{
  "id": "1209.0387",
  "version": "v1",
  "published": "2012-09-03T15:34:20.000Z",
  "updated": "2012-09-03T15:34:20.000Z",
  "title": "Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients",
  "authors": [
    "Marco Bramanti",
    "Giovanni Cupini",
    "Ermanno Lanconelli",
    "Enrico Priola"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a class of degenerate Ornstein-Uhlenbeck operators in $\\mathbb{R}^{N}$, of the kind [\\mathcal{A}\\equiv\\sum_{i,j=1}^{p_{0}}a_{ij}(x) \\partial_{x_{i}x_{j}}^{2}+\\sum_{i,j=1}^{N}b_{ij}x_{i}\\partial_{x_{j}}%] where $(a_{ij})$ is symmetric uniformly positive definite on $\\mathbb{R}^{p_{0}}$ ($p_{0}\\leq N$), with uniformly continuous and bounded entries, and $(b_{ij})$ is a constant matrix such that the frozen operator $\\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}(x_{0})$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\\infty$) of the kind:% [|\\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\\mathbb{R}% ^{N})}\\leq c{|\\mathcal{A}u|_{L^{p}(\\mathbb{R}^{N})}+|u|_{L^{p}(\\mathbb{R}% ^{N})}} for i,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% [|\\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\\leq c{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any $u\\in C_{0}^{\\infty}(S_{T}),$ where $S_{T}$ is the strip $\\mathbb{R}^{N}\\times[-T,T]$, $T$ small, and $L$ is the Kolmogorov-Fokker-Planck operator% [L\\equiv\\sum_{i,j=1}^{p_{0}}a_{ij}(x,t) \\partial_{x_{i}x_{j}}% ^{2}+\\sum_{i,j=1}^{N}b_{ij}x_{i}\\partial_{x_{j}}-\\partial_{t}%] with uniformly continuous and bounded $a_{ij}$'s.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-03T15:34:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35H10",
      "35B45",
      "35K70",
      "42B20"
    ],
    "keywords": [
      "degenerate ornstein-uhlenbeck operators",
      "variable coefficients",
      "symmetric uniformly positive definite",
      "constant matrix",
      "frozen operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.0387B"
    }
  }
}