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

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.

Sectorial Extensions, via Laplace Transforms, in Ultraholomorphic Classes Defined by Weight Functions

“…In the main result, Theorem 6.8, we prove the surjectivity of the Borel map in ultraholomorphic classes defined by weight functions ω, satisfying several standard assumptions and such that γ(ω) > 1, and in sectors of opening smaller than π(γ(ω) − 1). So, as it happens in Thilliez's result, the opening of the sectors for which the result applies is controlled by a growth index γ(ω), which has also been introduced in [11] and is studied in detail in [10]. Moreover, in [10] we consider other indices for ω and also their relation to the indices γ(M ) of Thilliez or ω(M ) introduced in [23].…”

On the Construction of Large Algebras Not Contained in the Image of the Borel Map

“…(ii) Let ω ∈ W be given. Then ω∼(ω ι ) ⋆ implies γ(ω) = +∞, with γ denoting the growth index studied in detail in [15] and used in the extension results in [17], [16] (the fact that ω has (ω 1 ) is equivalent to having γ(ω) > 0, see […”

Injectivity and surjectivity of the Stieltjes moment mapping in Gelfand–Shilov spaces

“…In [8,Ch. 2] and [10,Sect. 3], the connections between these indices, the growth properties usually imposed on sequences, and the theory of O-regular variation, have been thoroughly studied.…”

“…Subsequently, in [8] (see also [10]), a general procedure has been designed to obtain strongly regular sequences with preassigned positive values of γ(M) and ω(M). In particular, one can choose strongly regular sequences M with γ(M) ≤ 1 < ω(M) and thereby exclude both injectivity and surjectivity.…”

“…Namely, given two sequences of positive real numbers M = (M p ) p∈N and A = (A p ) p∈N , we consider the spaces S A M (0, ∞) and S M (0, ∞) consisting of all ϕ ∈ S(0, ∞) such that there exists h > 0 with Our aim is to improve and complete these results by including the spaces S A M (0, ∞) in our considerations, by dropping some hypotheses on M, specially moderate growth and (1.2), and by also studying the injectivity of the Stieltjes moment mapping. Our key tools are: a better understanding of the meaning of the different growth conditions usually imposed on the sequence M and their expression in terms of indices of Oregular variation, as developed in [10]; the use of the Fourier transform in order to translate our problems into the corresponding ones for the asymptotic Borel mapping in certain ultraholomorphic classes on the upper half-plane; the enhanced information obtained in [11] about the injectivity and surjectivity of the asymptotic Borel mapping for sequences M subject to minimal conditions. The paper is organized as follows.…”