{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$, and let $\\chi$, $\\psi$ and $\\Delta_j$, $j \\in \\Z$, be as in the [[Littlewood–Paley decomposition|D1PAT0K446]] on $\\R^n$.\nFor $m \\in \\Z$ and a bounded function $g$, let $S_m g$ be defined by $\\widehat{S_m g}(\\xi) = \\chi(2^{-m}\\xi) \\hat g(\\xi)$, where $\\hat{\\ }$ is the [[Fourier transform|DPVM34ADZV]] of tempered distributions.\nFor $f$ in the [[Lp space|SASP14MA1Z]] $L^2(\\R^n)$ and $g \\in L^\\infty(\\R^n)$, the *paraproduct* of $f$ by $g$ is\n$$\\Pi_g f = \\sum_{j \\in \\Z} (S_{j-3} g) \\, (\\Delta_j f),$$\nwhere the series converges in $L^2(\\R^n)$.\n\n## Remarks\n\nThe function $S_m g$ is the convolution of $g$ with $2^{mn} \\check\\chi(2^m \\cdot)$, a smooth average of $g$ at scale $2^{-m}$, so $\\abs{S_m g} \\le C \\norm{g}_\\infty$.\nThe Fourier transform of the $j$-th term is supported in $\\set{2^{j-2} \\le \\abs{\\xi} \\le 2^{j+2}}$, because $\\Delta_j f$ has frequencies between $2^{j-1}$ and $2^{j+1}$ and $S_{j-3} g$ has frequencies below $2^{j-2}$.\nSo the terms are almost orthogonal, which gives the convergence and the bound $\\norm{\\Pi_g f}_2 \\le C \\norm{g}_\\infty \\norm{f}_2$.\nThe shift $3$ is a convention; other authors use a different shift or a different cut-off.\n\nGrouping the terms of $fg = \\sum_{j, k} \\Delta_j f \\, \\Delta_k g$ according to $k \\le j - 3$, $j \\le k - 3$ or $\\abs{j - k} \\le 2$ gives Bony's decomposition\n$$fg = \\Pi_g f + \\Pi_f g + \\sum_{\\abs{j - k} \\le 2} \\Delta_j f \\, \\Delta_k g$$\nfor $f$ and $g$ both in $L^2(\\R^n) \\cap L^\\infty(\\R^n)$, since $\\sum_{k \\le m} \\Delta_k g = S_m g$.\n\nExample: if $g = c$ is a constant, then $\\hat g$ is a multiple of the point mass at $0$ and $\\chi(0) = 1$, so $S_m g = c$ for every $m$.\nHence $\\Pi_g f = c \\sum_j \\Delta_j f = c f$.\n\nNon-example: the term $\\Delta_j f \\, \\Delta_j g$, a product of two pieces at the same frequency, is not part of a paraproduct.\nIts Fourier transform can be supported near $0$, for instance when $\\widehat{\\Delta_j g}$ is the reflection of $\\widehat{\\Delta_j f}$, so high frequencies combine into low ones; such terms go into the remainder.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9RCFS9W90H",
    "kind": "definition",
    "msc": [
      "42B25",
      "42B20",
      "35S50"
    ],
    "name": "Paraproduct",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Paraproduct"
      }
    ],
    "tag": "640A1MTM4Z",
    "uses": [
      "D1PAT0K446",
      "SASP14MA1Z",
      "DPVM34ADZV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/640A1MTM4Z/",
  "links": [
    {
      "field": "defines",
      "target": "9RCFS9W90H"
    },
    {
      "field": "uses",
      "target": "D1PAT0K446"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/640A1MTM4Z.md",
  "msc": [
    "42B25",
    "42B20",
    "35S50"
  ],
  "name": "Paraproduct",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/640/640A1MTM4Z.md",
  "status": null,
  "tag": "640A1MTM4Z",
  "uri": "m2://640A1MTM4Z",
  "url": "https://tamnd.github.io/math2/wiki/paraproduct-definition/",
  "writing": {
    "tag": "640A1MTM4Z",
    "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
      }
    ]
  }
}