{
  "id": "2103.03112",
  "version": "v1",
  "published": "2021-03-04T15:34:16.000Z",
  "updated": "2021-03-04T15:34:16.000Z",
  "title": "A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space",
  "authors": [
    "Wei Chen",
    "Jingya Cui"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $M$ be the Doob maximal operator on a filtered measure space and let $v$ be an $A_p$ weight with $1<p<+\\infty$. We try proving that $\\lVert M f\\rVert _{L ^{p}(v) }\\leq p'[v]^{\\frac{1}{p-1}}_{A_p}\\lVert f\\rVert _{L ^{p} (v)},$ where $1/p+1/p'=1.$ But we do not find an approach which gives the constant $p'.$ Our results are as follows: $\\lVert M f\\rVert _{L ^{p}(v) }\\leq \\min\\{p^{\\frac{1}{p-1}},~a^{\\frac{2}{p}}\\eta^{(p'-1)}\\}p'[v]^{\\frac{1}{p-1}}_{A_p}\\lVert f\\rVert _{L ^{p} (v)}, $ where $a>1$ and $\\eta=\\frac{a}{a-1}$ are the constants in the construction of the principal sets. Furthermore, we show that $$\\lim\\limits_{p\\rightarrow+\\infty}p^{\\frac{1}{p-1}}=\\lim\\limits_{p\\rightarrow+\\infty}(\\min\\limits_{a>1}a^{\\frac{2}{p}}\\eta^{(p'-1)})=1.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-04T15:34:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G46"
    ],
    "keywords": [
      "doob maximal operator",
      "filtered measure space",
      "boundedness",
      "principal sets",
      "construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}