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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\C^n$ be a bounded [[pseudoconvex|AHYZMSBBNH]] open set.\nLet $H = \\set{z \\in \\C^n : z_n = 0}$, and let $\\psi$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $\\Omega$.\nThen there is a constant $C \u003e 0$, depending only on the diameter of $\\Omega$, with the following property.\nFor every holomorphic function $f$ on $\\Omega \\cap H$ with\n$$\\int_{\\Omega \\cap H} \\abs{f}^2 e^{-\\psi}\\, d\\lambda_{n-1} \u003c \\infty,$$\nthere is a [[holomorphic function|507C34W0ZT]] $F$ on $\\Omega$ with $F = f$ on $\\Omega \\cap H$ and\n$$\\int_\\Omega \\abs{F}^2 e^{-\\psi}\\, d\\lambda_n \\le C \\int_{\\Omega \\cap H} \\abs{f}^2 e^{-\\psi}\\, d\\lambda_{n-1}.$$\nHere $\\lambda_k$ denotes Lebesgue measure on $\\C^k$, and $\\Omega \\cap H$ is viewed as an open subset of $\\C^{n-1}$.\n\n## Remarks\n\nThe point is that $C$ does not depend on $\\psi$ or on $f$.\nThe proof solves a $\\bar\\partial$-equation with $L^2$ estimates for a weight that is singular along $H$, using a twisted form of the Bochner–Kodaira–Morrey–Kohn identity.\nOne extends $f$ smoothly with a cutoff in $z_n$ and corrects the extension by a solution $u$ of $\\bar\\partial u = g$ whose weight $e^{-\\psi}/\\abs{z_n}^2$ forces $u$ to vanish on $H$.\nThe sharp value of the constant was later found by Błocki and by Guan and Zhou, and it yields the Suita conjecture on the Bergman kernel of plane domains.\nExample: take $\\psi = 0$ and let $\\Omega$ be the unit ball.\nThen every holomorphic $f$ on the unit ball of $\\C^{n-1}$ with $\\int \\abs{f}^2 \u003c \\infty$ extends to a square integrable holomorphic function on the unit ball of $\\C^n$.\nFor the ball this is easy, because the trivial extension $F(z', z_n) = f(z')$ is defined on the ball and satisfies $\\int_\\Omega \\abs{F}^2 \\le \\pi \\int_{\\Omega \\cap H} \\abs{f}^2$.\nFor a general $\\Omega$ the point $(z', 0)$ need not lie in $\\Omega$ when $(z', z_n)$ does, so the trivial extension is not defined on $\\Omega$, and the theorem is needed.\n\n## Explanations\n\n### L1 human\n\nTake a holomorphic function on a slice of a domain cut by a hyperplane.\nThe theorem says it extends to a holomorphic function on the whole domain, and the extension is not much bigger than the original function.\nThe size is measured by weighted square integrals, and the bound holds uniformly for all plurisubharmonic weights.\n\n### L2 human\n\nThe theorem refines Cartan's theorem B, which gives an extension from a closed submanifold of a Stein manifold but without control of its size.\nBecause the constant is uniform in the weight, the theorem can be applied to singular weights such as $\\psi = c \\log \\abs{g}$.\nThis makes it a key tool in the theory of multiplier ideals: it is used in Demailly's approximation of plurisubharmonic functions and in Siu's proof of the invariance of plurigenera.\n",
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