{
  "id": "1512.08681",
  "version": "v1",
  "published": "2015-12-29T13:43:55.000Z",
  "updated": "2015-12-29T13:43:55.000Z",
  "title": "On the Boundedness of Multilinear Fractional Strong Maximal Operator with multiple weights",
  "authors": [
    "Mingming Cao",
    "Qingying Xue",
    "Kozo Yabuta"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "In this paper, we investigated the boundedness of multilinear fractional strong maximal operator $\\mathcal{M}_{\\mathcal{R},\\alpha}$ associated with rectangles or related to more general basis with multiple weights $A_{(\\vec{p},q),\\mathcal{R}}$. In the rectangles setting, we first gave an end-point estimate of $\\mathcal{M}_{\\mathcal{R},\\alpha}$, which not only extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extended the multilinear result of Grafakos, Liu, P\\'{e}rez and Torres ($\\alpha=0$) to the case $0<\\alpha<mn.$ Then, in one weight case, we gave several equivalent characterizations between $\\mathcal{M}_{\\mathcal{R},\\alpha}$ and $A_{(\\vec{p},q),\\mathcal{R}}$, by applying a different approach from what we have used before. Moreover, a sufficient condition for the two weighted norm inequality of $\\mathcal{M}_{\\mathcal{R},\\alpha}$ was presented and a version of vector-valued two weighted inequality for the strong maximal operator was established when $m=1$. In the general basis setting, we further studied the properties of the multiple weights $A_{(\\vec{p},q),\\mathcal{R}}$ conditions, including the equivalent characterizations and monotonic properties, which essentially extended one's previous understanding. Finally, a survey on multiple strong Muckenhoupt weights was given, which demonstrates the properties of multiple weights related to rectangles systematically.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-29T13:43:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "47G10"
    ],
    "keywords": [
      "multilinear fractional strong maximal operator",
      "multiple weights",
      "boundedness",
      "general basis",
      "equivalent characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151208681C"
    }
  }
}