Recent work showed that a theorem of Joris (that a function $f$ is smooth if two coprime powers of $f$ are smooth) is valid in a wide variety of ultradifferentiable classes $mathcal C$. The core of the proof was essentially $1$-dimensional. In certain cases a multidimensional version resulted from subtle reduction arguments, but general validity, notably in the quasianalytic setting, remained open. In this paper we give a uniform proof which works in all cases and dimensions. It yields the result even on infinite dimensional Banach spaces and convenient vector spaces. We also consider more general nonlinear conditions, namely general analytic germs $Phi$ instead of the powers, and characterize when $Phi circ f in mathcal C$ implies $f in mathcal C$.
We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.
We study the surjectivity of, and the existence of right inverses for, the asymptotic Borel map in Carleman-Roumieu ultraholomorphic classes defined by regular sequences in the sense of E. M. Dynkin. We extend previous results by J. Schmets and M. Valdivia, by V. Thilliez, and by the authors, and show the prominent role played by an index associated with the sequence and introduced by Thilliez. The techniques involve regular variation, integral transforms and characterization results of A. Debrouwere in a half-plane, steming from his study of the surjectivity of the moment mapping in general Gelfand-Shilov spaces.
The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.
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.
We characterize the validity of the Whitney extension theorem in the ultradifferentiable Roumieu setting with controlled loss of regularity. Specifically, we show that in the main Theorem 1.3 of [15] condition (1.3) can be dropped. Moreover, we clarify some questions that remained open in [15].
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 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 $mathbb{M}$ of positive real numbers. Injectivity had been solved in two of these cases by S. Mandelbrojt and B. Rodriguez-Salinas, respectively, and we completely solve the third one by means of the theory of proximate orders. A growth index $omega(mathbb{M})$ 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 $gamma(mathbb{M})$ (generally different from $omega(mathbb{M})$) for this problem, whose optimality was not established except for the Gevrey case. We considerably extend here their results, proving that $gamma(mathbb{M})$ is indeed optimal in some standard situations (for example, as far as $mathbb{M}$ is strongly regular) and puts apart the values of the opening of the sector for which surjectivity holds or not.
