{
  "id": "1911.10292",
  "version": "v1",
  "published": "2019-11-23T01:21:36.000Z",
  "updated": "2019-11-23T01:21:36.000Z",
  "title": "Nonlocal Poincaré Inequalities for Integral Operators with Integrable Nonhomogeneous Kernels",
  "authors": [
    "Mikil Foss"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The paper provides two versions of nonlocal Poincar\\'e-type inequalities for integral operators with a convolution-type structure and functions satisfying a zero-Dirichlet like condition. The inequalities extend existing results to a large class of nonhomogeneous kernels with supports that can vary discontinuously and need not contain a common set throughout the domain. The measure of the supports may even vanish allowing the zero-Dirichlet condition to be imposed on only a lower-dimensional manifold, with or without boundary. The conditions may be imposed on sets with co-dimension larger than one or even at just a single point. This appears to currently be the first such results in a nonlocal setting with integrable kernels. The arguments used are direct, and examples are provided demonstrating the explicit dependence of bounds for the Poincar\\'{e} constant upon structural parameters of the kernel and domain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-23T01:21:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D10",
      "45P05",
      "45E10",
      "47G10"
    ],
    "keywords": [
      "integral operators",
      "integrable nonhomogeneous kernels",
      "common set throughout",
      "nonlocal poincare-type inequalities",
      "inequalities extend existing results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}