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On the smoothness of slowly varying functions

Dalimil Peša

TL;DR

The paper proves that every slowly varying function $b:(0,\infty)\to(0,\infty)$ under the modern Definition $[DSV]$ is equivalent to a $c:(0,\infty)\to(0,\infty)$ that is also slowly varying and belongs to $\mathcal{C}^{\infty}$, i.e., $c \approx b$ and $c$ has continuous derivatives of all orders. The authors develop a mollification-based scheme combined with a careful decomposition of $b$ into parts, enabling the construction of a smooth, equivalent representative without losing the slowly varying structure. They introduce and leverage several auxiliary lemmas (e.g., for decomposing and recombining s.v. functions) to achieve the $\mathcal{C}^{\infty}$-regularization, and they discuss an open conjecture relating derivatives to integral transforms of $b$. This result clarifies how smoothness can be imposed in the analysis of function spaces and interpolation (notably Lorentz–Karamata spaces) while preserving the essential asymptotic behavior, with potential implications for endpoint descriptions and derivative-based characterizations.

Abstract

In this paper we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show, that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.

On the smoothness of slowly varying functions

TL;DR

The paper proves that every slowly varying function under the modern Definition is equivalent to a that is also slowly varying and belongs to , i.e., and has continuous derivatives of all orders. The authors develop a mollification-based scheme combined with a careful decomposition of into parts, enabling the construction of a smooth, equivalent representative without losing the slowly varying structure. They introduce and leverage several auxiliary lemmas (e.g., for decomposing and recombining s.v. functions) to achieve the -regularization, and they discuss an open conjecture relating derivatives to integral transforms of . This result clarifies how smoothness can be imposed in the analysis of function spaces and interpolation (notably Lorentz–Karamata spaces) while preserving the essential asymptotic behavior, with potential implications for endpoint descriptions and derivative-based characterizations.

Abstract

In this paper we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show, that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.
Paper Structure (3 sections, 8 theorems, 49 equations)

This paper contains 3 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Slowly varying functions---a survey
  3. Proof of the main result

Key Result

Theorem 1.2

Let $b$ be an s.v. function. Then there is a function $c : (0, \infty) \to (0, \infty)$ which satisfies $c \approx b$, is also s.v., and has continuous classical derivatives of all orders (i.e. $c \in \mathcal{C}^{\infty}$).

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • ...and 12 more