{
  "id": "2007.01468",
  "version": "v1",
  "published": "2020-07-03T02:55:54.000Z",
  "updated": "2020-07-03T02:55:54.000Z",
  "title": "Weak type $(p,p)$ bounds for Schrödinger groups via generalized Gaussian estimates",
  "authors": [
    "Zhijie Fan"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$, where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\\leq p_0 < 2$. It is known that the operator $(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s\\geq n|{1/ 2}-{1/p}| $ and $ p\\in (p_0, p_0')$ (see for example, \\cite{Blunck2, BDN, CCO, CDLY, DN, Mi1}). In this paper we study the endpoint case $p=p_0$ and show that for $s_0= n\\big|{1\\over 2}-{1\\over p_0}\\big|$, the operator $(I+L)^{-{s_0}}e^{itL} $ is of weak type $(p_{0},p_{0})$, that is, there is a constant $C>0$, independent of $t$ and $f$ so that \\begin{eqnarray*} \\mu\\left(\\left\\{x: \\big|(I+L)^{-s_0}e^{itL} f(x)\\big|>\\alpha \\right\\} \\right)\\leq C (1+|t|)^{n(1 - {p_0\\over 2}) } \\left( {\\|f\\|_{p_0} \\over \\alpha} \\right)^{p_0} , \\ \\ \\ t\\in{\\mathbb R} \\end{eqnarray*} for $\\alpha>0$ when $\\mu(X)=\\infty$, and $\\alpha>\\big(\\|f\\|_{p_{0}}/\\mu(X) \\big)^{p_{0}}$ when $\\mu(X)<\\infty$. Our results can be applied to Schr\\\"odinger operators with rough potentials and %second order elliptic operators with rough lower order terms, or higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-03T02:55:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized gaussian estimates",
      "weak type",
      "schrödinger groups",
      "higher order elliptic operators",
      "rough lower order terms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}