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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C^n$ be a [[pseudoconvex|AHYZMSBBNH]] open set.\nLet $\\varphi \\in C^2(\\Omega)$ be a real function, and let $c \\colon \\Omega \\to (0, \\infty)$ be a continuous function such that\n$$\\sum_{j,k=1}^n \\frac{\\partial^2 \\varphi}{\\partial z_j \\partial \\bar z_k}(z)\\, w_j \\bar w_k \\ge c(z) \\abs{w}^2 \\quad \\text{for all } z \\in \\Omega,\\ w \\in \\C^n,$$\nso that $\\varphi$ is strictly [[plurisubharmonic|FF2P7ER3QT]].\nLet $g = \\sum_{j=1}^n g_j \\, d\\bar z_j$ be a [[(0,1)-form|DC81TCR98Y]] with coefficients $g_j$ in $L^2_{\\mathrm{loc}}(\\Omega)$.\nSuppose that $\\bar\\partial g = 0$ in the sense of distributions and\n$$\\int_\\Omega \\frac{\\abs{g}^2}{c}\\, e^{-\\varphi}\\, d\\lambda \u003c \\infty,$$\nwhere $\\abs{g}^2 = \\sum_j \\abs{g_j}^2$ and $\\lambda$ is Lebesgue measure on $\\C^n = \\R^{2n}$.\nThen there is a function $u \\in L^2_{\\mathrm{loc}}(\\Omega)$ with $\\bar\\partial u = g$ in the sense of distributions and\n$$\\int_\\Omega \\abs{u}^2 e^{-\\varphi}\\, d\\lambda \\le \\int_\\Omega \\frac{\\abs{g}^2}{c}\\, e^{-\\varphi}\\, d\\lambda.$$\n\n## Remarks\n\nThe proof is a duality argument in weighted [[L2 spaces|SASP14MA1Z]].\nOne views $\\bar\\partial$ as a closed, densely defined operator between $L^2$ spaces with weight $e^{-\\varphi}$.\nThe key step is the a priori inequality $\\norm{\\bar\\partial^* f}^2 + \\norm{\\bar\\partial f}^2 \\ge \\int c \\abs{f}^2 e^{-\\varphi}$ for forms $f$ in the domains of both operators.\nThis Bochner–Kodaira–Morrey–Kohn type identity is where the plurisubharmonicity of $\\varphi$ enters.\nThe Riesz representation theorem then gives the solution $u$ together with the bound; pseudoconvexity is used to exhaust $\\Omega$ by sets on which the argument applies.\nThere is a version for any plurisubharmonic weight $\\varphi$ on a pseudoconvex $\\Omega$.\nFor every $\\bar\\partial$-closed $(0,1)$-form $g$ with $\\int \\abs{g}^2 e^{-\\varphi} \u003c \\infty$ there is $u$ with $\\bar\\partial u = g$ and $\\int \\abs{u}^2 e^{-\\varphi} (1 + \\abs{z}^2)^{-2} \\le \\int \\abs{g}^2 e^{-\\varphi}$.\nSimilar statements hold for $(p,q)$-forms and on Stein and complete Kähler manifolds with Hermitian line bundles.\nExample: for $\\varphi(z) = \\abs{z}^2$ one may take $c = 1$, so every $\\bar\\partial$-closed $g$ with $\\int \\abs{g}^2 e^{-\\abs{z}^2} \u003c \\infty$ is $\\bar\\partial u$ for some $u$ with $\\int \\abs{u}^2 e^{-\\abs{z}^2} \\le \\int \\abs{g}^2 e^{-\\abs{z}^2}$.\nPseudoconvexity cannot be dropped: an open set $\\Omega \\subseteq \\C^n$ on which every smooth $\\bar\\partial$-closed $(0,q)$-form, $1 \\le q \\le n - 1$, is $\\bar\\partial$ of a smooth form is pseudoconvex.\n\n## Explanations\n\n### L1 human\n\nThe equation $\\bar\\partial u = g$ asks for a function whose failure to be holomorphic is prescribed.\nHörmander's theorem says that on a pseudoconvex domain this equation always has a solution, and that the size of the solution is controlled by the size of the data.\nSize is measured by integrals with a weight $e^{-\\varphi}$, and choosing the weight cleverly is the art of using the theorem.\n\n### L2 human\n\nThe estimates turn the main existence problems of several complex variables into a single analytic tool.\nTo build a holomorphic function with given properties, one first builds a smooth function with these properties.\nThen one corrects it by a solution of a $\\bar\\partial$-equation whose weighted $L^2$ norm is small.\nIn this way one obtains Oka's solution of the Levi problem, the Cousin problems on pseudoconvex domains, and Cartan's theorem B for the structure sheaf.\nThe method is the basis of the Ohsawa–Takegoshi extension theorem and of the theory of multiplier ideals.\n",
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