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.

We prove an extension theorem for ultraholomorphic classes defined by so-called Braun-Meise-Taylor weight functions ω and transfer the proofs from the single weight sequence case from V. Thilliez [28] to the weight function setting. We are following a different approach than the results obtained in [11], more precisely we are working with real methods by applying the ultradifferentiable Whitney-extension theorem. We are treating both the Roumieu and the Beurling case, the latter one is obtained by a reduction from the Roumieu case.

We study the injectivity and surjectivity of the Borel map in three instances: in Roumieu-Carleman ultraholomorphic classes in unbounded sectors of the Riemann surface of the logarithm, and in classes of functions admitting, uniform or nonuniform, asymptotic expansion at the corresponding vertex. These classes are defined in terms of a log-convex sequence M of positive real numbers. Injectivity had been solved in two of these cases by S. Mandelbrojt and B. Rodríguez-Salinas, respectively, and we completely solve the third one by means of the theory of proximate orders. A growth index ω(M) turns out to put apart the values of the opening of the sector for which injectivity holds or not. In the case of surjectivity, only some partial results were available by J. Schmets and M. Valdivia and by V. Thilliez, and this last author introduced an index γ(M) (generally different from ω(M)) for this problem, whose optimality was not established except for the Gevrey case. We considerably extend here their results, proving that γ(M) is indeed optimal in some standard situations (for example, as far as M is strongly regular) and puts apart the values of the opening of the sector for which surjectivity holds or not.2010 MSC: Primary 30D60; secondary 30E05, 47A57, 34E05.According to V. Thilliez [29], if M is (lc), has (mg) and satisfies (snq), we say that M is strongly regular.Obviously, (mg) implies (dc), and (snq) implies (nq). Definition 2.2. For a sequence M we define the sequence of quotients m = (m p ) p∈N 0 by m p := M p+1 M p p ∈ N 0 .

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence M = (M p ) p∈N0 , have been put forward by A. Lastra, S. Malek and the second author [10], and their validity depends on the possibility of associating to M a nonzero proximate order. We provide several characterizations of this and other related properties, in which the concept of regular variation for functions and sequences plays a prominent role. In particular, we show how to construct well-behaved strongly regular sequences from nonzero proximate orders.

We consider r-ramification ultradifferentiable classes, introduced by J. Schmets and M. Valdivia in order to study the surjectivity of the Borel map, and later on also exploited by the authors in the ultraholomorphic context. We characterize quasianalyticity in such classes, extend the results of Schmets and Valdivia about the image of the Borel map in a mixed ultradifferentiable setting, and obtain a version of the Whitney extension theorem in this framework.