{
  "id": "1101.1016",
  "version": "v2",
  "published": "2011-01-05T16:01:16.000Z",
  "updated": "2011-09-15T16:12:58.000Z",
  "title": "Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces",
  "authors": [
    "Renjin Jiang"
  ],
  "comment": "Journal of Functional Analysis, to appear",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $(X,d)$ be a complete, pathwise connected metric measure space with locally Ahlfors $Q$-regular measure $\\mu$, where $Q>1$. Suppose that $(X,d,\\mu)$ supports a (local) $(1,2)$-Poincar\\'e inequality and a suitable curvature lower bound. For the Poisson equation $\\Delta u=f$ on $(X,d,\\mu)$, Moser-Trudinger and Sobolev inequalities are established for the gradient of $u$. The local H\\\"older continuity with optimal exponent of solutions is obtained.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-09-15T16:12:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C25",
      "31C45",
      "35B33",
      "35B65"
    ],
    "keywords": [
      "poisson equation",
      "gradient estimate",
      "suitable curvature lower bound",
      "pathwise connected metric measure space",
      "regular measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}