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A rather tricky question is the construction of wavelet bases on domains forsuitable function spaces (Sobolev, Besov, Triebel-Lizorkin type). In hismonograph from 2008, Triebel presented an approach how to construct wavelet(Riesz) bases in function spaces of Besov and Triebel-Lizorkin type on cellulardomains, in particular on the cube. However, he had to exclude essentialexceptional values of the smoothness parameter $s$, for instance the theoremsdo not cover the Sobolev space W_2^1(Q) on the n-dimensional cube Q for n atleast 2.  Triebel also gave an idea how to deal with those exceptional values for theTriebel-Lizorkin function space scale on the cube Q: He suggested to introducemodified function spaces for the critical values, the so-called reinforcedspaces. In this paper we start examining these reinforced spaces and transferthe crucial decomposition theorems necessary for establishing a wavelet basisfrom the non-critical values to analogous results for the critical cases nowdecomposing the reinforced function spaces of Triebel-Lizorkin type.

Wavelet decomposition techniques and Hardy inequalities for function spaces on cellular domains