{
  "id": "1711.08954",
  "version": "v1",
  "published": "2017-11-24T13:08:32.000Z",
  "updated": "2017-11-24T13:08:32.000Z",
  "title": "Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms",
  "authors": [
    "S. E. Boutiah",
    "A. Rhandi",
    "C. Tacelli"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^\\alpha)\\Delta +b|x|^{\\alpha-1}\\frac{x}{|x|}\\cdot\\nabla -|x|^\\beta$ satisfies $$ k(t,x,y) \\leq c_1e^{\\lambda_0 t+ c_2t^{-\\gamma}}\\left(\\frac{1+|y|^\\alpha}{1+|x|^\\alpha}\\right)^{\\frac{b}{2\\alpha}} \\frac{(|x||y|)^{-\\frac{N-1}{2}-\\frac{1}{4}(\\beta-\\alpha)}}{1+|y|^\\alpha} e^{-\\frac{\\sqrt{2}}{\\beta-\\alpha+2}\\left(|x|^{\\frac{\\beta-\\alpha+2}{2}}+ |y|^{\\frac{\\beta-\\alpha+2}{2}}\\right)} $$ for $t>0,\\,|x|,\\,|y|\\ge 1$, where $b\\in\\mathbb{R}$, $c_1,\\,c_2$ are positive constants, $\\lambda_0$ is the largest eigenvalue of the operator $A$, and $\\gamma=\\frac{\\beta-\\alpha+2}{\\beta+\\alpha-2}$, in the case where $N>2,\\,\\alpha>2$ and $\\beta>\\alpha -2$. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-24T13:08:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K08",
      "35J10",
      "47D08",
      "35K20",
      "47D07"
    ],
    "keywords": [
      "elliptic operators",
      "potential terms",
      "kernel estimates",
      "unbounded diffusion",
      "heat kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}