{
  "id": "2308.12936",
  "version": "v1",
  "published": "2023-08-24T17:25:08.000Z",
  "updated": "2023-08-24T17:25:08.000Z",
  "title": "On the regularity problem for parabolic operators and the role of half-time derivative",
  "authors": [
    "Martin Dindoš"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we present the following result on regularity of solutions of the second order parabolic equation $\\partial_t u - \\mbox{div} (A \\nabla u)+B\\cdot \\nabla u=0$ on cylindrical domains of the form $\\Omega=\\mathcal O\\times\\mathbb R$ where $\\mathcal O\\subset\\mathbb R^n$ is a uniform domain (it satisfies both corkscrew and Harnack chain conditions) and has uniformly $n-1$ rectifiable boundary. Let $u$ be a solution of such PDE in $\\Omega$ and the non-tangential maximal function of its gradient in spatial directions $\\tilde{N}(\\nabla u)$ belongs to $L^p(\\partial\\Omega)$ for some $p>1$. Furthermore, assume that for $u|_{\\partial\\Omega}=f$ we have that $D^{1/2}_tf\\in L^p(\\partial\\Omega)$. Then both $\\tilde{N}(D^{1/2}_t u)$ and $\\tilde{N}(D^{1/2}_tH_t u)$ also belong to $L^p(\\partial\\Omega)$, where $D^{1/2}_t$ and $H_t$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $L^p$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-24T17:25:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K10",
      "35K20",
      "35K40",
      "35K51"
    ],
    "keywords": [
      "parabolic operators",
      "parabolic regularity problem",
      "half-time derivative",
      "second order parabolic equation",
      "harnack chain conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}