{
  "id": "1304.3871",
  "version": "v2",
  "published": "2013-04-14T02:15:55.000Z",
  "updated": "2013-12-10T02:22:40.000Z",
  "title": "Morrey-Sobolev Spaces on Metric Measure Spaces",
  "authors": [
    "Yufeng Lu",
    "Dachun Yang",
    "Wen Yuan"
  ],
  "comment": "Potential Anal. (2013), DOI: 10.1007/s11118-013-9370-9",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\\mathscr{X},d,\\mu)$. The embedding of the Newton-Morrey-Sobolev space into the H\\\"older space is obtained if $\\mathscr{X}$ supports a weak Poincar\\'e inequality and the measure $\\mu$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Haj{\\l}asz gradient, the authors also introduce the Haj{\\l}asz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Haj{\\l}asz-Morrey-Sobolev space when $\\mu$ is doubling and $\\mathscr{X}$ supports a weak Poincar\\'e inequality. In particular, on the Euclidean space ${\\mathbb R}^n$, the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Haj{\\l}asz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when $(\\mathscr{X},d)$ is geometrically doubling and $\\mu$ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when $\\mu$ is doubling and satisfies some measure decay property, the authors further obtain the boundedness of some (fractional) maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces and Haj{\\l}asz-Morrey-Sobolev spaces.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-10T02:22:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "42B25",
      "42B35",
      "30L99"
    ],
    "keywords": [
      "metric measure space",
      "weak poincare inequality",
      "maximal operators",
      "morrey spaces",
      "newton-morrey-sobolev space coincides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.3871L"
    }
  }
}