Extended Gevrey regularity via weighted matrices

TL;DR

This paper compares two recent methods for analyzing the space between Gevrey classes and smooth functions, establishing their equivalence in the Beurling case and describing the structure in the Roumieu case.

Contribution

It demonstrates the equivalence of Gevrey spaces defined via two parameter sequences and weight matrices, clarifying their relationships in different cases.

Findings

Equivalence of spaces in the Beurling case via projective limits.

Identification of larger spaces in the Roumieu case.

Comparison of two approaches for Gevrey regularity.

Abstract

The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces $\mathcal G_t (U)$ and the space of smooth functions $C^{\infty}(U)$. The first approach in the style of Komatsu is based on the properties of two parameter sequences $M_p=p^{\tau p^{\sigma}}$, $\tau>0$, $\sigma>1$. The other one uses weight matrices defined by certain weight functions. We prove the equivalence of the corresponding spaces in the Beurling case by taking projective limits with respect to matrix parameters, while in the Roumieu case we need to consider a larger space then the one obtained as the inductive limit of extended Gevrey classes.