{
  "id": "1007.1103",
  "version": "v3",
  "published": "2010-07-07T10:48:17.000Z",
  "updated": "2011-03-08T09:45:02.000Z",
  "title": "On Sobolev regularity of mass transport and transportation inequalities",
  "authors": [
    "Alexander V. Kolesnikov"
  ],
  "comment": "21 pages; 34 references. minor changes",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study Sobolev a priori estimates for the optimal transportation $T = \\nabla \\Phi$ between probability measures $\\mu=e^{-V} \\ dx$ and $\\nu=e^{-W} \\ dx$ on $\\R^d$. Assuming uniform convexity of the potential $W$ we show that $\\int \\| D^2 \\Phi\\|^2_{HS} \\ d\\mu$, where $\\|\\cdot\\|_{HS}$ is the Hilbert-Schmidt norm, is controlled by the Fisher information of $\\mu$. In addition, we prove similar estimate for the $L^p(\\mu)$-norms of $\\|D^2 \\Phi\\|$ and obtain some $L^p$-generalizations of the well-known Caffarelli contraction theorem. We establish a connection of our results with the Talagrand transportation inequality. We also prove a corresponding dimension-free version for the relative Fisher information with respect to a Gaussian measure.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-03-08T09:45:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "35B45"
    ],
    "keywords": [
      "sobolev regularity",
      "mass transport",
      "fisher information",
      "well-known caffarelli contraction theorem",
      "talagrand transportation inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.1103K"
    }
  }
}