{
  "id": "2006.10326",
  "version": "v1",
  "published": "2020-06-18T07:31:58.000Z",
  "updated": "2020-06-18T07:31:58.000Z",
  "title": "The square root of a parabolic operator",
  "authors": [
    "El Maati Ouhabaz"
  ],
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "Let L(t) = --div (A(x, t)$\\nabla$ x) for t $\\in$ (0, $\\tau$) be a uniformly elliptic operator with boundary conditions on a domain $\\Omega$ of R d and $\\partial$ = $\\partial$ $\\partial$t. Define the parabolic operator L = $\\partial$ + L on L 2 (0, $\\tau$, L 2 ($\\Omega$)) by (Lu)(t) := $\\partial$u(t) $\\partial$t + L(t)u(t). We assume a very little of regularity for the boundary of $\\Omega$ and assume that the coefficients A(x, t) are measurable in x and piecewise C $\\alpha$ in t for some $\\alpha$ > 1 2. We prove the Kato square root property for $\\sqrt$ L and the estimate $\\sqrt$ L u L 2 (0,$\\tau$,L 2 ($\\Omega$)) $\\approx$ $\\nabla$ x u L 2 (0,$\\tau$,L 2 ($\\Omega$)) + u H 1 2 (0,$\\tau$,L 2 ($\\Omega$)) + $\\tau$ 0 u(t) 2 L 2 ($\\Omega$) dt t 1/2. We also prove L p-versions of this result. Keywords: elliptic and parabolic operators, the Kato square root property, maximal regularity, the holomorphic functional calculus, non-autonomous evolution equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-18T07:31:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic operator",
      "kato square root property",
      "holomorphic functional calculus",
      "non-autonomous evolution equations",
      "maximal regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}