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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $M$ be a smooth real hypersurface in $\\C^n$.\nLet $p \\in M$, and let $\\rho$ be a smooth real function on a neighbourhood $V$ of $p$ with $M \\cap V = \\set{\\rho = 0}$ and $d\\rho \\ne 0$ on $M \\cap V$.\nAssume that the [[Levi form|80MTNXWW80]] of $\\rho$ at $p$, restricted to the complex tangent space of $M$ at $p$, has at least one positive eigenvalue.\nThen there is a neighbourhood $U \\subset V$ of $p$ with the following property.\nLet $f$ be a [[CR function|JFKYYK2CEB]] of class $C^1$ on $M \\cap V$.\nThen there is a unique continuous function $F$ on $U \\cap \\set{\\rho \\le 0}$ with $F = f$ on $M \\cap U$ that is [[holomorphic|507C34W0ZT]] on $U \\cap \\set{\\rho \u003c 0}$.\n\n## Remarks\n\nThe theorem is due to Hans Lewy.\nOne proof fills the side $\\set{\\rho \u003c 0}$ near $p$ with small analytic discs whose boundaries lie in $M$.\nThe values of $f$ on the boundary circles define $F$ by the Cauchy integral, and the CR equations make $F$ holomorphic.\nIf the Levi form at $p$ has eigenvalues of both signs, extensions exist on both sides.\nThen $f$ is the restriction of a function holomorphic on a full neighbourhood of $p$.\nFor the unit sphere with $\\rho = \\abs{z}^2 - 1$, the Levi form is positive, so CR functions extend locally into the ball.\nThe hypothesis on the Levi form is needed.\nOn the [[Levi-flat hypersurface|S4842D7MAE]] $\\set{\\operatorname{Im} z_1 = 0}$, the function $f(z) = g(\\operatorname{Re} z_1)$ is CR for every real $C^1$ function $g$.\nIf $f$ extended holomorphically to one side, the Schwarz reflection principle in $z_1$ would make $g$ real-analytic, so a $g$ that is $C^1$ but not real-analytic gives no extension.\n\n## Explanations\n\n### L1 human\n\nA CR function lives only on a hypersurface, but it satisfies the part of the Cauchy–Riemann equations that makes sense there.\nThe theorem says that when the hypersurface curves in a complex direction, such a function is automatically the boundary value of a holomorphic function on one side.\nSo the data on the thin hypersurface secretly carry a holomorphic function on an open set.\n\n### L2 human\n\nThe Lewy extension theorem is a local form of the Hartogs phenomenon for CR functions.\nIt turns questions about CR functions into questions about holomorphic functions on one-sided neighbourhoods.\nIt is the starting point of the extension theory of CR functions, which treats submanifolds of higher codimension as well.\nA global analogue for the boundary of a bounded domain with connected boundary is the [[Hartogs–Bochner extension theorem|8QYVM1JTMF]].\n",
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  "name": "Lewy extension theorem",
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