{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "7TVFY6NZDB"
    },
    {
      "field": "about",
      "target": "R1PHMVBJ9J"
    },
    {
      "field": "uses",
      "target": "R1PHMVBJ9J"
    }
  ],
  "body": "\n## Summary\n\nThe weak $L^p$ space $L^{p,\\infty}(\\mu)$ contains the measurable functions whose [[distribution function|HQVXKEMN5N]] decays at least as fast as $t^{-p}$.\nIt contains [[$L^p(\\mu)$|SASP14MA1Z]], by Chebyshev's inequality.\nIt is the natural target space for operators that fail to be bounded on $L^1$, such as the [[Hardy–Littlewood maximal function|VX1A56P5KQ]].\nIt is the [[Lorentz space|3240RP14MT]] $L^{p,q}$ with $q = \\infty$.\n\n## Explanations\n\n### L1 human\n\nA function is in $L^p$ when $\\abs{f}^p$ has finite integral.\nWeak $L^p$ asks only that the set where $\\abs{f}$ exceeds $t$ has measure at most $C t^{-p}$.\nThis allows functions that just barely fail to be in $L^p$.\nOn $\\R$, the function $1/\\abs{x}$ is not integrable near $0$, but the set where it exceeds $t$ is an interval of length $2/t$, so it lies in weak $L^1$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46E30",
      "42B35"
    ],
    "name": "Weak Lp space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lorentz_space"
      }
    ],
    "specializes": [
      "3240RP14MT"
    ],
    "tag": "H4C888M5AZ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/H4C888M5AZ/",
  "links": [
    {
      "field": "specializes",
      "target": "3240RP14MT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/H4C888M5AZ.md",
  "msc": [
    "46E30",
    "42B35"
  ],
  "name": "Weak Lp space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/H4C/H4C888M5AZ.md",
  "status": null,
  "tag": "H4C888M5AZ",
  "uri": "m2://H4C888M5AZ",
  "url": "https://math2.tamnd.com/wiki/weak-lp-space/",
  "writing": {
    "tag": "H4C888M5AZ",
    "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
      }
    ]
  }
}