On the equivalence between moderate growth-type conditions in the weight matrix setting

TL;DR

This paper extends the concept of moderate growth conditions from single weight sequences to weight matrices, revealing that equivalences break down in the matrix setting and focusing on those linked to weight functions.

Contribution

It generalizes the known moderate growth conditions to weight matrices and analyzes their equivalences, especially for matrices associated with weight functions.

Findings

Mixed conditions are not generally equivalent in the matrix setting.

Focus on weight matrices related to (associated) weight functions.

Provides insights into ultradifferentiable function classes.

Abstract

We study the generalizations of the known equivalent reformulations of condition moderate growth from the single weight sequence to the weight matrix setting. This condition, also known in the literature under the name stability under ultradifferentiable operators, plays a significant role in the theory of ultradifferentiable (and ultraholomorophic) function classes defined in terms of weight sequences and its generalization becomes relevant when dealing with classes defined by weight matrices. In the matrix setting, we prove that the different mixed conditions are in general not equivalently satisfied anymore and we focus on weight matrices associated with (associated) weight functions.