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

We describe the growth envelope of Besov and Triebel-Lizorkin spaces B pq(R) and F σ pq(R) with generalized smoothness, i.e. instead of the usual scalar regularity index σ ∈ R we consider now the more general case of a sequence σ = {σj}j∈N0 . We take under consideration the range of the parameters σ, p, q which, in analogy to the classical terminology, we call sub-critical.