A comparison of two different ways to define classes of ultradifferentiable functions

Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes

“…Moreover, according to the Denjoy‐Carleman theorem the ultradifferentiable functions can also be classified in quasianalytic and non‐quasianalytic classes, where, roughly speaking, the quasianalytic ones are functions whose Fourier transform has a stronger decay at infinity. We refer to 2, 10, 16, 19 for the different ways to introduce these classes and to 8 for an exhaustive comparison between them. We emphasize that the Gevrey classes are particular cases of non‐quasianalytic ultradifferentiable functions of Roumieu type.…”

Ultra‐differentiable classes and intersection theorems

“…With these notations we can recall the definition of ultra‐differentiable functions with respect to ω. (See , , , .) Definition Let ω be a weight function as in Definition and denote by φ the function associated with ω by .…”

“… The aim of the paper is to study the relation between ultra‐differentiable classes of functions defined in terms of estimates on derivatives on one hand and in terms of growth properties of Fourier transforms of suitably localized functions in the class on the other hand. We establish this relation for the ultra‐differentiable classes introduced in , , and show that the classes of , , can be regarded as inhomogeneous Gevrey classes in the sense of . We also discuss a number of properties of the weight functions used to define the respective classes and of their Young conjugates.…”

On some generalizations of Gevrey classes

“… ω( t ) = t α , 0 < α < 1. In this case the class \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}${\mathcal E}_{\lbrace \omega \rbrace }(\mathbf{R}^n)$\end{document} defined by condition (1.3) coincides with the classical Gevrey class $G^s({\bf R}^n)$ for s ≔ 1/α; ω( t ) = (log(1 + t )) β , β > 1; Let $(M_{p})_{p\in {\bf N}_{0}}$ be a sequence of positive numbers satisfying the conditions (M1), (M2), (M3) of Komatsu 13, then there exists a concave weight function κ such that \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal { E}_{(M_{j})}({\bf R}^n)=\mathcal { E}_{(\kappa )}({\bf R}^n)$\end{document}, where \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal { E}_{(M_{j})}({\bf R}^n)$\end{document} is the set of all the functions \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$f \in \mathcal { C}^{\infty }({\bf R}^n)$\end{document} such that for any h > 0 and any compact subset K of ${\bf R}^n$ .Bonet et al in 3 characterized the sequences $(M_p)_{p \in {\bf N}_{0}}$ for which there exists a weight function ω such that \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal { E}_{(M_{j})}({\bf R}^n)=\mathcal { E}_{(\omega )}({\bf R}^n)$\end{document} (in both the Beurling and Roumieu case). …”

“…Bonet et al in 3 characterized the sequences $(M_p)_{p \in {\bf N}_{0}}$ for which there exists a weight function ω such that \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal { E}_{(M_{j})}({\bf R}^n)=\mathcal { E}_{(\omega )}({\bf R}^n)$\end{document} (in both the Beurling and Roumieu case).…”