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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $S \\subseteq \\R^n$ be a smooth hypersurface whose [[Gaussian curvature|SQVQ5DQT6W]] is nonzero at every point.\nLet $\\sigma$ be the surface measure on $S$, and let $\\psi \\colon \\R^n \\to \\C$ be a [[smooth function|0V33BNKZ9S]] with compact support.\nLet $\\mu = \\psi \\, d\\sigma$, and let its [[Fourier transform|DPVM34ADZV]] be\n$$\\hat\\mu(\\xi) = \\int_S e^{-2\\pi i x \\cdot \\xi} \\psi(x) \\, d\\sigma(x).$$\nThen there is a constant $C$ such that\n$$\\abs{\\hat\\mu(\\xi)} \\le C (1 + \\abs{\\xi})^{-(n-1)/2} \\quad \\text{for all } \\xi \\in \\R^n.$$\n\n## Remarks\n\nBy a partition of unity, one may assume that the support of $\\psi$ meets $S$ in a small piece.\nThis piece is the graph of a smooth function $h$ over a ball in $\\R^{n-1}$.\nThen $\\hat\\mu(\\xi)$ is an [[oscillatory integral|D5HZKE8E1A]] in $n - 1$ variables with phase $-2\\pi (x' \\cdot \\xi' + h(x')\\xi_n)$.\nIf $\\xi$ is far from the normal directions to the piece, the phase has no critical point and the integral decays faster than any power.\nIf $\\xi$ is close to a normal direction, the critical points are nondegenerate, because the curvature hypothesis makes the Hessian of $h$ invertible.\nThen stationary phase gives the decay $\\abs{\\xi}^{-(n-1)/2}$.\n\nExample: for the unit sphere $S^2 \\subseteq \\R^3$ and $\\psi = 1$ near $S^2$, a computation in polar coordinates gives\n$$\\hat\\sigma(\\xi) = 2\\pi \\int_{-1}^{1} e^{-2\\pi i \\abs{\\xi} t} \\, dt = \\frac{2 \\sin(2\\pi\\abs{\\xi})}{\\abs{\\xi}},$$\nwhich decays like $\\abs{\\xi}^{-1} = \\abs{\\xi}^{-(3-1)/2}$ and no faster.\n\nThe curvature hypothesis is needed.\nIf $S$ is the hyperplane $\\set{x_n = 0}$, which has curvature zero, then $e^{-2\\pi i x \\cdot \\xi} = 1$ on $S$ when $\\xi = (0, \\ldots, 0, \\xi_n)$.\nSo $\\hat\\mu(0, \\ldots, 0, \\xi_n) = \\int_S \\psi \\, d\\sigma$ for every $\\xi_n$, and there is no decay when this integral is not $0$.\n\n## Explanations\n\n### L1 human\n\nA measure spread on a curved surface, like the sphere, has a Fourier transform that tends to zero at infinity at a definite rate.\nThe more the surface curves in all directions, the more the waves cancel, and the rate is half the dimension of the surface.\nA flat surface gives no decay in the direction perpendicular to it.\n\n### L2 human\n\nThis decay is the starting point of the [[Stein–Tomas restriction theorem|ZZVWHNFY6F]], whose proof only uses the bound $\\abs{\\hat\\mu(\\xi)} \\lesssim \\abs{\\xi}^{-(n-1)/2}$.\nIt also gives the dispersive decay of waves and the asymptotics of Bessel functions, since $\\hat\\sigma$ for the sphere is a Bessel function.\nWhen some principal curvatures vanish, the decay is weaker and is measured by the number of nonvanishing curvatures.\n",
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