  A rather tricky question is the construction of wavelet bases on domains for
suitable function spaces (Sobolev, Besov, Triebel-Lizorkin type). In his
monograph from 2008, Triebel presented an approach how to construct wavelet
(Riesz) bases in function spaces of Besov and Triebel-Lizorkin type on cellular
domains, in particular on the cube. However, he had to exclude essential
exceptional values of the smoothness parameter $s$, for instance the theorems
do not cover the Sobolev space W_2^1(Q) on the n-dimensional cube Q for n at
least 2.
  Triebel also gave an idea how to deal with those exceptional values for the
Triebel-Lizorkin function space scale on the cube Q: He suggested to introduce
modified function spaces for the critical values, the so-called reinforced
spaces. In this paper we start examining these reinforced spaces and transfer
the crucial decomposition theorems necessary for establishing a wavelet basis
from the non-critical values to analogous results for the critical cases now
decomposing the reinforced function spaces of Triebel-Lizorkin type.
