{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "FFM7Y61KRT"
    }
  ],
  "body": "\n## Statement\n\nLet $n \\ge 1$, let $1 \\le p \\le 2$, and let $q$ be the conjugate exponent, $\\frac{1}{p} + \\frac{1}{q} = 1$.\nLet $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ be the [[Fourier transform|DPVM34ADZV]] of $f \\in L^1(\\R^n)$.\nThen for every $f \\in L^1(\\R^n) \\cap L^p(\\R^n)$,\n$$\\norm{\\hat f}_{L^q(\\R^n)} \\le \\norm{f}_{L^p(\\R^n)}.$$\nConsequently the Fourier transform extends uniquely to a bounded linear map from [[$L^p(\\R^n)$|SASP14MA1Z]] to $L^q(\\R^n)$ of norm at most $1$.\n\n## Remarks\n\nThe two endpoint cases are known: for $p = 1$, $\\abs{\\hat f(\\xi)} \\le \\norm{f}_1$ for every $\\xi$, and for $p = 2$, the [[Plancherel theorem|HPZHRZ0502]] gives $\\norm{\\hat f}_2 = \\norm{f}_2$.\nThe [[Riesz–Thorin theorem|AGRKRG1TMJ]] interpolates between them and gives the inequality for $1 \u003c p \u003c 2$.\n\nThe constant $1$ is not sharp for $1 \u003c p \u003c 2$; the sharp constant is given by the Babenko–Beckner inequality.\nThe restriction $p \\le 2$ is necessary: for $p \u003e 2$ there are functions in $L^p(\\R^n)$ whose Fourier transform, as a tempered distribution, is not a function.\n\nThere is a version for Fourier series on the circle $\\mathbb{T} = \\R / \\Z$ with normalized measure: for $1 \\le p \\le 2$ and $f \\in L^p(\\mathbb{T})$, $\\left(\\sum_{k \\in \\Z} \\abs{\\hat f(k)}^q\\right)^{1/q} \\le \\norm{f}_{L^p(\\mathbb{T})}$.\n\nExample: for $f(x) = e^{-\\pi x^2}$ on $\\R$, $\\hat f = f$, and $\\norm{f}_r = r^{-1/(2r)}$ for every $r \u003e 0$.\nSo $\\norm{\\hat f}_q / \\norm{f}_p = \\left( p^{1/p} / q^{1/q} \\right)^{1/2}$, and the theorem says that this number is at most $1$.\n\n## Explanations\n\n### L1 human\n\nThe Fourier transform cannot make a function much bigger in a suitable average sense.\nIf $f$ is $p$-th power integrable with $p$ between $1$ and $2$, then its transform is $q$-th power integrable for the conjugate exponent $q$, with norm no larger.\n\n### L2 human\n\nThe inequality is the first example of an $L^p$ estimate obtained by interpolation, and it is the model for the Riesz–Thorin method.\nIt shows that the Fourier transform is defined as a function on $L^p$ for $1 \\le p \\le 2$.\nIt is used in restriction theory, in the theory of Fourier multipliers and in the entropic uncertainty principle.\nOn locally compact abelian groups the same statement holds, with the dual group in place of the frequency space.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "DPVM34ADZV"
    ],
    "kind": "statement",
    "msc": [
      "42B10",
      "42A38",
      "42A16"
    ],
    "name": "Hausdorff–Young inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q5682823",
      "wikipedia": "Hausdorff–Young inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hausdorff%E2%80%93Young_inequality"
      }
    ],
    "tag": "HXNJH6W23S",
    "uses": [
      "DPVM34ADZV",
      "SASP14MA1Z",
      "AGRKRG1TMJ",
      "HPZHRZ0502"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/HXNJH6W23S/",
  "links": [
    {
      "field": "about",
      "target": "DPVM34ADZV"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    },
    {
      "field": "uses",
      "target": "AGRKRG1TMJ"
    },
    {
      "field": "uses",
      "target": "HPZHRZ0502"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HXNJH6W23S.md",
  "msc": [
    "42B10",
    "42A38",
    "42A16"
  ],
  "name": "Hausdorff–Young inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HXN/HXNJH6W23S.md",
  "status": null,
  "tag": "HXNJH6W23S",
  "uri": "m2://HXNJH6W23S",
  "url": "https://tamnd.github.io/math2/wiki/hausdorff-young-inequality/",
  "writing": {
    "tag": "HXNJH6W23S",
    "kind": "statement",
    "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
      }
    ]
  }
}