{
  "id": "1512.06577",
  "version": "v1",
  "published": "2015-12-21T10:50:14.000Z",
  "updated": "2015-12-21T10:50:14.000Z",
  "title": "The annular decay property and capacity estimates for thin annuli",
  "authors": [
    "Anders Björn",
    "Jana Björn",
    "Juha Lehrbäck"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP",
    "math.MG"
  ],
  "abstract": "We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also characterize the $1$-annular decay property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-21T10:50:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31E05",
      "30L99",
      "31C15",
      "31C45"
    ],
    "keywords": [
      "annular decay property",
      "thin annuli",
      "capacity estimates",
      "lower bounds",
      "poincare inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151206577B"
    }
  }
}