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  "body": "\n## Statement\n\nLet $U \\colon \\R \\to \\R$ be a nondecreasing, right-continuous [[monotone function|0WFF6AZPBP]] with $U(x) = 0$ for $x \u003c 0$.\nAssume that its Laplace–Stieltjes transform\n$$\\widehat{U}(s) = \\int_{[0, \\infty)} e^{-sx} \\, dU(x),$$\na [[Riemann–Stieltjes integral|8NMH42GJRK]] over $[0, \\infty)$ that includes a possible jump of $U$ at $0$, is finite for every $s \u003e 0$.\nLet $\\rho \\ge 0$ and $c \u003e 0$ be real numbers, let $L$ be a [[slowly varying function|EHJ7D5CRZS]], and let $\\Gamma$ be the [[gamma function|0V3FZMVB4K]].\nThen the following are equivalent:\n\n1. $\\widehat{U}(s) \\sim c \\, s^{-\\rho} L(1/s)$ as $s \\to 0^+$;\n2. $U(x) \\sim \\dfrac{c}{\\Gamma(1 + \\rho)} \\, x^{\\rho} L(x)$ as $x \\to \\infty$.\n\nHere $\\sim$ denotes [[asymptotic equivalence|0QR92XBV6H]].\n\n## Remarks\n\nThe direction from 2 to 1 is an Abelian statement and follows by integration by parts and standard estimates for slowly varying functions.\nThe direction from 1 to 2 is the Tauberian part, where monotonicity of $U$ is essential.\nKaramata's proof shows that $\\int_0^\\infty g(e^{-sx}) \\, dU(x)$ has the expected behaviour for polynomials $g$, then for the indicator function of $[e^{-1}, 1]$ by one-sided polynomial approximation, which recovers $U(1/s)$.\n\nExample: let $a_n \\ge 0$ and $U(x) = \\sum_{n \\le x} a_n$.\nThen $\\widehat{U}(s) = \\sum_{n \\ge 0} a_n e^{-sn}$, and the case $\\rho = 1$, $c = 1$, $L = 1$ is the [[Hardy–Littlewood Tauberian theorem|N42JDBT0VA]] with $x = e^{-s}$.\n\nMonotonicity cannot be dropped in the Tauberian direction: the counterexample to the Hardy–Littlewood theorem with $a_n = 1 + (-1)^n n$ also violates statement 2 here.\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform turns a growing function into a function of $s$ that blows up as $s \\to 0$.\nKaramata's theorem says that, for a nondecreasing function, a power-law growth (up to a slowly varying factor) of one side is exactly matched by a power-law blow-up of the other.\nSo one can read off the growth of $U$ from its transform near $0$, and back.\n\n### L2 human\n\nThe theorem is the standard form of Tauberian theorems for Laplace–Stieltjes transforms, and it is central in the theory of regular variation.\nIn probability it relates the tail of a distribution or the growth of a renewal function to the behaviour of its Laplace transform near $0$.\nIts proof by polynomial approximation, through the [[Weierstrass approximation theorem|0JQE130DHW]], gave a short route to the Hardy–Littlewood theorem.\n",
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