{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R150",
  "content_hash": null,
  "slug": "cm3ih-claim-melas-upper-bound",
  "type": "claim",
  "title": "Melas's sharp weak-type theorem gives H_3 at most (11+sqrt(61))/4",
  "summary": "The sharp weak (1,1) constant for the centered maximal operator yields the rigorous upper bound 4.702562418976664.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Melas proved that the one-dimensional centered Hardy-Littlewood maximal operator has sharp weak \\((1,1)\\) constant\n\\[\nC=\\frac{11+\\sqrt{61}}{12}.\n\\]\nApply the inequality to \\(f=\\mathbf 1_A\\). For every \\(t<1/3\\),\n\\[\n|\\{M_c\\mathbf 1_A\\geq1/3\\}|\n\\leq |\\{M_c\\mathbf 1_A>t\\}|\n\\leq \\frac{C}{t}|A|.\n\\]\nLetting \\(t\\) increase to \\(1/3\\) gives\n\\[\nH_3\\leq 3C=\\frac{11+\\sqrt{61}}4\n=4.702562418976663\\ldots.\n\\]\nThe theorem applies to every integrable function, so the restriction to three intervals needs no extra hypothesis.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "all measurable subsets of the real line of finite positive measure, including unions of at most three bounded intervals"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/math/0311452",
      "locator": "Antonios D. Melas, The best constant for the centered Hardy-Littlewood maximal inequality, Annals of Mathematics 157 (2003), Theorem 1; DOI 10.4007/annals.2003.157.647"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0311452",
    "locator": "Antonios D. Melas, The best constant for the centered Hardy-Littlewood maximal inequality, Annals of Mathematics 157 (2003), Theorem 1; DOI 10.4007/annals.2003.157.647"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R149",
      "title": "The exact value of H_3 remains unresolved in this record",
      "object_type": "attempt",
      "relation": "constrains",
      "direction": "outgoing"
    },
    {
      "slug": "centered-maximal-three-interval-halo",
      "title": "centered maximal three interval halo",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}