{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary with a smooth positive density $\\mu$.\nLet $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$.\nFor a smooth vector bundle $V$ over $M$, let $\\Hom(V, V)$ also denote the bundle over $M \\times M$ with fibre $\\Hom(V_y, V_x)$ at $(x, y)$.\nA linear operator $R \\colon \\Gamma(V) \\to \\Gamma(V)$ is *smoothing* if there is a smooth section $K$ of this bundle such that\n$$(R u)(x) = \\int_M K(x, y) \\, u(y) \\, d\\mu(y) \\quad \\text{for all } u \\in \\Gamma(V).$$\nLet $P \\colon \\Gamma(E) \\to \\Gamma(F)$ be a [[differential operator|CGBX1N9V8H]].\nA *parametrix* for $P$ is a continuous linear operator $Q \\colon \\Gamma(F) \\to \\Gamma(E)$ such that $QP - \\id$ and $PQ - \\id$ are smoothing.\n\n## Remarks\n\nAn operator $Q$ with only $QP - \\id$ smoothing is a left parametrix, and one with only $PQ - \\id$ smoothing is a right parametrix.\nTwo parametrices of the same operator differ by a smoothing operator.\nFor constant-coefficient operators on $\\R^n$, a parametrix corresponds to a distribution $e$ with $P e = \\delta + \\omega$ for a smooth function $\\omega$, an approximate [[fundamental solution|K2NK9A8M5F]].\nAn elliptic operator of order $m$ has a parametrix that is a [[pseudo-differential operator|FF27N0BWTP]] of order $-m$.\n\nExample: on the circle $S^1 = \\R / 2\\pi\\Z$ with $d\\mu = dx$, let $P = -d^2/dx^2$ on functions, and define\n$$Q\\Bigl(\\sum_{n \\in \\Z} c_n e^{inx}\\Bigr) = \\sum_{n \\ne 0} \\frac{c_n}{n^2} \\, e^{inx}.$$\nThen $PQu = QPu = u - c_0$, where $c_0 = \\frac{1}{2\\pi} \\int_0^{2\\pi} u(y) \\, dy$.\nSo $QP - \\id$ and $PQ - \\id$ are both the operator with the constant smooth kernel $K(x, y) = -\\frac{1}{2\\pi}$, and $Q$ is a parametrix.\n\nNon-example: the zero operator is not a parametrix for $P = -d^2/dx^2$.\nHere $PQ - \\id = -\\id$, and the identity is not smoothing.\nAn operator with a continuous kernel extends to a compact operator on $L^2(S^1)$, while the identity of an infinite-dimensional Hilbert space is not compact.\n\n## Explanations\n\n### L1 human\n\nA parametrix inverts an operator up to an error that has a smooth kernel, an error that makes every input smooth.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Z97HVFD6JE",
    "kind": "definition",
    "msc": [
      "58J40",
      "35S05",
      "35A08"
    ],
    "name": "Parametrix",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Parametrix"
      }
    ],
    "tag": "C24D9MWRV3",
    "uses": [
      "CGBX1N9V8H",
      "P4DZM975BV",
      "F45VQ47GPV",
      "K2NK9A8M5F",
      "FF27N0BWTP"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/C24D9MWRV3/",
  "links": [
    {
      "field": "defines",
      "target": "Z97HVFD6JE"
    },
    {
      "field": "uses",
      "target": "CGBX1N9V8H"
    },
    {
      "field": "uses",
      "target": "P4DZM975BV"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "K2NK9A8M5F"
    },
    {
      "field": "uses",
      "target": "FF27N0BWTP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/C24D9MWRV3.md",
  "msc": [
    "58J40",
    "35S05",
    "35A08"
  ],
  "name": "Parametrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C24/C24D9MWRV3.md",
  "status": null,
  "tag": "C24D9MWRV3",
  "uri": "m2://C24D9MWRV3",
  "url": "https://tamnd.github.io/math2/wiki/parametrix-definition/",
  "writing": {
    "tag": "C24D9MWRV3",
    "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
      }
    ]
  }
}