{
  "id": "0803.0509",
  "version": "v1",
  "published": "2008-03-04T18:31:52.000Z",
  "updated": "2008-03-04T18:31:52.000Z",
  "title": "On a class of hypoelliptic operators with unbounded coefficients in ${\\matbb R}^N$",
  "authors": [
    "B. Farkas",
    "L. Lorenzi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a class of non-trivial perturbations ${\\mathscr A}$ of the degenerate Ornstein-Uhlenbeck operator in ${\\mathbb R}^N$. In fact we perturb both the diffusion and the drift part of the operator (say $Q$ and $B$) allowing the diffusion part to be unbounded in ${\\mathbb R}^N$. Assuming that the kernel of the matrix $Q(x)$ is invariant with respect to $x\\in {\\mathbb R}^N$ and the Kalman rank condition is satisfied at any $x\\in{\\mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernstein's method we prove that we can associate a semigroup $\\{T(t)\\}$ of bounded operators (in the space of bounded and continuous functions) with the operator ${\\mathscr A}$. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup $\\{T(t)\\}$ both in isotropic and anisotropic spaces of (H\\\"older-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${\\mathscr A}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-03-04T18:31:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K65",
      "35J70",
      "35B65",
      "35K15"
    ],
    "keywords": [
      "hypoelliptic operators",
      "unbounded coefficients",
      "kalman rank condition",
      "degenerate ornstein-uhlenbeck operator",
      "continuous functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.0509F"
    }
  }
}