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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $S^{n-1} \\subseteq \\R^n$ be the unit [[sphere|T62369RFH6]], and let $\\sigma$ be its surface measure.\nLet $\\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.\nLet $1 \\le p \u003c \\frac{2n}{n+1}$, let $p'$ be given by $1/p + 1/p' = 1$, and let $1 \\le q \\le \\frac{n-1}{n+1} p'$.\nThen there is a constant $C$ such that\n$$\\Big( \\int_{S^{n-1}} \\abs{\\hat f(\\xi)}^q \\, d\\sigma(\\xi) \\Big)^{1/q} \\le C \\norm{f}_{L^p(\\R^n)}$$\nfor every [[Schwartz function|CG9JEATW3Q]] $f$ on $\\R^n$.\n\n## Remarks\n\nThe conditions on $p$ and $q$ are necessary.\nThe condition $p \u003c 2n/(n+1)$ comes from the function $\\widehat{d\\sigma}$.\nIt decays exactly like $\\abs{x}^{-(n-1)/2}$, so it lies in $L^{p'}$ only for $p' \u003e 2n/(n-1)$.\nBy duality, the conjecture requires $\\widehat{d\\sigma}$ to lie in $L^{p'}$.\nThe condition $q \\le \\frac{n-1}{n+1}p'$ comes from the Knapp example, a function whose Fourier transform is concentrated on a small cap of the sphere.\n\nStatus: the conjecture holds for $n = 2$ and is open for every $n \\ge 3$.\nThe [[Stein–Tomas restriction theorem|ZZVWHNFY6F]] proves the case $q = 2$, where the range is $p \\le 2(n+1)/(n+3)$.\nThe conjecture implies the Kakeya conjecture that every [[Kakeya set|6PEN7AKC7M]] in $\\R^n$ has Hausdorff dimension $n$.\n\n## Explanations\n\n### L1 human\n\nThe conjecture says exactly when the Fourier transform of a function in $L^p$ can be restricted to the sphere, with a bound in an $L^q$ norm on the sphere.\nThe two conditions on $p$ and $q$ are known to be needed, and the question is whether they are enough.\nIt is one of the main open problems of harmonic analysis, linked to the geometry of thin tubes in Kakeya sets.\n",
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    "about": [
      "DPVM34ADZV"
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    "history": {
      "posed_by": "Elias Stein"
    },
    "kind": "conjecture",
    "msc": [
      "42B10",
      "42B20"
    ],
    "name": "Restriction conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Restriction_conjecture"
      }
    ],
    "tag": "9CE11G48JY",
    "uses": [
      "DPVM34ADZV",
      "SASP14MA1Z",
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      "CG9JEATW3Q"
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/9CE11G48JY/",
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  "markdown": "https://tamnd.github.io/math2/node/9CE11G48JY.md",
  "msc": [
    "42B10",
    "42B20"
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  "name": "Restriction conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9CE/9CE11G48JY.md",
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  "tag": "9CE11G48JY",
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  "url": "https://tamnd.github.io/math2/wiki/restriction-conjecture/",
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    "tag": "9CE11G48JY",
    "kind": "conjecture",
    "level": "W0",
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