Using Littlewood Paley decomposition, Triebel classified most of function spaces into three index function spaces: Besov spaces and Triebel Lizorkin spaces. But such spaces contain neither real interpolation spaces of two Sobolev spaces L^p(Lorentz spaces), nor dual space and predual space of Triebel Lizorkin spaces F^{α,q}_1; the authors did not know how to give a uniform description for Triebel Lizorkin spaces and Lorentz spaces. Using wavelets, the authors can give all these spaces a uniform description: Triebel Lizorkin Lorentz spaces, BesovLorentz spaces and dual space and predual space of F^{α,q}_1; furthermore, the authors study also some properties for these spaces.
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