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  "body": "\n## Problem\n\nLet $n \\ge 2$, let $S \\subseteq \\R^n$ be a compact smooth hypersurface, and let $\\sigma$ be its surface measure.\nLet $\\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.\nFor which pairs of exponents $1 \\le p, q \\le \\infty$ is there a constant $C$ such that\n$$\\Big( \\int_S \\abs{\\hat f(\\xi)}^q \\, d\\sigma(\\xi) \\Big)^{1/q} \\le C \\norm{f}_{L^p(\\R^n)}$$\nfor every [[Schwartz function|CG9JEATW3Q]] $f$ on $\\R^n$?\n\n## Remarks\n\nFor $p = 1$ the estimate holds for all $q$, since $\\abs{\\hat f} \\le \\norm{f}_{L^1}$ everywhere.\nFor $p = 2$ it fails for every $q$, since $\\hat f$ is then an arbitrary $L^2$ function and has no meaningful values on a set of measure zero.\nWhen $S$ is a piece of a hyperplane, it fails for every $p \u003e 1$.\nWhen $S$ has nonzero Gaussian curvature, the [[Stein–Tomas restriction theorem|ZZVWHNFY6F]] gives the estimate for $q = 2$ and $1 \\le p \\le 2(n+1)/(n+3)$.\nFor the sphere, the conjectured answer is the [[restriction conjecture|9CE11G48JY]]; it is known for $n = 2$ and open for $n \\ge 3$.\n\n## Explanations\n\n### L1 human\n\nThe Fourier transform of a general function is only known up to sets of measure zero, so its values on a surface seem to have no meaning.\nThe problem asks for which $L^p$ spaces the Fourier transform can nevertheless be restricted to a curved surface, with a bound on the result.\nThe answer depends on the curvature of the surface and is a central open question of harmonic analysis.\n",
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