{
  "id": "1612.07144",
  "version": "v1",
  "published": "2016-12-15T11:13:35.000Z",
  "updated": "2016-12-15T11:13:35.000Z",
  "title": "$L^p$ mapping properties for nonlocal Schrödinger operators with certain potential",
  "authors": [
    "Woocheol Choi",
    "Yong-Cheol Kim"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "In this paper, we consider nonlocal Schr\\\"odinger equations with certain potentials $V$ given by an integro-differential operator $L_K$ as follows; \\begin{equation*}L_K u+V u=f\\,\\,\\text{ in $\\BR^n$ }\\end{equation*} where $V\\in\\rh^q$ for $q>\\f{n}{2s}$ and $0<s<1$. We denote the solution of the above equation by $\\cS_V f:=u$, which is called {\\it the inverse of the nonlocal Schr\\\"odinger operator $L_K+V$ with potential $V$}; that is, $\\cS_V=(L_K+V)^{-1}$. Then we obtain a weak Harnack inequality of weak subsolutions of the nonlocal equation \\begin{equation}\\begin{cases}L_K u+V u=0\\,\\,&\\text{ in $\\Om$,} \\quad u=g\\,\\,&\\text{ in $\\BR^n\\s\\Om$,} \\end{cases}\\end{equation} where $g\\in H^s(\\BR^n)$ and $\\Om$ is a bounded open domain in $\\BR^n$ with Lipschitz boundary, and also get an improved decay of a fundamental solution $\\fe_V$ for $L_K+V$. Moreover, we obtain $L^p$ and $L^p-L^q$ mapping properties of the inverse $\\cS_V$ of the nonlocal Schr\\\"odinger operator $L_K+V$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-15T11:13:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G20",
      "45K05",
      "47G20",
      "45K05",
      "35J60",
      "35B65",
      "35D10",
      "60J75"
    ],
    "keywords": [
      "nonlocal schrödinger operators",
      "mapping properties",
      "weak harnack inequality",
      "fundamental solution",
      "bounded open domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}