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.