:=-(-\\Delta)^{\\beta/2}$. In other words, $$\\mathcal{L}^{b}f(x):=\\lim_{\\varepsilon\\to 0}\\mathcal{A}(d,-\\beta)\\int_{|z|>\\varepsilon}\\left(f(x+z)-f(x)\\right) j^b(x, z)\\,dz,$$ where $j^b(x, z):= \\mathcal{A}(d,-\\alpha) |z|^{-(d+\\alpha)}+ \\mathcal{A}(d,-\\beta) b(x, z)|z|^{-(d+\\beta)}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)\\geq 0$ on $\\mathbb{R}^d\\times \\mathbb{R}^d$, there is a conservative Feller process $X^{b}$ having $\\mathcal{L}^b$ as its infinitesimal generator. In this paper we establish, under certain conditions on $b$, a uniform boundary Harnack principle for harmonic functions of $X^b$ (or equivalently, of $\\mathcal{L}^b$) in any $\\kappa$-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of $X^{b}$ in open sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-09T03:05:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "non-local operator",
      "harmonic functions",
      "uniform boundary harnack principle",
      "establish uniform gradient estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150102023C"
    }
  }
}