Karamata's Tauberian theorem

Remarks

The direction from 2 to 1 is an Abelian statement and follows by integration by parts and standard estimates for slowly varying functions. The direction from 1 to 2 is the Tauberian part, where monotonicity of $U$ is essential. Karamata'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 $U(1/s)$.

Example: let $a_n \ge 0$ and $U(x) = \sum_{n \le x} a_n$. Then $\widehat{U}(s) = \sum_{n \ge 0} a_n e^{-sn}$, and the case $\rho = 1$, $c = 1$, $L = 1$ is the Hardy–Littlewood Tauberian theorem with $x = e^{-s}$.

Monotonicity 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.

Going deeper

For a student

The theorem is the standard form of Tauberian theorems for Laplace–Stieltjes transforms, and it is central in the theory of regular variation. In probability it relates the tail of a distribution or the growth of a renewal function to the behaviour of its Laplace transform near $0$. Its proof by polynomial approximation, through the Weierstrass approximation theorem, gave a short route to the Hardy–Littlewood theorem.