Karamata's Slowly Varying Functions

• Slowly varying functions in the Karamata sense are positive measurable functions that exhibit asymptotic flatness, meaning L(vx)/L(x) consistently approaches 1 for any fixed v > 0.
• They admit an integral representation that quantifies deviations through a vanishing ε(t), enabling extensions to ψ-locally constant functions for broader modeling applications.
• Their applications span probability theory, extreme value analysis, and function space theory, underpinning limit theorems and refined asymptotic estimates in various analytical settings.

A slowly varying function in the Karamata sense is a positive measurable function $L(x)$, usually defined for $x$ large, with the property that for every fixed $v>0$,

$\lim_{x \to \infty} \frac{L(vx)}{L(x)} = 1.$

This fundamental concept, introduced by Jovan Karamata, underpins the modern theory of regular variation, Tauberian theorems, asymptotic analysis, and plays a central role across probability, analysis, and partial differential equations. The Karamata sense both captures the essence of asymptotic “flatness” (variation that is sub-power in growth or decay) and provides a functional calculus for explicitly handling such phenomena, with a precise integral representation, various extensions (e.g. ψ-locally constant functions), and deep connections to limit theorems and function space theory.

1. Definition and Integral Representation

A function $L(x)$ is (Karamata) slowly varying at infinity if the dilation ratio converges to 1 at every fixed scaling factor, as above. Every positive measurable slowly varying function admits Karamata’s integral representation:

$$L(x) = c(x) \exp\left\{ \int_1^x \frac{\varepsilon(t)}{t} \, dt \right\}, \quad x \geq 1$$

where $c(x) \to c > 0$ and $\varepsilon(t) \to 0$ as $t \to \infty$ (Borovkov et al., 2010). This form demonstrates that all asymptotic deviations from constancy occur only through the vanishing $\varepsilon(t)$. Typical examples include $L(x) = (\log x)^\alpha$, compositions with iterated logarithms, or any function diverging to infinity (or zero) slower than any positive or negative power of $x$.

An important uniformity property known as Karamata’s Uniform Convergence Theorem asserts that

$$\sup_{v \in [a, b]} \left| \frac{L(vx)}{L(x)} - 1 \right| \to 0 \text{ as } x \to \infty \qquad \forall 0 < a < b < \infty$$

(Mijajlović et al., 2022). This ensures the regularity needed for limit theorems and function space embeddings.

2. Extensions: ψ-Locally Constant Functions

The notion of slow variation is strictly multiplicative. The paper “On an Extension of the Concept of Slowly Varying Function with Applications to Large Deviation Limit Theorems” (Borovkov et al., 2010) generalizes this via ψ-locally constant functions (ψ-l.c.f.). For a nondecreasing $\psi(x)=o(x)$,

$$\lim_{x \to \infty} \frac{g(x + v \psi(x))}{g(x)} = 1 \qquad (\forall v \in \mathbb{R})$$

with $x + v\psi(x)$ tending to infinity. Classical cases correspond to $\psi(x)=1$ (locally constant) and $\psi(x)=x$ (slowly varying). Under technical conditions on $\psi$, every ψ-l.c.f. $g$ admits an analogous integral representation:

$$g(x) = c(x) \exp\left\{ \int_1^x \frac{\varepsilon(t)}{\psi(t)} dt \right\}, \quad c(x)\to c\in(0,\infty),~\varepsilon(t)\to 0$$

This framework interpolates between pure multiplicative and additive scaling, and is essential when modeling phenomena (notably, heavy-tailed random variables) lacking strict multiplicative homogeneity.

3. Role in Probability Theory and Extreme Value Analysis

Slowly varying functions govern the tails of distributions beyond the pure power-law (regularly varying) regime, and are crucial in the asymptotic analysis of sums and maxima. In the context of large deviation theory, for i.i.d. random variables $X_1,\ldots$ with tail $\bar F(x)=P(X>x)$

• If $\bar F(x)$ is regularly varying with index $\alpha<0$, then for large $x$,

$P\left(S_n \geq x\right) \sim n \bar F(x)$

where $S_n = X_1+\cdots+X_n$ (Borovkov et al., 2010).

• The result extends to $\psi$-locally constant $\bar F$ (not exactly regularly varying) under standard upper-power conditions, yielding the same asymptotic, thus covering broader classes of “almost” power-law tails.

In stochastic recursions and random difference equations $R_{n+1} = A_{n+1} R_n + B_{n+1}$, the stationary distribution’s tail is expressible through integrated slowly varying functions:

$$P[R > x] \sim -\frac{1}{\mathbb{E}\log A} \int_{\log x}^\infty P[\log(A \vee B) > y]dy$$

(Dyszewski, 2014). Integrated slowly varying tails appear naturally in perpetuities, risk theory, and models of financial extremes.

In Markov branching processes with infinite variance, the infinitesimal generating function takes the form

$f(s) = (1-s)^{1+\nu} L\left(\frac{1}{1-s}\right)$

with $L$ slowly varying. Precise survival probabilities and limit theorems then depend critically on the properties of $L$ and its remainder (Imomov et al., 2021).

