Complements on Growth Envelopes of Spaces with Generalized Smoothness in the Sub-Critical Case

Abstract

We describe the growth envelope of Besov and Triebel-Lizorkin spaces $B_{p,q}^\sigma (\mathbb R^n)$ and $F_{p,q}^\sigma (\mathbb R^n)$ with generalized smoothness, i.e.\ instead of the usual scalar regularity index $\sigma \in \mathbb R$ we consider now the more general case of a sequence $\sigma = \{\sigma_j\}_{j \in\mathbb N_0}$. We take under consideration the range of the parameters $\sigma, p,q$ which, in analogy to the classical terminology, we call sub-critical.