{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $f \\colon \\R^n \\to \\C$ be locally integrable for [[Lebesgue measure|2Z2QVWZ262]] on [[Euclidean space|0XPR1E85W0]].\nLet $Q \\subseteq \\R^n$ be a cube with sides parallel to the axes, and let $\\abs{Q}$ be its volume.\nLet $f_Q = \\frac{1}{\\abs{Q}} \\int_Q f \\, dx$ be the mean of $f$ on $Q$, a [[Lebesgue integral|DWJF7ABPR1]].\nThe function $f$ has *bounded mean oscillation* if\n$$\\norm{f}_{\\mathrm{BMO}} = \\sup_Q \\frac{1}{\\abs{Q}} \\int_Q \\abs{f(x) - f_Q} \\, dx \u003c \\infty,$$\nwhere the supremum is over all such cubes $Q$.\nThe space of these functions is denoted $\\mathrm{BMO}(\\R^n)$.\n\n## Remarks\n\n$\\norm{\\cdot}_{\\mathrm{BMO}}$ vanishes exactly on the constant functions, so it is a norm on BMO modulo constants, which is a Banach space.\nThe same definition with arcs of the unit circle in place of cubes gives $\\mathrm{BMO}(\\mathbb{T})$.\nEvery bounded function lies in BMO with $\\norm{f}_{\\mathrm{BMO}} \\le 2 \\sup \\abs{f}$.\nThe John–Nirenberg inequality says that a BMO function is exponentially integrable on each cube after subtracting its mean.\n\nExample: $f(x) = \\log \\abs{x}$ on $\\R$ lies in BMO.\nFor $\\lambda \u003e 0$ one has $f(\\lambda x) = f(x) + \\log \\lambda$, and adding a constant does not change the mean oscillation.\nSo it suffices to bound the mean oscillation on intervals of length $1$, where a direct estimate works.\n\nNon-example: $f(x) = x$ on $\\R$.\nOn $Q = [0, L]$ one has $f_Q = L/2$ and $\\frac{1}{L} \\int_0^L \\abs{x - L/2} \\, dx = L/4$, which is unbounded in $L$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "QJYTXFT8Y4",
    "kind": "definition",
    "msc": [
      "30H35",
      "42B35"
    ],
    "name": "Bounded mean oscillation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bounded_mean_oscillation"
      }
    ],
    "tag": "MRDZAR4X0N",
    "uses": [
      "0XPR1E85W0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/MRDZAR4X0N/",
  "links": [
    {
      "field": "defines",
      "target": "QJYTXFT8Y4"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/MRDZAR4X0N.md",
  "msc": [
    "30H35",
    "42B35"
  ],
  "name": "Bounded mean oscillation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MRD/MRDZAR4X0N.md",
  "status": null,
  "tag": "MRDZAR4X0N",
  "uri": "m2://MRDZAR4X0N",
  "url": "https://math2.tamnd.com/wiki/bounded-mean-oscillation-definition/",
  "writing": {
    "tag": "MRDZAR4X0N",
    "kind": "definition",
    "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
      }
    ]
  }
}