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  "body": "\n## Statement\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]] and let $H^\\infty$ be the algebra of bounded [[holomorphic functions|0DG0Z72CQD]] on $\\mathbb{D}$.\nLet $f_1, \\ldots, f_n \\in H^\\infty$, and assume there is $\\delta \u003e 0$ such that\n$$\\abs{f_1(z)} + \\cdots + \\abs{f_n(z)} \\ge \\delta \\quad \\text{for all } z \\in \\mathbb{D}.$$\nThen there are $g_1, \\ldots, g_n \\in H^\\infty$ such that\n$$f_1 g_1 + \\cdots + f_n g_n = 1 \\quad \\text{on } \\mathbb{D}.$$\n\n## Remarks\n\nThe hypothesis is necessary: if $\\sum f_j g_j = 1$ with $\\abs{g_j} \\le C$, then $\\sum \\abs{f_j} \\ge 1/C$ on $\\mathbb{D}$.\nThe theorem is equivalent to the density of $\\mathbb{D}$ in the space of [[maximal ideals|09J6KGT65G]] of the Banach algebra $H^\\infty$.\nHere $z \\in \\mathbb{D}$ corresponds to the ideal of functions vanishing at $z$.\nThe name comes from this picture: without the theorem, there could be a \"corona\" of maximal ideals away from the closure of the disc.\n\nProof idea: Carleson proved the theorem in 1962 with a construction that led to [[Carleson measures|SQHGJ9C0QE]].\nA shorter proof due to Wolff solves the problem first with smooth functions, then corrects them by solving a $\\bar\\partial$-equation with bounded solutions, using a Carleson measure estimate.\n\nExample: for $n = 1$ the theorem says that if $\\abs{f_1} \\ge \\delta$ on $\\mathbb{D}$, then $1/f_1$ is bounded and holomorphic, which is clear.\n\n## Explanations\n\n### L1 human\n\nIf you have finitely many bounded holomorphic functions that never get small at the same time, the theorem says you can combine them to get the constant $1$.\nThe combination uses bounded holomorphic coefficients.\nFor polynomials the analogous fact is the statement that polynomials with no common zero generate the unit ideal.\nIn the disc, \"no common zero\" must be replaced by \"not all close to zero\", because zeros can escape to the boundary.\n\n### L2 human\n\nIn Banach algebra language, the theorem identifies where the disc sits inside the maximal ideal space of $H^\\infty$: it is dense.\nIts proof introduced [[Carleson measures|SQHGJ9C0QE]] and interpolating sequences, which became central tools in function theory.\nThe analogue for the unit ball in $\\C^n$ with $n \\ge 2$ is a well known open problem.\n",
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