{
  "id": "2111.03513",
  "version": "v2",
  "published": "2021-11-05T14:11:42.000Z",
  "updated": "2021-11-29T15:51:01.000Z",
  "title": "Upper and lower bounds for Dunkl heat kernel",
  "authors": [
    "Jacek Dziubański",
    "Agnieszka Hejna"
  ],
  "comment": "17 pages, we corrected some typos",
  "categories": [
    "math.FA"
  ],
  "abstract": "On $\\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(\\alpha) > 0$, and the associated measure $$ dw(\\mathbf x)=\\prod_{\\alpha\\in R}|\\langle \\mathbf x,\\alpha\\rangle|^{k(\\alpha)}\\, d\\mathbf x, $$ let $h_t(\\mathbf x,\\mathbf y)$ denote the heat kernel of the semigroup generated by the Dunkl Laplace operator $\\Delta_k$. Let $d(\\mathbf x,\\mathbf y)=\\min_{\\sigma\\in G} \\| \\mathbf x-\\sigma(\\mathbf y)\\|$, where $G$ is the reflection group associated with $R$. We derive the following upper and lower bounds for $h_t(\\mathbf x,\\mathbf y)$: for all $c_l>1/4$ and $0<c_u<1/4$ there are constants $C_l,C_u>0$ such that $$ C_{l}w(B(\\mathbf{x},\\sqrt{t}))^{-1}e^{-c_{l}\\frac{d(\\mathbf{x},\\mathbf{y})^2}{t}} \\Lambda(\\mathbf x,\\mathbf y,t) \\leq h_t(\\mathbf{x},\\mathbf{y}) \\leq C_{u}w(B(\\mathbf{x},\\sqrt{t}))^{-1}e^{-c_{u}\\frac{d(\\mathbf{x},\\mathbf{y})^2}{t}} \\Lambda(\\mathbf x,\\mathbf y,t), $$ where $\\Lambda(\\mathbf x,\\mathbf y,t)$ can be expressed by means of some rational functions of $\\| \\mathbf x-\\sigma(\\mathbf y)\\|/\\sqrt{t}$. An exact formula for $\\Lambda(\\mathbf x,\\mathbf y,t)$ is provided.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-11-29T15:51:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "44A20",
      "35K08",
      "33C52",
      "43A32",
      "39A70"
    ],
    "keywords": [
      "dunkl heat kernel",
      "lower bounds",
      "dunkl laplace operator",
      "multiplicity function",
      "normalized root system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}