Triebel Consistent Decomposition Of Space And Nature

Besov Space and Triebel space, two types of important function spaces, formed in the 1960's to 1980's, and have achieved extensive application in Partial Differential Equations in recent years. These two spaces are the mutual interaction between dyadic decomposition and lq(Lp), Lp(lq). The former is modulation space, and the latter is the space which the author takes emphasize in.In this article, the author first introduces Schwartz function with its Fourier trans-form, tempered distribution with its Fourier transform, and multipliers with their prop-erties.Then I introduce Besov Space, Triebel space, and Modulation space with their properties. Finally, the author defines the space which is the interaction between dyadic decomposition and Lp(lq), and introduces some properties of the space, such as Embed-ding property, Completeness, Equivalent Norm, Translation and Dilation properties, and gives detailed proof.