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  "body": "\n## Statement\n\nLet $\\delta \u003e 0$ and $1 \\le p \\le \\infty$.\nLet $S^\\delta$ be the [[Bochner–Riesz operator|AFS2DCWCD3]] of order $\\delta$ on $\\R^2$, the operator with multiplier $(1 - \\abs{\\xi}^2)_+^\\delta$.\nThen $S^\\delta$ is bounded on $L^p(\\R^2)$ if and only if\n$$\\Big| \\frac{1}{p} - \\frac{1}{2} \\Big| \u003c \\frac{1 + 2\\delta}{4}.$$\nIn particular, $S^\\delta$ is bounded on $L^p(\\R^2)$ for every $\\delta \u003e 0$ and every $4/3 \\le p \\le 4$.\n\n## Remarks\n\nLennart Carleson and Per Sjölin proved the sufficiency of the condition in 1972.\nThe necessity holds in every dimension $n$.\nThe kernel of $S^\\delta$ decays like $\\abs{x}^{-(n + 1)/2 - \\delta}$ with oscillation, and it lies in $L^p$ only when $1/p - 1/2 \u003c (1 + 2\\delta)/(2n)$.\nBoundedness of $S^\\delta$ on $L^p$ forces its kernel into $L^p$.\nSince $S^\\delta$ is self-adjoint, boundedness on $L^p$ is equivalent to boundedness on $L^{p'}$, and this gives the other half of the condition.\nThe proof of sufficiency decomposes the multiplier into pieces near the circle $\\abs{\\xi} = 1$ and uses an $L^4$ estimate for oscillatory integrals whose phase has curved level sets.\n\nExample: for $\\delta = 1/4$ the condition reads $\\abs{1/p - 1/2} \u003c 3/8$, that is $8/7 \u003c p \u003c 8$.\nWithout $\\delta \u003e 0$ the result fails: for $\\delta = 0$, [[Fefferman's ball multiplier theorem|4A5MJY4XPV]] shows that $S^0$ is bounded only on $L^2(\\R^2)$.\n\nThe same condition in dimension $n$, namely $\\abs{1/p - 1/2} \u003c (1 + 2\\delta)/(2n)$, is the Bochner–Riesz conjecture; the theorem settles it for $n = 2$, while it is open for $n \\ge 3$.\n\n## Explanations\n\n### L1 human\n\nIn the plane, rebuilding a function from its frequencies by a sharp round cut fails in every $L^p$ norm except $L^2$.\nThis theorem says that fading the frequencies out with any tiny amount of smoothness, any $\\delta \u003e 0$, already works for all $p$ between $4/3$ and $4$.\nIt also gives the exact range of $p$ for each $\\delta$.\n\n### L2 human\n\nThe theorem is the two-dimensional case of the Bochner–Riesz problem, and its proof has the same structure as the two-dimensional case of the [[restriction conjecture|9CE11G48JY]].\nFor $\\delta \u003e 0$ and $4/3 \\le p \\le 4$, it gives $L^p$ convergence of the Bochner–Riesz means $S^\\delta_R f \\to f$ for $f \\in L^p(\\R^2)$.\nIn higher dimensions, the [[Stein–Tomas restriction theorem|ZZVWHNFY6F]] gives a partial range, and the full range is open.\n",
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