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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $S \\subseteq \\R^n$ be a compact smooth hypersurface, possibly with boundary, whose [[Gaussian curvature|SQVQ5DQT6W]] is nonzero at every point.\nLet $\\sigma$ be the surface measure on $S$, and let $\\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.\nLet\n$$1 \\le p \\le \\frac{2(n + 1)}{n + 3}.$$\nThen there is a constant $C$ such that\n$$\\Big( \\int_S \\abs{\\hat f(\\xi)}^2 \\, d\\sigma(\\xi) \\Big)^{1/2} \\le C \\norm{f}_{L^p(\\R^n)}$$\nfor every [[Schwartz function|CG9JEATW3Q]] $f$ on $\\R^n$.\nSo the restriction map $f \\mapsto \\hat f|_S$ extends to a bounded operator from $L^p(\\R^n)$ to $L^2(S, \\sigma)$.\n\n## Remarks\n\nBy duality, the estimate is equivalent to $\\norm{\\widehat{g \\, d\\sigma}}_{L^{p'}(\\R^n)} \\le C \\norm{g}_{L^2(S)}$, with $1/p + 1/p' = 1$.\nA $TT^*$ argument reduces this to the bound $\\norm{f * \\hat\\sigma}_{L^{p'}} \\le C^2 \\norm{f}_{L^p}$.\nOne splits $\\hat\\sigma$ into dyadic pieces in $\\abs{x}$.\nEach piece is bounded from $L^1$ to $L^\\infty$ by the [[decay of the Fourier transform of surface measure|SYP1ATSTTH]], and on $L^2$ by the size of its Fourier transform.\nInterpolating these two bounds, the pieces sum when $p \u003c 2(n+1)/(n+3)$.\nPeter Tomas proved the range $p \u003c 2(n + 1)/(n + 3)$, and Elias Stein obtained the endpoint.\n\nExample: for the circle in $\\R^2$ the range is $1 \\le p \\le 6/5$, and for the sphere in $\\R^3$ it is $1 \\le p \\le 4/3$.\n\nThe exponent $2(n + 1)/(n + 3)$ is sharp for the $L^2(S)$ norm on the left.\nThe Knapp example takes $\\hat f$ to be a smooth bump adapted to a cap of $S$ of radius $\\delta$.\nSuch a cap lies in a slab of thickness about $\\delta^2$.\nComparing both sides as $\\delta \\to 0$ forces $p \\le 2(n+1)/(n+3)$.\nWithout curvature, for a flat piece of a hyperplane, no estimate with $p \u003e 1$ holds.\n\n## Explanations\n\n### L1 human\n\nFor a general function in $L^p$, its Fourier transform is only defined almost everywhere, so its values on a surface of measure zero seem meaningless.\nThe theorem says that the Fourier transform does have a meaningful restriction to a curved surface, such as a sphere.\nThis holds with a norm bound when $p$ is close enough to $1$.\nCurvature is what makes this work: on a flat plane, it fails.\n\n### L2 human\n\nThis is the main known case of the [[Fourier restriction problem|FFCH15ZNZG]] with $L^2$ norm on the surface, and it is sharp in that setting.\nIt is used to prove Strichartz estimates for the Schrödinger and wave equations, which are restriction estimates for paraboloids and cones.\nThe [[restriction conjecture|9CE11G48JY]] asks for more general $L^q(S)$ norms and a larger range of $p$.\n",
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