{
  "id": "1804.07950",
  "version": "v1",
  "published": "2018-04-21T11:46:41.000Z",
  "updated": "2018-04-21T11:46:41.000Z",
  "title": "Defect of compactness for Sobolev spaces on manifolds with bounded geometry",
  "authors": [
    "Leszek Skrzypczak",
    "Cyril Tintarev"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on manifolds different from M, that are induced by a limiting procedure. The profiles satisfy an inequality of Plancherel type, and a similar relation, related to the Brezis-Lieb Lemma, holds for Lebesgue norms of profiles on the respective manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-21T11:46:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "46B50",
      "58J99",
      "35B44",
      "35A25"
    ],
    "keywords": [
      "sobolev space",
      "bounded geometry",
      "compactness",
      "elementary concentrations",
      "profile decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}