Iterates of systems of operators in spaces of $\omega$-ultradifferentiable functions

TL;DR

This paper investigates the relationships between spaces of $ ext{omega}$-ultradifferentiable functions defined via iterates of systems of linear PDE operators, providing necessary and sufficient conditions for inclusion and generalizing classical iterate theorems.

Contribution

It introduces a comprehensive framework for understanding inclusions of ultradifferentiable function spaces based on operator systems and weight functions, extending classical results.

Findings

Derived necessary and sufficient conditions for space inclusion

Generalized classical Theorem of the Iterates

Established criteria relating systems and weight functions

Abstract

Given two systems $P=(P_j(D))_{j=1}^N$ and $Q=(Q_j(D))_{j=1}^M$ of linear partial differential operators with constant coefficients, we consider the spaces ${\mathcal E}_\omega^P$ and ${\mathcal E}_\omega^Q$ of $\omega$-ultradifferentiable functions with respect to the iterates of the systems $P$ and $Q$ respectively. We find necessary and sufficient conditions, on the systems and on the weights $\omega(t)$ and $\sigma(t)$, for the inclusion ${\mathcal E}_\omega^P\subseteq{\mathcal E}_\sigma^Q$. As a consequence we have a generalization of the classical Theorem of the Iterates.