{
  "id": "0807.4020",
  "version": "v1",
  "published": "2008-07-25T08:11:35.000Z",
  "updated": "2008-07-25T08:11:35.000Z",
  "title": "Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators",
  "authors": [
    "M. Bramanti",
    "G. Cupini",
    "E. Lanconelli",
    "E. 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}\\partial_{x_{i}x_{j}}^{2} +\\sum_{i,j=1}^{N}b_{ij}x_{i}\\partial_{x_{j}}% \\] where $(a_{ij}) ,(b_{ij}) $ are constant matrices, $(a_{ij}) $ is symmetric positive definite on $\\mathbb{R} ^{p_{0}}$ ($p_{0}\\leq N$), and $(b_{ij}) $ is such that $\\mathcal{A}$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\\infty$) of the kind:% \\[ \\Vert \\partial_{x_{i}x_{j}}^{2}u\\Vert_{L^{p}(\\mathbb{R}% ^{N})}\\leq c\\{\\Vert \\mathcal{A}u\\Vert_{L^{p}(\\mathbb{R}^{N})}+\\Vert u\\Vert_{L^{p}(\\mathbb{R}% ^{N})}\\} \\text{for}i,j=1,2,...,p_{0}% \\] and corresponding weak (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue $L^{p}$ spaces, for complete H\\\"{o}rmander's operators $ \\sum X_{i}^{2}+X_{0},$ proved in absence of a structure of homogeneous group. We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% \\[ \\Vert \\partial_{x_{i}x_{j}}^{2}u\\Vert_{L^{p}(S)}\\leq c\\Vert Lu\\Vert_{L^{p}(S)}% \\] for any $u\\in C_{0}^{\\infty}(S) ,$ where $S$ is the strip $\\mathbb{R}^{N}\\times[ -1,1] $ and $L$ is the Kolmogorov-Fokker-Planck operator $\\mathcal{A}-\\partial_{t}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-25T08:11:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35H10",
      "35K70",
      "42B20"
    ],
    "keywords": [
      "degenerate ornstein-uhlenbeck operators",
      "constant matrices",
      "symmetric positive definite",
      "first case",
      "global estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.4020B"
    }
  }
}