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

TL;DR

This paper explores the equivalence of moderate growth conditions for weight sequences and functions in the mixed setting, providing new characterizations crucial for weight matrix and function analysis.

Contribution

It offers a novel characterization of moderate growth conditions in the weight matrix setting using associated weight functions, extending previous results.

Findings

Established a new characterization of moderate growth in terms of weight functions.

Clarified the limitations of generalizing conditions to the mixed setting.

Enhanced understanding of weight matrix and weight function relationships.

Abstract

We continue the study of the known equivalent reformulations of the classical moderate growth condition for weight sequences in the mixed setting; i.e. when dealing with two different sequences. This approach is becoming crucial in the weight matrix setting and also, in particular, when dealing with weight functions in the sense of Braun-Meise-Taylor. It is known that a full generalization to the mixed setting fails, more precisely the condition comparing the growth of the corresponding sequences of quotients and roots is not clear. In the main result we prove a new characterization of this property in terms of the associated weight function; i.e. when the given weight function is based on a weight sequence.