{
  "id": "0910.3266",
  "version": "v1",
  "published": "2009-10-17T13:10:59.000Z",
  "updated": "2009-10-17T13:10:59.000Z",
  "title": "Dirichlet Heat Kernel Estimates for $Δ^{α/2}+ Δ^{β /2}$",
  "authors": [
    "Zhen-Qing Chen",
    "Panki Kim",
    "Renming Song"
  ],
  "comment": "33 page",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "For $d\\geq 1$ and $0<\\beta<\\alpha<2$, consider a family of pseudo differential operators $\\{\\Delta^{\\alpha} + a^\\beta \\Delta^{\\beta/2}; a \\in [0, 1]\\}$ that evolves continuously from $\\Delta^{\\alpha/2}$ to $ \\Delta^{\\alpha/2}+ \\Delta^{\\beta/2}$. It gives arise to a family of L\\'evy processes \\{$X^a, a\\in [0, 1]\\}$, where each $X^a$ is the sum of independent a symmetric $\\alpha$-stable process and a symmetric $\\beta$-stable process with weight $a$. For any $C^{1,1}$ open set $D$, we establish explicit sharp two-sided estimates (uniform in $a\\in [0,1]$) for the transition density function of the subprocess $X^{a, D}$ of $X^a$ killed upon leaving the open set $D$. The infinitesimal generator of $X^{a, D}$ is the non-local operator $\\Delta^{\\alpha} + a^\\beta \\Delta^{\\beta/2}$ with zero exterior condition on $D^c$. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for $X^{a, D}$ and uniform boundary Harnack principle for $X^a$ in $D$ with explicit decay rate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-17T13:10:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "47G20",
      "60J75"
    ],
    "keywords": [
      "dirichlet heat kernel estimates",
      "explicit sharp two-sided estimates",
      "uniform sharp green function estimates",
      "open set",
      "sharp heat kernel estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.3266C"
    }
  }
}