{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2R4H0RFQZZ"
    },
    {
      "field": "uses",
      "target": "375ZGE0V2W"
    },
    {
      "field": "uses",
      "target": "4A5MJY4XPV"
    },
    {
      "field": "uses",
      "target": "F5V7CXW30E"
    },
    {
      "field": "uses",
      "target": "Y1E9SJC3Y0"
    },
    {
      "field": "about",
      "target": "375ZGE0V2W"
    },
    {
      "field": "about",
      "target": "4A5MJY4XPV"
    },
    {
      "field": "about",
      "target": "F5V7CXW30E"
    },
    {
      "field": "defines",
      "target": "41D1KJWM78"
    }
  ],
  "body": "\n## Summary\n\nAn $L^p$ multiplier is a bounded function $m$ on $\\R^n$ such that the operator $f \\mapsto (m \\hat f)^\\vee$ is bounded on the [[$L^p$ space|SASP14MA1Z]] $L^p(\\R^n)$.\nIt is a [[Fourier multiplier|ZQEFCY61T7]] that respects the size of functions measured in the $L^p$ norm.\nFor $p = 2$ every bounded function is an $L^p$ multiplier, but for $p \\ne 2$ the question is subtle.\nMany operators of analysis and PDE are multiplier operators, such as partial sums of Fourier integrals and Riesz transforms.\nSo criteria for $L^p$ multipliers are a central topic of harmonic analysis.\n\n## Explanations\n\n### L1 human\n\nA multiplier operator changes a function by turning each of its frequencies up or down by a factor $m(\\xi)$.\nThe question is whether this change can blow up the size of the function.\nIn the $L^2$ norm, the size is the total energy over all frequencies, so a bounded factor can never blow it up.\nIn other $L^p$ norms, sharp cuts in the frequency picture can make the function much larger, so not every bounded $m$ works.\nFor example, $m(\\xi) = e^{2\\pi i \\xi h}$ only shifts a function by $h$, so it keeps every $L^p$ norm and is an $L^p$ multiplier for all $p$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "42B15",
      "42A45"
    ],
    "name": "Lp multiplier",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Multiplier_(Fourier_analysis)"
      }
    ],
    "specializes": [
      "ZQEFCY61T7"
    ],
    "tag": "RAGR38Q40B"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/RAGR38Q40B/",
  "links": [
    {
      "field": "specializes",
      "target": "ZQEFCY61T7"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/RAGR38Q40B.md",
  "msc": [
    "42B15",
    "42A45"
  ],
  "name": "Lp multiplier",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RAG/RAGR38Q40B.md",
  "status": null,
  "tag": "RAGR38Q40B",
  "uri": "m2://RAGR38Q40B",
  "url": "https://tamnd.github.io/math2/wiki/lp-multiplier/",
  "writing": {
    "tag": "RAGR38Q40B",
    "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
      }
    ]
  }
}