{
  "id": "1108.2515",
  "version": "v2",
  "published": "2011-08-11T20:40:24.000Z",
  "updated": "2013-08-08T17:39:07.000Z",
  "title": "Estimates for the Poisson kernel and the evolution kernel on nilpotent meta-abelian groups",
  "authors": [
    "Richard Penney",
    "Roman Urban"
  ],
  "comment": "28 pages; this is a shorter version; some sections of the previous version (on skew-product formula) have already appeared in print in J. Evol. Equ. 12, No. 2 (2012), 327-351",
  "journal": "Studia Math. 219(1):69--96, 2013",
  "categories": [
    "math.FA",
    "math.AP",
    "math.PR"
  ],
  "abstract": "Let $S$ be a semi direct product $S=N\\rtimes A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic with $\\R^k,$ $k>1.$ We consider a class of second order left-invariant differential operators on $S$ of the form $\\mathcal L_\\alpha=L^a+\\Delta_\\alpha,$ where $\\alpha\\in\\R^k,$ and for each $a\\in\\R^k,$ $L^a$ is left-invariant second order differential operator on $N$ and $\\Delta_\\alpha=\\Delta-<\\alpha,\\nabla>,$ where $\\Delta$ is the usual Laplacian on $\\R^k.$ Using some probabilistic techniques (e.g., skew-product formulas for diffusions on $S$ and $N$ respectively) we obtain an upper bound for the Poisson kernel for $\\mathcal L_\\alpha.$ We also give an upper estimate for the transition probabilities of the evolution on $N$ generated by $L^{\\sigma(t)},$ where $\\sigma$ is a continuous function from $[0,\\infty)$ to $\\R^k.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-08T17:39:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A85",
      "31B05",
      "22E25",
      "22E30",
      "60J25",
      "60J60"
    ],
    "keywords": [
      "nilpotent meta-abelian groups",
      "poisson kernel",
      "evolution kernel",
      "left-invariant second order differential operator",
      "second order left-invariant differential operators"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.2515P"
    }
  }
}