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  "body": "\n## Statement\n\nLet $0 \u003c p \\le \\infty$, and let $f$ be a function in the [[Hardy space|PNNNFJVF6W]] $H^p$ of the [[unit disc|8G5ABXRK85]] $\\mathbb{D}$, with $f$ not identically $0$.\nThen $f = I F$, where $I$ is an [[inner function|NN8XH5RJYB]] and $F$ is an [[outer function|KGJD4A4ATY]] that lies in $H^p$.\nThe factors are unique up to constants of modulus $1$, and $\\abs{F^*} = \\abs{f^*}$ almost everywhere on the circle.\nMoreover $I = c B S_\\mu$, where $\\abs{c} = 1$ and $B$ is the [[Blaschke product|9N6R5ADJXT]] of the zeros of $f$, with multiplicity.\nThe last factor is\n$$S_\\mu(z) = \\exp\\left(-\\int_0^{2\\pi} \\frac{e^{it} + z}{e^{it} - z} \\, d\\mu(t)\\right)$$\nfor a finite positive Borel measure $\\mu$ on the circle that is singular with respect to Lebesgue measure.\n\n## Remarks\n\nProof idea: the zeros of $f$ satisfy the Blaschke condition, by Jensen's formula.\nDividing by their Blaschke product $B$ gives a function $g = f/B$ in $H^p$ with no zeros and the same norm.\nThe outer function $F$ with modulus $\\abs{g^*}$ dominates $g$ in the sense that $\\abs{g} \\le \\abs{F}$, because $\\log \\abs{g(z)}$ is at most the Poisson integral of $\\log \\abs{g^*}$ for $g \\in H^p$.\nThen $g/F$ is bounded, has no zeros and has boundary modulus $1$, and such a function is a singular inner function $c S_\\mu$.\n\nExample: $f(z) = z (2 + z)$ is bounded, its inner factor is $z$ and its outer factor is $2 + z$.\n\nThe hypothesis $f \\not\\equiv 0$ is needed, since the zero function has no inner factor.\n\n## Explanations\n\n### L1 human\n\nEvery function in a Hardy space splits into two parts.\nThe outer part is fixed by the size of the function on the boundary circle.\nThe inner part has size $1$ on the circle and holds the zeros of the function and a hidden \"singular\" part.\nThis is the disc analogue of writing a number as a sign times a size.\n\n### L2 human\n\nThe factorization is the main structural tool for Hardy spaces.\nIt reduces many questions to zero-free functions, for which one can take logarithms and roots: for example $f = B g$ with $g$ zero-free gives $g^{p/2} \\in H^2$, which transfers results from $H^2$ to $H^p$.\nIt underlies [[Beurling's theorem|2V1DXXM9FA]] on shift-invariant subspaces, where outer functions are exactly the cyclic vectors of the shift on $H^2$.\nThe same factorization holds in the [[Nevanlinna class|M3WQTN3FH2]] if one allows quotients of singular inner functions.\n",
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