Indices of O-regular variation for weight functions and weight sequences

TL;DR

This paper explores the role of O-regular variation in weight functions and sequences used in defining various functional spaces, revealing that many conditions are equivalent and can be characterized by growth indices.

Contribution

It demonstrates that diverse conditions on weights in functional analysis are unified under the framework of O-regular variation and introduces indices to quantify their regularity.

Findings

Many weight conditions are equivalent and relate to O-regular variation.

Indices of regularity effectively characterize qualitative properties.

The framework simplifies understanding of weight functions and sequences in functional spaces.

Abstract

A plethora of spaces in Functional Analysis (Braun-Meise-Taylor and Carleman ultradifferentiable and ultraholomorphic classes; Orlicz, Besov, Lipschitz, Lebesque spaces, to cite the main ones) are defined by means of a weighted structure, obtained from a weight function or sequence subject to standard conditions entailing desirable properties (algebraic closure, stability under operators, interpolation, etc.) for the corresponding spaces. The aim of this paper is to stress or reveal the true nature of these diverse conditions imposed on weights, appearing in a scattered and disconnected way in the literature: they turn out to fall into the framework of O-regular variation, and many of them are equivalent formulations of one and the same feature. Moreover, we study several indices of regularity/growth for both functions and sequences, which allow for the rephrasing of qualitative properties in terms of quantitative statements.