Iterates of systems of operators in spaces of $ω$-ultradifferentiable 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 ${\mathcal E}_ω^P$ and ${\mathcal E}_ω^Q$ of $ω$-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 $ω(t)$ and $σ(t)$, for the inclusion ${\mathcal E}_ω^P\subseteq{\mathcal E}_σ^Q$. As a consequence we have a generalization of the classical Theorem of the Iterates.