4. Slowly Varying Functions in Function Spaces

Slow variation underpins the construction of generalized smoothness scales, most prominently in Besov, Triebel–Lizorkin, Sobolev–Hörmander, and Lorentz–Karamata spaces:

• Refined Sobolev/Hörmander scales: $$H^{s,\varphi}(\mathbb{R}^n) = \{w\in\mathcal{S}'(\mathbb{R}^n) : \int_{\mathbb{R}^n} \langle\xi\rangle^{2s} \varphi(\langle\xi\rangle)^2 |\hat w(\xi)|^2 d\xi < \infty\}$$ with $\varphi$ slowly varying (Chepurukhina et al., 2014, Chepurukhina et al., 2015, Kasirenko et al., 2018). These spaces interpolate between standard Sobolev norms, providing finer control over intermediate smoothness via, e.g., iterated logarithms.
• Generalized Besov/Triebel–Lizorkin scales: In $B^{s,\psi}_{p,q}(\mathbb{R}^d)$ and $F^{s,\psi}_{p,q}(\mathbb{R}^d)$, dyadic weights take the form $2^{js}\psi(2^j)$, with $\psi$ slowly varying (Harrison et al., 3 Oct 2025). This allows for regularity “just above critical,” essential for well-posedness in borderline PDE problems such as the Euler equations.
• Lorentz–Karamata and Small Lebesgue Spaces: Norms feature slowly varying weights, e.g.,

$$\|f\|_{L^{p,q;b}} = \left( \int_0^\infty \Big[ t^{1/p-1/q} b(t) f^*(t) \Big]^q dt \right)^{1/q}$$

with $b$ slowly varying (Peša, 2020, Bathory, 2017, Neves et al., 2019). This permits precise endpoint embeddings and optimal interpolation results, including control over “borderline” operator bounds.

Crucially, every slowly varying function in this setting is equivalent to a smooth ($C^\infty$) slowly varying function, ensuring the analytic flexibility needed for real interpolation and embedding theorems (Peša, 2023).

5. Slowly Varying Functions in Tauberian Theory and Asymptotic Optimality

Slowly varying functions represent the precise barrier in many Tauberian theorems. For instance, the Ingham–Karamata theorem asserts the o(1) remainder under minimal Laplace transform analyticity assumptions, and this o(1) remainder is best possible:

• The class of continuously differentiable functions $\tau$ with $\tau'(x) = o(1)$ and entire Laplace transforms forms a Fréchet space $V_{-\infty}$. For any function $\rho(x)\to 0$ (even a slowly varying $\rho$), for every $\tau$ there exists another $\tau$ with $|\tau(x)|/\rho(x)\not\to 0$. Sharpened remainders cannot be forced by analytic extension alone, even with entire Laplace transforms (Callewaert et al., 22 Jul 2024).
• Any quantitative improvement (e.g., $|\tau(x)| = O(\rho(x))$ for all $\tau$) would require additional growth conditions; slowly varying $\rho(x)$ represents the "softest" possible quantitative bound sub-o(1).

This phenomenon is notably universal: in analytic number theory, operator semigroup theory, and harmonic analysis, slowly varying functions are the limiting factor for remainder theorems linked to Laplace methods or Fourier analysis.

6. Functional Calculus, Extensions, and Regularity

Karamata’s integral representation and functional calculus for slow variation extend to operator theory and numerical analysis:

• The operator $\mathcal{L}$,

$$(\mathcal{L} h)(x) = \frac{1}{\ln x} \int_1^x \frac{h(t)}{t} dt,$$

maps functions tending to a constant to slowly varying functions, is invertible, and preserves the space of regularly varying functions (Mijajlović et al., 2022).

• In high-frequency asymptotic and numerical analysis, families of "slowly varying" functions are constructed by factoring out high oscillatory exponentials (carrying the eikonal) in integral kernels:

$G(x,y) = e^{-s\,\tau(x,y)} \cdot g(x,y),$

with $g$ slowly varying in $x,y$ and amenable to efficient polynomial (e.g., Chebyshev) approximation (Lin et al., 7 Aug 2024).

The flexibility of the slowly varying framework supports such functional-analytic approaches while still retaining robust asymptotic control for both theoretical and computational purposes.

7. Summary of Key Properties and Applications

Aspect	Formulation / Role	Reference
Classical definition	$\lim_{x\to\infty} L(vx)/L(x) = 1$, $\forall v>0$	(Borovkov et al., 2010)
Integral representation	$L(x) = c(x)\exp\{\int_1^x \varepsilon(t)/t\,dt\}$, $c(x)\to c>0$, $\varepsilon(t)\to 0$	(Borovkov et al., 2010)
Uniform convergence property	$\sup_{v\in[a,b]}|L(vx)/L(x) - 1| \to 0$ as $x\to\infty$	(Mijajlović et al., 2022)
Ψ-locally constant generalization	$g(x+v\psi(x))/g(x)\to 1$, with suitable $\psi(x)=o(x)$	(Borovkov et al., 2010)
Fundamental in function space theory	Appears as weights in Sobolev, Besov, Lorentz–Karamata spaces	(Chepurukhina et al., 2014, Peša, 2020, Harrison et al., 3 Oct 2025)
Obstructs o(1) improvement in Tauberian	No quantitative remainder $O(\rho(x))$ is attainable with only analytic Laplace extension	(Callewaert et al., 22 Jul 2024)
Smooth approximation	Every (modern) slowly varying function is equivalent to a $C^\infty$ slowly varying one	(Peša, 2023)

Slowly varying functions in the Karamata sense thus form a central organizing structure in both classical and modern analysis, enabling the explicit treatment of sub-power asymptotics, sharp limit results in probability, Tauberian and operator theory, and the fine-grained construction of function spaces necessary for both qualitative and quantitative PDE theory. Their generalizations (ψ-locally constant functions), integral representations, and robust analytic properties are indispensable in the rigorous asymptotic treatment of problems where the main asymptotic term is modulated only by extremely mild ("slow") variation—guaranteeing optimality, and clarifying the precise boundaries of quantitative remainder bounds in analysis.