Wavelet decomposition techniques and Hardy inequalities for function spaces on cubes

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 ss, for instance the theorems do not cover the Sobolev space W21(Q) on the nn-dimensional cube QQ for n≥2n≥2.Triebel also gave an idea how to deal with those exceptional values for the Triebel–Lizorkin function space scale on the nn-dimensional cube QQ: he suggested to introduce modified function spaces for the critical values, the so-called reinforced spaces which are subsets of the classical Triebel–Lizorkin function spaces on the cube. 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.