{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "744826NNN9"
    },
    {
      "field": "uses",
      "target": "SAZ0ESVBVP"
    }
  ],
  "body": "\n## Summary\n\nLet $f$ be a locally integrable function on $\\R^n$.\nIts Hardy–Littlewood maximal function $Mf$ gives at each point the largest average of $\\abs{f}$ over balls centred there.\nIt dominates $\\abs{f}$ almost everywhere and controls all local averages of $f$ at once.\nThe maximal operator $f \\mapsto Mf$ is bounded on $L^p$ for $1 \u003c p \\le \\infty$ and of weak type $(1,1)$.\nThese bounds give the Lebesgue differentiation theorem and are a basic tool in harmonic analysis.\n\n## Explanations\n\n### L1 human\n\nAt each point, look at all balls centred there and average $\\abs{f}$ over each ball.\nThe maximal function records the biggest of these averages.\nIt is a worst-case summary of how large $f$ is near the point at every scale.\nFor the indicator of $[0, 1]$ on the line, the maximal function at a point $x \u003e 1$ is $1/(2x)$, so it decays slowly far away.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "42B25",
      "28A15"
    ],
    "name": "Hardy–Littlewood maximal function",
    "same_as": {
      "wikidata": "Q3075250",
      "wikipedia": "Hardy–Littlewood maximal function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hardy–Littlewood_maximal_function"
      }
    ],
    "tag": "VX1A56P5KQ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/VX1A56P5KQ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/VX1A56P5KQ.md",
  "msc": [
    "42B25",
    "28A15"
  ],
  "name": "Hardy–Littlewood maximal function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VX1/VX1A56P5KQ.md",
  "status": null,
  "tag": "VX1A56P5KQ",
  "uri": "m2://VX1A56P5KQ",
  "url": "https://math2.tamnd.com/wiki/hardy-littlewood-maximal-function/",
  "writing": {
    "tag": "VX1A56P5KQ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}