{
  "id": "2203.03891",
  "version": "v1",
  "published": "2022-03-08T07:29:32.000Z",
  "updated": "2022-03-08T07:29:32.000Z",
  "title": "Heat kernel estimates for non-local operator with multisingular critical killing",
  "authors": [
    "Renming Song",
    "Peixue Wu",
    "Shukun Wu"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "Given a $C^{1,\\beta}$ regular open set $D$ with boundary $\\partial D = \\bigcup_{k=1}^d \\bigcup_{j=1}^{m_k} \\Gamma_{k,j}$, where for any $1\\le k\\le d$, $ 1 \\le j \\le m_k$, $\\Gamma_{k,j}$ is a $C^{1,\\beta}$ submanifold without boundary of codimension $k$ and $\\{\\Gamma_{k,j}\\}_{1\\le k\\le d, 1\\le j \\le m_k}$ are disjoint. We show that the heat kernel $p^D(t,x,y)$ of the following non-local operator with multi-singular critical killing potential \\begin{align*} \\big((\\Delta|_D)^{\\alpha/2} - \\kappa\\big)(f)(x):= \\text{p.v.}\\mathcal{A}(d,-\\alpha) \\int_D \\frac{f(y)-f(x)}{|y-x|^{d+\\alpha}}dy - \\sum_{k=1}^d \\sum_{j=1}^{m_k} \\lambda_{k,j} \\delta_{\\Gamma_{k,j}}(x)^{-\\alpha}f(x), \\end{align*} where $ \\lambda_{k,j}>0, \\alpha \\in (0,2), \\beta \\in ((\\alpha-1)_+,1]$, has the following estimates: for any given $T>0$, \\begin{align*} p^D(t,x,y) \\asymp p(t,x,y) \\prod_{k=1}^d \\prod_{j=1}^{m_k} (\\frac{\\delta_{\\Gamma_{k,j}}(x)}{t^{1/\\alpha}} \\wedge 1)^{p_{k,j}}(\\frac{\\delta_{\\Gamma_{k,j}}(y)}{t^{1/\\alpha}} \\wedge 1)^{p_{k,j}}, \\forall t\\in (0,T), x,y\\in D, \\end{align*} where $p(t,x,y)$ is the heat kernel of the $\\alpha$-stable process on $\\mathbb R^d$, and $p_{k,j}$ are determined by $\\lambda_{k,j}$ via a strictly increasing function $\\lambda = C(d,k,\\alpha,p)$. Our method is based on the result established in [Cho et al. Journal de Math\\'ematiques Pures et Appliqu\\'ees 143(2020): 208-256] and a detailed analysis of $C^{1,\\beta}$ manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-08T07:29:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heat kernel estimates",
      "non-local operator",
      "multisingular critical killing",
      "regular open set",
      "multi-singular critical killing potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}