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.

37 pages

References in corpus (1)

• Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes

Cited by in corpus (16)

• On subadditivity-like conditions for associated weight functions
• On generalized definitions of ultradifferentiable classes
• The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces
• Optimal flat functions in Carleman-Roumieu ultraholomorphic classes in sectors
• Equality of ultradifferentiable classes by means of indices of mixed O-regular variation
• Factorization in Denjoy-Carleman classes associated to representations of $({\mathbb{R}}^d,+)$
• On the construction of large Algebras not contained in the image of the Borel map
• Ultradifferentiable classes of entire functions
• Surjectivity of the asymptotic Borel map in Carleman-Roumieu ultraholomorphic classes defined by regular sequences
• On inclusion relations of weighted $L^p$-type spaces defined in terms of weight function matrices
• Ellipticity and the problem of iterates in Denjoy-Carleman classes
• On the class of almost subadditive weight functions
• Ultraholomorphic sectorial extensions of Beurling type
• Surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes defined by sequences with shifted moments
• On the conjugate weight function and ultradifferentiable classes of entire functions
• On Orlicz classes defined in terms of associated weight functions