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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $B = \\set{\\xi \\in \\R^n : \\abs{\\xi} \\le 1}$ be the closed unit ball, and let $\\mathbf{1}_B$ be its indicator function.\nThen $\\mathbf{1}_B$ is an [[$L^p$ multiplier|RAGR38Q40B]] on $\\R^n$ if and only if $p = 2$.\n\n## Remarks\n\nFor $p = 2$, the indicator is bounded, so the Plancherel theorem gives the bound.\nFor $p \\ne 2$, Charles Fefferman's 1971 proof uses [[Kakeya sets|6PEN7AKC7M]] of small measure in the plane.\nBy a lemma of Yves Meyer, a bound for the ball multiplier would give a vector-valued bound for the multipliers of many half-planes with different normal directions.\nA Kakeya construction gives many thin rectangles whose union has small area while their translates in the direction of their long sides are disjoint, and this contradicts that vector-valued bound.\nBy duality, it suffices to disprove the case $p \u003e 2$.\n\nThe theorem contrasts with dimension one, where the indicator of an interval is an $L^p$ multiplier for all $1 \u003c p \u003c \\infty$, since it is a combination of modulated Hilbert transforms.\nIt also contrasts with cubes: the indicator of a cube is a product of indicators of intervals, so it is an $L^p$ multiplier for $1 \u003c p \u003c \\infty$.\n\nA consequence is that the spherical partial sums $(\\mathbf{1}_{RB} \\hat f)^\\vee$ of the Fourier integral of $f \\in L^p(\\R^n)$ need not converge to $f$ in $L^p$ when $p \\ne 2$ and $n \\ge 2$.\n\n## Explanations\n\n### L1 human\n\nTake a function in several variables and keep only its frequencies inside a ball, throwing the rest away.\nFor the energy norm $L^2$ this is harmless.\nThe theorem says that for every other $L^p$ norm, this sharp round cut can make functions arbitrarily large compared to what they were.\nThe curved boundary of the ball is the problem: cutting along a flat face, as with a cube, is fine.\n\n### L2 human\n\nThis result explains why one studies the smoothed [[Bochner–Riesz means|AFS2DCWCD3]] instead of sharp spherical partial sums in higher dimensions.\nIts proof brought the Kakeya problem into Fourier analysis, a link that runs through the [[restriction conjecture|9CE11G48JY]] as well.\n",
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