{
  "id": "1105.5987",
  "version": "v1",
  "published": "2011-05-30T13:23:55.000Z",
  "updated": "2011-05-30T13:23:55.000Z",
  "title": "A Characterization of the boundedness of the median maximal function on weighted L^p spaces",
  "authors": [
    "Henri Martikainen",
    "Tuomas Orponen"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We introduce and study the median maximal function \\mathcal{M} f, defined in the same manner as the classical Hardy-Littlewood maximal function, only replacing integral averages of f by medians throughout the definition. This change has a qualitative impact on the mapping properties of the maximal operator: in contrast with the Hardy-Littlewood operator, which is not bounded on L^1, we prove that \\mathcal{M} is bounded on L^p(w) for all 0 < p < \\infty, if and only if w \\in A_{\\infty}. The characterization is purely qualitative and does not give the dependence on [w]_{A_{\\infty}}. However, the sharp bound \\|\\mathcal{M}\\|_{L^1(w) \\to L^1(w)} \\lesssim [w]_{A_1} is established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-30T13:23:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B35"
    ],
    "keywords": [
      "median maximal function",
      "characterization",
      "boundedness",
      "classical hardy-littlewood maximal function",
      "medians throughout"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.5987M"
    }
  }
}