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

TL;DR

The paper addresses the challenge of extending the classical moderate growth condition $(\operatorname{mg})$ to mixed settings with two weight sequences, showing that a full quotient/root-type generalization for weight matrices is not generally achievable. It introduces a new characterization of the generalized mg property in terms of the associated weight function $\omega_{\mathbf{M}}$ when $\omega_{\mathbf{M}}$ is derived from a weight sequence, and leverages recent results on the generalized lower Legendre conjugate to establish this link. A key contribution is proving that the generalized mg condition $(\operatorname{genmgintro})$ is preserved under equivalence of weight sequences and can be expressed via $\omega_{\mathbf{M}}$, with corollaries connecting to the Braun-Meise-Taylor weight function setting through $\mathcal{M}_{\omega}$. The work thus strengthens the bridge between weight sequence, weight matrix, and weight function formalisms for ultradifferentiable spaces, providing a concrete criterion in terms of $\omega_{\mathbf{M}}$ and highlighting the role of the Legendre conjugate in obtaining these characterizations.

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.