{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\abs{z}$ be the Euclidean norm on $\\C^n$.\nThe *Lelong class* $\\mathcal{L} = \\mathcal{L}(\\C^n)$ is the set of [[plurisubharmonic functions|FF2P7ER3QT]] $u$ on $\\C^n$ for which there is a constant $C_u \\in \\R$ with\n$$u(z) \\le \\log(1 + \\abs{z}) + C_u \\quad \\text{for all } z \\in \\C^n .$$\nThe subclass $\\mathcal{L}^+$ consists of the $u \\in \\mathcal{L}$ for which there is also a constant $c_u \\in \\R$ with $u(z) \\ge \\log(1 + \\abs{z}) + c_u$ for all $z \\in \\C^n$.\n\n## Remarks\n\nExample: let $p$ be a polynomial on $\\C^n$ of degree $d \\ge 1$ and $u = \\frac{1}{d} \\log\\abs{p}$.\nThen $u$ is plurisubharmonic, and $\\abs{p(z)} \\le M (1 + \\abs{z})^d$ for some $M \u003e 0$, so $u(z) \\le \\log(1 + \\abs{z}) + \\frac{1}{d} \\log M$; hence $u \\in \\mathcal{L}$.\nIt is not in $\\mathcal{L}^+$ when $p$ has a zero, since $u = -\\infty$ there.\nThe function $\\log(1 + \\abs{z}^2)/2$ is in $\\mathcal{L}^+$.\nNon-example: $u(z) = \\abs{z_1}$ is plurisubharmonic, but $\\abs{z_1} - \\log(1 + \\abs{z})$ is unbounded above, so $u \\notin \\mathcal{L}$.\nFor $n = 1$, the logarithmic potentials $\\int \\log\\abs{z - w} \\, d\\mu(w)$ for compactly supported probability measures $\\mu$ are in $\\mathcal{L}$.\nSome authors write $\\log^+\\abs{z} = \\max(\\log\\abs{z}, 0)$ in place of $\\log(1 + \\abs{z})$; the class is the same, since the two functions differ by a bounded amount.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "X4ZHQG9995",
    "kind": "definition",
    "msc": [
      "32U35",
      "32U15"
    ],
    "name": "Lelong class",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Plurisubharmonic_function"
      }
    ],
    "tag": "SV6A17MEYZ",
    "uses": [
      "FF2P7ER3QT"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/SV6A17MEYZ/",
  "links": [
    {
      "field": "defines",
      "target": "X4ZHQG9995"
    },
    {
      "field": "uses",
      "target": "FF2P7ER3QT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SV6A17MEYZ.md",
  "msc": [
    "32U35",
    "32U15"
  ],
  "name": "Lelong class",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SV6/SV6A17MEYZ.md",
  "status": null,
  "tag": "SV6A17MEYZ",
  "uri": "m2://SV6A17MEYZ",
  "url": "https://tamnd.github.io/math2/wiki/lelong-class-definition/",
  "writing": {
    "tag": "SV6A17MEYZ",
    "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
      }
    ]
  }
}